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337			<h1 class="entry-title"><a href="https://jkmsmkj.fyi/2024/05/30/disappointing-failure/" rel="bookmark">Disappointing Failure</a></h1>
338						<div class="entry-meta">
339				<span class="posted-on">On <a href="https://jkmsmkj.fyi/2024/05/30/disappointing-failure/" rel="bookmark"><time class="entry-date published updated" datetime="2024-05-30T21:44:32+05:30">May 30, 2024</time></a></span><span class="byline"> By <span class="author vcard"><a class="url fn n" href="https://jkmsmkj.fyi/author/jkmsmkj/">JKMSMKJ</a></span></span><span class="cat-links">In <a href="https://jkmsmkj.fyi/category/conference/" rel="category tag">Conference</a>, <a href="https://jkmsmkj.fyi/category/mathematics/" rel="category tag">Mathematics</a>, <a href="https://jkmsmkj.fyi/category/physics/" rel="category tag">Physics</a>, <a href="https://jkmsmkj.fyi/category/timepass/" rel="category tag">TimePass</a></span><span class="comments-link"><a href="https://jkmsmkj.fyi/2024/05/30/disappointing-failure/#respond">Leave a comment</a></span>			</div><!-- .entry-meta -->
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343			<p align="justify">A week or so ago, an article appeared in Quanta Magazine that discussed disappointments / failures in the field of mathematics and of mathematicians: <a href="https://www.quantamagazine.org/how-failure-has-made-mathematics-stronger-20240522/" target="_blank">How Failure Has Made Mathematics Stronger | Quanta Magazine</a>. It is an insightful interview with D. Calegari, a topologist (a mathematician who studies topology, study of “shapes” and their deformations) who talks about failures, a <em>taboo</em> subject in academia. I highly recommend you to read this interview even if you are not in academia because one can always “learn” better ways to deal with disappointments and failures (which are, you will agree, quite abundant in general). I feel that disappointments arising from the realization that “nobody cares about your work” transcends academia and reading about how others deal with that “successfully” can be insightful and eye-opening. It is also worth one’s time, especially when it is not in the format of a self-help book, or being preached from a high moral ground.</p>
344<p align="justify">In addition, the introduction of the above interview contains a hyperlink for an essay Calegari had written recently about the importance of failures: <a href="https://www.ams.org/journals/notices/202309/noti2782/noti2782.html" target="_blank">Disappointment | Notices of the American Mathematical Society</a>. In this essay, I especially liked his recollection of preparing for IMO, its qualification exam, its result and the aftermath. I feel that sort of analytical storytelling is worth everyone’s time. So have a go at that and we will continue this post shortly after that.</p>
345<p align="left">⋯</p>
346<p align="left">⋯ ⋯</p>
347<p align="left">⋯ ⋯ ⋯</p>
348<p align="left">⋯ ⋯ ⋯ ⋯</p>
349<p align="left">Welcome back!</p>
350<p align="justify">In the AMS essay, the author (correctly) claims that disappointment takes many forms and gives a (partial) list of how &amp; when it can kick in. That list stuck a chord with me and in the rest of this post I will try to elaborate some of those list items from my own personal history. None of them will be as inspiring as his “IMO phase” but nonetheless, it is high time to list my failures and disappointments somewhere and what better place than my own physics blog, so here I go:</p>
351<p align="justify">(1) being un- or under-acknowledged in a colleague’s paper or talk: This has “happened a lot” simply because of human nature I guess. As my PhD advisor used to say, “People just do their own thing and ignore what others are doing”. In general, scientists acknowledge others’ work in scientific papers via citations (that is, add a reference to the previously published papers at appropriate places in one’s own paper). But when there’s “competition”, the appropriate places tend to reduce from main text of the paper to a bunch of citations squished together in the introduction to mere footnotes. Specially, the worst culprits are those that stoop to the last method, adding a citation as if on second thought, “Oh! and here are these other authors who did a similar thing before”, when the correct way would be to cite in the main text with the phrase, “Oh! these authors pioneered this method and we have neither a way of improving it nor finding new applications for it!”. Well, I go even beyond these worst culprits by omitting to cite them altogether (&amp; not just in future papers!), because who cares! (Well, this is a bit of an exaggeration but <a href="https://blog.jkmsmkj.fyi/2019/12/inextensible-and-infallible.html" target="_blank">here’s a slightly more real recollection</a>.)</p>
352<p align="justify">(2) being scooped: I was scooped once within a week of starting to work on a topic. What I had done till then comprised just a couple of pages of the scooping paper and what I had planned to do was only half of it. So I was not that disappointed or disheartened, because that paper did more than what I had planned and the rest was done in the way I would have done. If that paper had appeared a month later, then that would have been quite a real disappointment. Though, it was heartening to know that I have scooped others a couple of times (with no explicit intention, obviously)!</p>
353<p align="justify">(3) missing out on a job/prize/conference invitation: No for a prize or a conference invitation. But definitely for a job. This was the case of a postdoc job at <a href="https://chep.iisc.ac.in/index.html" target="_blank">IISc, Bangalore</a> and the “selection committee” had deemed I was “too old” to be hired for that position even when the IISc people supporting my application told me they “had recommended me highly” to the committee. So yes, that was a great disappointment. And after that a few faculty jobs vanished in a similar fashion, though nobody cared to tell me the exact reason as people from IISc had done. Thus there was much less disappointment in these latter cases, as I started to care less &amp; less!</p>
354<p align="justify">(4) having a prospective student work with someone else: Well, as hinted above, I’m not a faculty member and even as a postdoc, I have only worked with a few students and never have I thought they “belong” to me. This goes back to my PhD days at the <a href="http://insti.physics.sunysb.edu/itp/www/people" target="_blank">YITP</a>, Stony Brook University (SBU). The YITP had quite an open culture regarding who worked with whom and on what! Obviously, one had an official PhD advisor but one was allowed (even encouraged) to work with other professors, postdocs, students on whatever problems / topics one fancied. No professor seemed possessive of their students. Once when I went to meet my advisor after a gap of 4 months, I remember he did not ask what have I been up to or where have I been all this time! We just talked about our “running” project (which had been on hiatus for the last 4 months). At the end of that discussion, when I told him that I have just finished a paper with others (yes, <a href="https://arxiv.org/abs/1211.1388" target="_blank">this paper</a>
354 was completed in such a short time and is one of the few that I am really proud of!), he simply replied, “I thought so!” and that was that. [Though, flipping this scenario around, I had taken (in my second year at SBU) an astronomy course whose professor was “impressed” with my course work and final presentation that he offered to become my advisor if my YITP oral exam didn’t go as planned. So he might have been a little bit disappointed when he lost me as a prospective student.]</p>
355<p align="justify">(5) having a potential advisor turn me down as a student: Again a no. The first professor I approached became my advisor, both for my Masters at <a href="https://www.iitkgp.ac.in/department/PH/" target="_blank">IITKGP</a> and Doctorate at SBU. So nothing more to add here. Though, if my PhD advisor hadn’t agreed, I’d have been very, very disappointed!</p>
356<p align="justify">(6) having a promising line of attack on a problem fail to pan out: Oh, this has happened countless number of times, as is common when one does research, which is the point of the above articles! Though, one that happened while I was a postdoc at <a href="https://www.phys.ntu.edu.tw/enphysics/Default.html" target="_blank">NTU</a> stands out a lot among other “failures”, because that line of attack wasted nearly 6 months of our time. Not just mine, but also my colleague’s. And on top of that I’d seen that line of attack not work in my early PhD days! So it was a double whammy of disappointment as it dawned on me that not only have I wasted others’ time but also that it was preventable because of my “prior experience”!</p>
357<p align="justify">(7) discovering an error in an amazing proof: This hasn’t happened to me directly as such because I’m not a mathematician. But some things related to the previous point could easily be stated here in the kind of research that I have done.</p>
358<p align="justify">(8) having a paper go unread or a book go unreviewed: No book to talk about. And no disappointment for my papers going unread because, even as Calegari agrees, nobody cares. So no point in being disappointed about that. I do my work because I like it being done, not because somebody else will notice / read it!</p>
359<p align="justify">(9) seeing an admired senior colleague behave badly: Not really… Though I’ve seen a few “skirmishes” and I want to believe there was nothing personal about them. Taking the “couldn’t care less” attitude further, I respect their work more than them personally.</p>
360<p align="justify">(10) realizing that I haven’t lived up to my own standards of behavior: Oh! This has happened a countless number of times. Most of my disappointments are in this category, basically. But there’s not much to elaborate…</p>
361<p align="justify">(11) discovering that success, when it came, was not all I hoped it would be: Well, as far as academic success is concerned, I haven’t reached (and never will) the pinnacle of being a tenured professor. So I can’t say what “success” would have felt like there and if it would have been what I’d have hoped for or not. Graduating from IITKGP with an institute silver medal did feel like “success”. Getting admitted into SBU did feel like “success”. But after that, “success” has been few and far between. Graduating from SBU didn’t feel like “success” and doing three postdocs after that didn’t feel like “success” either. Many colleagues (and brilliant colleagues at that, better than me in all aspects) whom I have known for the last decade or so not “making it” in academia has not felt like “success” either. To say the least, it has instead felt like a personal failure, a crushing weight on my own self. One such disappointment was shared recently by a colleague <a href="https://4gravitons.com/2024/01/19/well-that-didnt-work/" target="_blank">here</a>, <a href="https://4gravitons.com/2024/01/26/why-we-are-leaving-france-the-misadventures-of-a-trailing-spouse/" target="_blank">here</a> &amp; <a href="https://4gravitons.com/2024/03/01/france-for-non-eu-sp
361ouses-of-eu-citizens-to-get-your-rights-dont-follow-the-rules/" target="_blank">here</a>. You know, sometimes it feels like the world is against you but in this case, it is literally true… well, at least a country of the world!</p>
362<p align="justify">It is also disappointing that this post will not have a hyperlink at the end.</p>
363
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371<article id="post-231" class="post-231 post type-post status-publish format-standard hentry category-physics category-timepass">
372			<div class="entry-wrapper">
373		<header class="entry-header">
374			<h1 class="entry-title"><a href="https://jkmsmkj.fyi/2022/08/26/to-explain-the-world/" rel="bookmark">To Explain the&nbsp;World</a></h1>
375						<div class="entry-meta">
376				<span class="posted-on">On <a href="https://jkmsmkj.fyi/2022/08/26/to-explain-the-world/" rel="bookmark"><time class="entry-date published" datetime="2022-08-26T14:12:31+05:30">August 26, 2022</time><time class="updated" datetime="2022-08-26T14:32:04+05:30">August 26, 2022</time></a></span><span class="byline"> By <span class="author vcard"><a class="url fn n" href="https://jkmsmkj.fyi/author/jkmsmkj/">JKMSMKJ</a></span></span><span class="cat-links">In <a href="https://jkmsmkj.fyi/category/physics/" rel="category tag">Physics</a>, <a href="https://jkmsmkj.fyi/category/timepass/" rel="category tag">TimePass</a></span><span class="comments-link"><a href="https://jkmsmkj.fyi/2022/08/26/to-explain-the-world/#respond">Leave a comment</a></span>			</div><!-- .entry-meta -->
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378
379		<div class="entry-content">
380			<p align="justify">Today is <a href="https://blog.jkmsmkj.fyi/2020/08/happy-wittens-day.html" target="_blank" rel="noopener">Ed Witten’s birthday</a>, apparently. I recently completed <a href="https://en.wikipedia.org/wiki/Steven_Weinberg" target="_blank" rel="noopener">S. Weinberg</a>’s book “<a href="https://www.amazon.com/Explain-World-Discovery-Modern-Science-ebook/dp/B00KFG18Q4" target="_blank" rel="noopener">To Explain the World</a>”, so I will instead collect some excerpts from that book here. Let’s start with one from near the beginning of the book.</p>
381<blockquote>
382<p align="justify">There remains a poetic element in modern physics. We do not write in poetry; much of the writing of physicists barely reaches the level of prose. But we seek beauty in our theories, and use aesthetic judgments as a guide in our research.</p>
383</blockquote>
384<p align="justify">I like the phrase “barely reaches the level of prose”. It really made me laugh out loud, while reminding me of a quote by <a href="https://www.amazon.com/Astonishing-Hypothesis-Scientific-Search-Soul/dp/0684801582" target="_blank" rel="noopener">Francis Crick</a>:</p>
385<blockquote>
386<p align="justify">There is no form of prose more difficult to understand and more tedious to read than the average scientific paper.</p>
387</blockquote>
388<p align="justify">Moving on, Weinberg also cites the preface to the “<a href="https://archive.org/details/treatiseonlight031310mbp/page/n3/mode/2up" target="_blank" rel="noopener">Treatise on Light</a>” by <a href="https://en.wikipedia.org/wiki/Christiaan_Huygens" target="_blank" rel="noopener">C. Huygens</a>:</p>
389<blockquote>
390<p align="justify">There will be seen [in this book] demonstrations of those kinds which do not produce as great a certitude as those of Geometry, and which differ much therefrom, since whereas the Geometers prove their Propositions by fixed and incontestable Principles, here the Principles are verified by the conclusions to be drawn from them; the nature of these things not allowing of this being done otherwise.</p>
391</blockquote>
392<p align="justify">I like this “way of doing science” as it is how I think about theoretical physics research, especially high energy physics. Of course, the sentiment above is as valid for any theorist or experimentalist or phenomenologist or someone in-between. Because in my view, the phrase “⋯Principles are verified by the conclusions⋯” is so broad that the “conclusions” include everything from experimental results (Gedanken or physical or any other kind one could think of) to results in terms of equations (Abstract or useful or any other kind one could think of). This broad view further enforced by the next phrase “⋯nature of these things not allowing of this being done otherwise”, puts into perspective that the “dated” view of scientific method (Hypothesis → Experiment → Confirmation? → Theory or Re-Hypothesize!) is indeed “dated”. The scientific method should instead be thought of as “Principles → ??? → Consequences”. [A better discussion along similar lines can be found <a href="https://4gravitons.com/2021/07/02/digging-for-buried-insight/" target="_blank" rel="noopener">here</a>.] The “???” could be anything from a one-page mathematical proof to a simple thought-experiment to a backyard contraption to a billion-dollar lab experiment to a decades-long development of a theoretical framework. The former view of a scientific method is good enough to be discussed in a nursery class where kids don’t even realize that something could be subtracted from 0 whereas the latter is what actually happens in the real world of scientific research. Though, even the real researchers could be in denial of this fact most of the time, because the early nursery education is quite hard to override, even by the time one becomes a faculty member. A personal example of this: after I finished my talk on “3d supersymmetric localization” at <a href="https://www.bose.res.in/" target="_blank" rel="noopener">SNBose</a> and asked one of the senior professors what he thought about the talk, he replied, “I am a <em>real</em> physicist”! I laughed out loud, and thought, “Oh, he’s <em>that kind</em> of a physicist!”. By “<em>that kind</em>” I, of course, mean someone who hasn’t been able to move past the above-mentioned nursery education.</p>
393<p align="justify">Someone at this point will surely interject and say something like, “what about the real world?” or “what about this thing’s usefulness?” or “<a href="https://www.youtube.com/watch?v=Fs9asU2y9Pg" target="_blank" rel="noopener">what about the real world applications?</a>” or “when will I actually need this in my real life?”. For all those interjectors, I point my index finger towards the future posts which I will write on this blog slowly and steadily, but surely, as I continue <a href="https://www.theguardian.com/commentisfree/2022/aug/06/d
393ara-o-briain-mock-the-week" target="_blank" rel="noopener">burning various bridges</a>. Because why not? In short, here are my two cents: Have you seen the real world recently? Why the hell would I ever think about it, except in my <a href="https://www.netflix.com/title/81150303" target="_blank" rel="noopener">nightmares</a>? <a href="https://jkmsmkj.wordpress.com/2021/06/20/superspace/" target="_blank" rel="noopener">Supersymmetry and Superspace</a> is a much better and nicer place to live in and it turns out to be quite useful to avoid the real world, and that’s all I need in my real life!</p>
394<p align="justify">That went off in a different direction than intended so let’s get back on track with another excerpt from the book.</p>
395<blockquote>
396<p align="justify">The search for knowledge of practical value can serve as a corrective to uncontrolled speculation, but explaining the world has value in itself, whether or not it leads directly to anything useful.</p>
397</blockquote>
398<p align="justify">This again echoes what I said above or rather, what I said above echoes these thoughts of Weinberg. Of course, it’s the latter.</p>
399<blockquote>
400<p align="justify">We get intense pleasure when something has been successfully explained, as when Newton explained Kepler&#8217;s laws of planetary motion along with much else. The scientific theories and methods that survive are those that provide such pleasure, whether or not they fit any preexisting model of how science ought to be done.</p>
401</blockquote>
402<p align="justify">Weinberg so eloquently expresses the need to throw away the “outdated” view of the scientific method, which according to him, had outlived its usefulness already at the time of Newton. And finally,</p>
403<blockquote>
404<p align="justify">The rejection of Newton&#8217;s theories by the followers of Descartes and Leibniz suggests a moral for the practice of science: it is never safe simply to reject a theory that has as many impressive successes in accounting for observation as Newton&#8217;s had. Successful theories may work for reasons not understood by their creators, and they always turn out to be approximations to more successful theories, but they are never simply mistakes.</p>
405</blockquote>
406<p align="justify">Well-said but it seems this moral is not taken seriously even by many self-appointed real physicists in this day and age!</p>
407<p align="justify">That’s quite a long post which I definitely did not intend for it to be. So let’s leave it there, as I move on to another book of Weinberg:</p>
408<p align="justify"><a href="https://www.amazon.com/Third-Thoughts-Steven-Weinberg-ebook/dp/B07DGDHS44">Third Thoughts</a></p>
409
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417<article id="post-211" class="post-211 post type-post status-publish format-standard hentry category-hep-th category-timepass tag-arxix tag-science-communication">
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420			<h1 class="entry-title"><a href="https://jkmsmkj.fyi/2022/01/10/communicating-science-challenge/" rel="bookmark">Communicating Science Challenge</a></h1>
421						<div class="entry-meta">
422				<span class="posted-on">On <a href="https://jkmsmkj.fyi/2022/01/10/communicating-science-challenge/" rel="bookmark"><time class="entry-date published" datetime="2022-01-10T21:25:00+05:30">January 10, 2022</time><time class="updated" datetime="2022-07-25T08:05:20+05:30">July 25, 2022</time></a></span><span class="byline"> By <span class="author vcard"><a class="url fn n" href="https://jkmsmkj.fyi/author/jkmsmkj/">JKMSMKJ</a></span></span><span class="cat-links">In <a href="https://jkmsmkj.fyi/category/physics/hep-th/" rel="category tag">HEP-TH</a>, <a href="https://jkmsmkj.fyi/category/timepass/" rel="category tag">TimePass</a></span><span class="comments-link"><a href="https://jkmsmkj.fyi/2022/01/10/communicating-science-challenge/#comments">2 Comments</a></span>			</div><!-- .entry-meta -->
423					</header><!-- .entry-header -->
424
425		<div class="entry-content">
426			<p align="justify"><a href="https://4gravitons.com/2022/01/07/the-arxiv-scicomm-challenge" target="_blank" rel="noopener">4gravitons</a> has proposed a challenge: To explain scientific papers appearing on <a href="https://arxiv.org/" target="_blank" rel="noopener">arXiv</a> on a given day in a given (sub)field to a general audience. The challenge is directed towards fellow scie
426nce communicators so that definitely excludes me. But one of my old year thoughts (turned to a new year resolution) for this blog was to discuss papers on arXiv that interested me, on a monthly basis (apart from some physic(ist)s-related rants <a href="https://jkmsmkj.wordpress.com/2021/12/20/ism-2021-review/" target="_blank" rel="noopener">like this</a>) in an effort to keep this blog (sort of) active. So, despite not being a science communicator, I feel like the above-mentioned challenge is one way to get into discussing papers this month. So here we go…</p>
427<p align="justify"><strong>Warning as given by 4gravitons:</strong> <em>I’m looking at papers in the “<a href="https://arxiv.org/archive/hep-th" target="_blank" rel="noopener">High Energy Physics – Theory</a>” area, announced 10 Jan, 2022. I haven’t read these papers, just their abstracts, so apologies if I got your paper wrong!</em></p>
428<p align="justify"><strong>Extra Disclaimer: </strong><em>I am not an expert in any of the (sub)topics covered by the 11 papers appearing below. I am familiar with enough ‘hep-th jargon’ to have a vague idea about what each paper might be about and that’s what is reflected in my explanations. So don’t take what follows too seriously, or to put it another way, take it seriously at your own risk.</em></p>
429<p align="left">[1] <a href="https://arxiv.org/abs/2201.02200">arXiv:2201.02200</a>: A <img src="https://s0.wp.com/latex.php?latex=p&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=p&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=p&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="p" class="latex" />-Adic Matter in a Closed Universe</p>
430<p align="justify">The author looks at dynamics of an <a href="https://www.quantamagazine.org/how-the-towering-p-adic-numbers-work-20201019/" target="_blank" rel="noopener">exotic kind of matter</a> and its implications to cosmology. They find by employing Einstein’s theory of gravity that a Universe filled with this kind of matter would be closed (like the surface of a sphere) and would expand exponentially. They also discuss the possibility of this kind of matter as a dark matter candidate. [Our Universe with ordinary + dark matter (+dark energy) is <a href="https://en.wikipedia.org/wiki/Lambda-CDM_model" target="_blank" rel="noopener">more or less flat and expanding but not exponentially</a>.]</p>
431<p align="left">[2] <a href="https://arxiv.org/abs/2201.02201">arXiv:2201.02201</a>: Anisotropic Special Relativity</p>
432<p align="justify">Einstein’s theory of Special Relativity treats all directions of space and time on equal footing. As the title of this paper suggests, the author here considers breaking the isotropy of space and choosing one preferred direction in space. Using such a setup, they go on to discuss the consequences for quantum field theory, which is the framework used to describe our current understanding of fundamental particles and forces known as the Standard Model. [Not sure how far they reach in ‘reworking’ QFT using ASR as I’ve not read the paper.]</p>
433<p align="left">[3] <a href="https://arxiv.org/abs/2201.02206">arXiv:2201.02206</a>: Precision Bootstrap for the <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=1" class="latex" /> Super-Ising Model</p>
434<p align="justify">This is a note detailing a computational technique used to determine some physical properties of a 3d lattice of ‘spins’ (variables that can take two values like {+1, -1}) in a better way than previously used techniques. Such 3d systems are useful in studying phase transitions (like ice → water → steam), modelling phenomena like ferromagnetism, spin glasses, and even <a href="https://en.wikipedia.org/wiki/Ising_model#Application_to_neuroscience" target="_blank" rel="noopener">neurons</a>
434! So having the ability to compute stuff easily and efficiently is always welcome. [I don’t think I can make it any more interesting.]</p>
435<p align="left">[4] <a href="https://arxiv.org/abs/2201.02407">arXiv:2201.02407</a>: More on Topological Hydrodynamic Modes</p>
436<p align="justify">The authors discuss fluid/gravity correspondence, a version of holography where fluid dynamics (or hydrodynamics) is described by a gravity theory living in one dimension higher. Specifically, they find a holographic description of a hydrodynamic phenomenon they had discovered earlier. What they had discovered were special kind of very-low-energy (virtually zero) fluid excitations (or modes) by ‘weakly’ breaking energy-momentum conservation in the hydrodynamic system. In this paper, by choosing a special rotating frame of reference (think of a merry-go-round; using Einstein’s equivalence principle, this rotating system can be thought of as a non-rotating system with a gravitational force), they could find existence of the same very-low-energy modes in the holographic setup too. They also extend their holographic construction to include ‘electric’ charges and find similar very-low-energy modes. [Usually, any paper whose title starts with “More on” or “Note on” is more or less an addendum to a previous paper without much of an independent story. I have also been known to succumb to this practice more than once in the past and will do so again later this year.]</p>
437<p align="left">[5] <a href="https://arxiv.org/abs/2201.02412">arXiv:2201.02412</a>: <img src="https://s0.wp.com/latex.php?latex=U%281%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=U%281%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=U%281%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="U(1)" class="latex" /> Fields from Qubits: an Approach via D-theory Algebra</p>
438<p align="justify">A framework for building algorithms for quantum computing is being explored in this paper. The authors pick a simple quantum mechanical toy model and discuss ideas to extend it to more complex models in various dimensions ranging from 1d to 4d (our world). One of the motivations for considering this seems to be the possible generalization to QCD (quantum chromodynamics; the theoretical framework describing strong nuclear force where exact analytical calculations are quite hard to perform and usually discretized spacetime lattices are used to perform calculations that are expected to give sensible answers in a continuum limit) such that lattice computations for QCD become more ‘robust’. [Don’t know why but this paper reminded me of one of my <a href="https://1drv.ms/b/s!AhGovQArU6bHgtpGy-KSv9nWQKWY-g" target="_blank" rel="noopener">parody</a> papers that I wrote ages ago, where one simple equation surprisingly described all of physics!]</p>
439<p align="left">[6] <a href="https://arxiv.org/abs/2201.02493">arXiv:2201.02493</a>: Solving formally the Auxiliary System of <img src="https://s0.wp.com/latex.php?latex=O%28N%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=O%28N%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=O%28N%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="O(N)" class="latex" /> Non Linear Sigma Model</p>
440<p align="justify">I don’t think I’ll attempt to explain this. Can someone else give it a try? [What? I’m still only at the half-way mark… Man, this is harder than I expected even after taking into account that explaining things is hard!]</p>
441<p align="left">[7] <a href="https://arxiv.org/abs/2201.02500">arXiv:2201.02500</a>: Entanglement Entropy in Horndeski Gravity</p>
442<p align="justify">Entanglement Entropy is a measure of quantum entanglement between two subsystems of a quantum system. Holographic EE is a quantity computed for a certain gravitational system by considering a subregion in a spacetime with boundary, which encodes the EE of a non-gravitational matter system living on that boundary (with the two subsystems required for EE being the chosen subregi
442on on the boundary and the rest of the boundary). The author chooses a general theory of 4d gravity (<a href="https://en.wikipedia.org/wiki/Horndeski%27s_theory" target="_blank" rel="noopener">Horndeski</a>) and computes HEE for an infinite strip subregion of the boundary. Certain information theoretical and thermodynamical aspects of this setup are then discussed. [Isn’t it always a good idea to refrain from citing papers in the abstract? I have been guilty of doing that myself but that one was unavoidable!]</p>
443<p align="left">[8] <a href="https://arxiv.org/abs/2201.02524">arXiv:2201.02524</a>: On some new types of membrane solutions</p>
444<p align="justify">The author provides new solutions (of certain extensions of Einstein’s equations of gravity) in M-theory (mother of all string theories) with two time dimensions and discusses their properties, like no black hole formation. [Usually, one deals with only one time dimension.]</p>
445<p align="left">[9] <a href="https://arxiv.org/abs/2201.02572">arXiv:2201.02572</a>: Topological invariant of 4-manifolds based on a 3-group</p>
446<p align="justify">This paper claims to be a generalization of another paper from 2007, which has been flagged in the “arXiv admin note”. Anyway, the authors consider an exotic extension of Maxwell’s theory of electromagnetism to study properties of 4d spacetimes. They construct a quantity ‘Z’ which is a topological invariant, by which it is meant that this Z does not depend on the specifics of the 4d spacetimes (for example, how distances are measured on them) but just on the overall shape (topology) of the spacetime. In a certain sense, it means that Z can be used to classify different types of 4d spacetimes (for example, 2d surfaces can be classified by counting the <a href="https://en.wikipedia.org/wiki/Genus_(mathematics)" target="_blank" rel="noopener">number of holes</a>). [Also note that they give a nice way to avoid using citations in the abstract!]</p>
447<p align="left">[10] <a href="https://arxiv.org/abs/2201.02575">arXiv:2201.02575</a>: The spatial Functional Renormalization Group and Hadamard states on cosmological spacetimes</p>
448<p align="justify">The authors study the ‘flow’ of a special state living on cosmologically relevant spatial and temporal scales. They give a concrete mathematical realization of such a state and study how the state flows from high energy (ultraviolet; let’s say near Big Bang) to low energy (infrared; let’s say our humdrum life now). They find that choosing certain states of low energy in pre-inflationary period (inflation is the epoch of exponential expansion in our cosmological past, very close to the Big Bang) and allowing them to flow using their ‘flow equations’ leads to concretely computable property of the state in the kind of spacetime that we are familiar with now!  [So many familiar words, yet their exact order renders them challenging to comprehend! Hopefully, I have gotten the gist of the abstract right.]</p>
449<p align="left">[11] <a href="https://arxiv.org/abs/2201.02595">arXiv:2201.02595</a>: Celestial holography meets twisted holography: 4d amplitudes from chiral correlators</p>
450<p align="justify">This paper introduces a new program for computing scattering amplitudes (actually, certain parts of it; the whole thing is expected to give the probability for a given scattering event to take place) in 4d field theories (involving photons, W/Z bosons, gravitons, etc.) from certain 6d theories. They recognize these parts of the scattering amplitudes as something originating in different kinds of 4d theories and/or 2d theories. This connection allows them to reproduce well-known results efficiently as well as compute some new results for certain scattering amplitudes. In addition, they also make contact with other programs like “celestial holography” and “twisted holography” that have become quite popular recently. [For the last statement, I refer to my previous post reviewing ISM 2021 and links therein.]</p>
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459<article id="post-203" class="post-203 post type-post status-publish format-standard has-post-thumbnail hentry category-conference category-string-theory category-timepass tag-ism-2021 tag-review tag-strings">
460				<a href="https://jkmsmkj.fyi/2021/12/20/ism-2021-review/" title="ISM 2021 Review"><img width="550" height="669" src="https://jkmsmkj.fyi/wp-content/uploads/2021/12/barstack2.png?w=550" class="attachment-lovecraft-post-image size-lovecraft-post-image wp-post-image" alt="Distribution of talks!" decoding="async" srcset="https://jkmsmkj.fyi/wp-content/uploads/2021/12/barstack2.png 550w, https://jkmsmkj.fyi/wp-content/uploads/2021/12/barstack2.png?w=123 123w, https://jkmsmkj.fyi/wp-content/uploads/2021/12/barstack2.png?w=247 247w" sizes="(max-width: 550px) 100v
460w, 550px" data-attachment-id="202" data-permalink="https://jkmsmkj.fyi/2021/12/20/ism-2021-review/barstack2-png/" data-orig-file="https://jkmsmkj.fyi/wp-content/uploads/2021/12/barstack2.png" data-orig-size="550,669" data-comments-opened="1" data-image-title="BarStack2.png" data-image-description="" data-image-caption="&lt;p&gt;Distribution of talks!&lt;/p&gt;
461" data-large-file="https://jkmsmkj.fyi/wp-content/uploads/2021/12/barstack2.png?w=550" /></a>
462		<div class="entry-wrapper">
463		<header class="entry-header">
464			<h1 class="entry-title"><a href="https://jkmsmkj.fyi/2021/12/20/ism-2021-review/" rel="bookmark">ISM 2021 Review</a></h1>
465						<div class="entry-meta">
466				<span class="posted-on">On <a href="https://jkmsmkj.fyi/2021/12/20/ism-2021-review/" rel="bookmark"><time class="entry-date published" datetime="2021-12-20T22:11:05+05:30">December 20, 2021</time><time class="updated" datetime="2025-06-06T18:13:58+05:30">June 6, 2025</time></a></span><span class="byline"> By <span class="author vcard"><a class="url fn n" href="https://jkmsmkj.fyi/author/jkmsmkj/">JKMSMKJ</a></span></span><span class="cat-links">In <a href="https://jkmsmkj.fyi/category/conference/" rel="category tag">Conference</a>, <a href="https://jkmsmkj.fyi/category/physics/string-theory/" rel="category tag">String Theory</a>, <a href="https://jkmsmkj.fyi/category/timepass/" rel="category tag">TimePass</a></span><span class="comments-link"><a href="https://jkmsmkj.fyi/2021/12/20/ism-2021-review/#respond">Leave a comment</a></span>			</div><!-- .entry-meta -->
467					</header><!-- .entry-header -->
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469		<div class="entry-content">
470			<p align="justify">The past week saw 6 days of the latest edition of “<a href="https://iitr.ac.in/ism21/#/" target="_blank" rel="noopener">Indian Strings Meeting</a>” hosted by <a href="https://www.iitr.ac.in/" target="_blank" rel="noopener">IIT Roorkee</a>. Well, more like hosted by <a href="https://www.iitr.ac.in/~PH/aalokfph" target="_blank" rel="noopener">Aalok Misra</a> with some background support from his ‘tech team’ taking care of <a href="https://iitr.ac.in/ism21/#/Zoom" target="_blank" rel="noopener">Zoom calls and YouTube live streams</a> and <a href="https://iitr.ac.in/ism21/#/Gallery" target="_blank" rel="noopener">Webpage updates</a>. I could not attend the Zcalls because of connection problems (at my end I guess) as audio and/or shared screen would randomly drop so I mostly watched the YT live streams. The latter was quite a barren region with &lt;20 viewers at any given time compared to the former which was bustling with &lt;50 participants (i.e., whenever I could join and see and hear something relevant). I wonder what happened to the ~400 registered <a href="https://iitr.ac.in/ism21/#/Participants" target="_blank" rel="noopener">participants</a> on the website!</p>
471<p align="justify">Anyway, enough with the personnel logistics, let’s get on with the real logistics that is most relevant in a conference review. That of how many talks were there and of what kind. I will categorize the talks based on four topics extracted from the six review talks, two more topics for 3d field theories and non-3d field theories and one last ‘miscellaneous’ topic (as expected) to catch all those that do not fit into the aforementioned six topics. I do feel like I should mention a disclaimer here, before we have a look at the table, that this categorization of talks is my personal subjective opinion and not to be taken as that of the speakers or Aalok or <a href="https://iitr.ac.in/ism21/#/Noc" target="_blank" rel="noopener">ISM’s NOC</a> or anyone else for that matter. If anyone does not agree with this categorization, they are welcome to comment below or write their own post and share it as a comment below. (No guarantee that it will change my opinion but if it does, I will be happy to edit things around here accordingly.) Here goes the full table with raw data mined from <a href="https://iitr.ac.in/ism21/#/Schedule" target="_blank" rel="noopener">here</a>:</p>
472<table border="1" cellspacing="0" cellpadding="1">
473<tbody>
474<tr>
475<td align="center" width="20%">Topic</td>
476<td align="center" width="20%">Review Talks (60m)</td>
477<td align="center" width="20%">Research Talks (30m)</td>
478<td align="center" width="20%">Short Presentations (15m)</td>
479<td align="center" width="20%">Total Time (h)</td>
480</tr>
481<tr>
482<td width="20%">(A)dS/CFT + Holography</td>
483<td width="20%">RG1 + EP6 + DG6 = 3</td>
484<td width="20%">TT1 + KN1 + SA2 + SD2 + RL3 + KS3 = 6</td>
485<td width="20%">SM6 + AS6 = 2</td>
486<td align="center" width="20%">6½</td>
487</tr>
488<tr>
489<td width="20%">BMS Symmetry + Flat Space Holography</td>
490<td width="20%">LD2</td>
491<td width="20%">CK1 + AB2 + MM2 + AB2 + RB4 + AYS5 + PP5 = 7</td>
492<td width="20%">PP5 + DG6 = 2</td>
493<td align="center" width="20%">5</td>
494</tr>
495<tr>
496<td width="20%">Black Holes + Information Theory</td>
497<td width="20%">DH3</td>
498<td width="20%">AM1 + AV2 + OP3 + DS3 + NK4 + AA4 + PR6 + JC6 + JKB6 + RS6 + SP6 = 11</td>
499<td width="20%">AK1 + AKP2 + SM2 + SK3 + AG4 + HP4 + GB5 + AB5 = 8</td>
500<td align="center" width="20%">8½</td>
501</tr>
502<tr>
503<td width="20%">Amplitudes + (OTO) Correlators</td>
504<td width="20%">JP6</td>
505<td width="20%">AK1 + AZ1 + AS3 + AR5 + AM5 + SK6 = 6</td>
506<td width="20%">SD1 + APS1 + SK2 + TS3 + PH5 + AM6 = 6</td>
507<td align="center" width="20%">5½</td>
508</tr>
509<tr>
510<td width="20%">3d (S)CFTs</td>
511<td width="20%"></td>
512<td width="20%">AM1 + SG2 + KC2 + SM3 + SJ5 = 5</td>
513<td width="20%">RRJ1 + NP4 = 2</td>
514<td align="center" width="20%">3</td>
515</tr>
516<tr>
517<td width="20%">Other Field Theories</td>
518<td width="20%"></td>
519<td width="20%">RG2 + YT3 + CG3 + ZK3 + SD4 + VBS4 + NS4 = 7</td>
520<td width="20%">DM4</td>
521<td align="center" width="20%">3¾</td>
522</tr>
523<tr>
524<td width="20%">Miscellaneous</td>
525<td width="20%"></td>
526<td width="20%">AM1 + BS3 + SG3 + KR5 + LA5 + NS5 + AM6 = 7</td>
527<td width="20%">MS3 + MM3 = 2</td>
528<td align="center" width="20%">4</td>
529</tr>
530<tr>
531<td align="right" width="20%">Total:</td>
532<td align="center" width="20%">6</td>
533<td align="center" width="20%">49</td>
534<td align="center" width="20%">23</td>
535<td align="center" width="20%">36¼</td>
536</tr>
537</tbody>
538</table>
539<p align="justify">Let’s acknowledge that the notation used in the table is self-explanatory: the letters are the initials of the speakers (no effort is put to break the degeneracy of AM’s, et al) and the number denotes the day on which they spoke. I have doubly cross-checked the data on the official webpage and my categorization above for consistency. Normally, I don’t put anything up on my blogs, unless it is quadruply checked but since even the official data is not rigorously vetted, I don’t think I need to exert myself that much in this post. For example, a speaker (<a href="https://katalog.uu.se/profile/?id=N18-2168" target="_blank" rel="noopener">Parijat Dey</a> from Uppsala University, Sweden) appears in the detailed list of talks but not in the schedule table, which I guess happened because two speakers cancelled their talks on the fifth day which were then moved to the sixth day! Also, at least one 15m talk appears in the list of 30m talks (or vice-versa) and of course, the big elephant in the room is that <a href="https://youtu.be/PVQTu6m4RZ4" target="_blank" rel="noopener">Enrico Pajer</a>’s hour-long review talk appears in a 30m timeslot and it was treated as something in-between! So given that the raw data is not too trustworthy, I think the doubly checked table appearing above is more than enough for my purposes here.</p>
540<p align="justify">I also have some pet peeves about the formatting/styling employed for the “Schedule” webpage linked above. I mean, who in the ‘tech team’ had the ‘brilliant’ idea of sorting the detailed list of talks wrt length of the talks and then further sorting alphabetically (which is also not quite correct) wrt last names of the speakers? This leads to the following bad user experience: when someone clicks a speaker’s name in a particular timeslot of the schedule table, they are catapulted at warp-speeds through the long webpage to the speaker details but now they’re lost (&amp; disoriented) because they don’t see any other speaker details around the timeslot they were just looking at a moment ago! This problem could have been avoided if the speakers were sorted chronologically and not alphabetically (basically, focus on talks and not speakers). Also, that awfully long webpage could have been shortened by light-years if a tabbed interface were used for the six days, with each day’s tab showing the chronological details of the talks. I also find the insistence on “speakers of” review/research talks in the headings for the corresponding sections but not for the short presentations pretty hilarious (see the screenshot below). As if the people giving those short presentations are not worthy of being even identified with a common noun. Talk about inequalities prevalent in academia! This problem also would not have risen if the talks were chronologically listed. A lot fixed with a simple change in one’s viewpoint.</p>
541<p align="justify"><img style="border:0 currentcolor;margin-right:auto;margin-left:auto;float:none;display:block;background-image:none" title="ISM 2021 Speakers?" src="https://jkmsmkj.fyi/wp-content/uploads/2021/12/ism2021speakers.jpg?w=550&#038;h=367" alt="ISM 2021 Speakers?" width="550" height="367" border="0" /></p>
542<p align="justify">Anyway, enough of my rant about webpage development (<span style="color:#c0504d">I mean I could have done a better job myself of formatting/organizing that page…</span> <span style="color:#f79646">Come on, man! Move on! Your rant just spilled on to the next para…</span> <span style="color:#9bbb59">and I will have to start a new clean para again!</span>).</p>
543<p align="justify">That table above presents quite a boring view so let’s look at a colourful chart based on it instead:</p>
544<p><img style="border:0 currentcolor;margin-right:auto;margin-left:auto;float:none;display:block;background-image:none" title="Stacked Bar Chart 1" src="https://jkmsmkj.fyi/wp-content/uploads/2021/12/barstack.png?w=550&#038;h=673" alt="Stacked Bar Chart 1" width="550" height="673" border="0" /></p>
545<p align="justify">We see that the topic of black holes and related stuff like entanglement entropy, information paradox was extremely popular taking up more than eight hours, i.e., more than a day in a six-days-long conference. The three other main topics in this conference were “(A)dS/CFT &amp; holography”, “S-matrices, scattering amplitudes &amp; correlators (incl. OTOCs)”, and “BMS symmetry &amp; flat space holography”, which took up roughly a day each. The talks on field theories in various dimensions took up another day with most people focussing on 3d (super)conformal field theories but some quite interesting talks on 2d and 4d theories too. Disappointingly, my personal favourite “5d SCFTs” barely got mentioned. Finally, the miscellany of talks included a talk on string phenomenology, a couple on constructing actions for Weyl supergravity multiplets, and a few on mathematical aspects like new Lie algebras, machine learning, etc. Before we move on, let us also see the above data split day-wise. Can you spot the only topic which was talked about every single day? The answer won’t surprise you!</p>
546<p><img loading="lazy" style="margin:0 auto;border:0 currentcolor;float:none;display:block;background-image:none" title="Stacked Bar Chart 2" src="https://jkmsmkj.fyi/wp-content/uploads/2021/12/barstack2.png?w=550&#038;h=669" alt="Stacked Bar Chart 2" width="550" height="669" border="0" /></p>
547<p align="justify">Now, I guess I should lay down my own personal views on ISM 2021. First (confession), I didn’t watch all the talks but most of them to be able to have some motivation to write this post. Second, the constant shifts of the schedule by at least 15m (and going up to 45m as the day progressed) really twisted my holography (reference to DG6’s talk if you don’t get my drift). Third, I was looking forward to <a href="https://youtu.be/x4aucLKFyZ8" target="_blank" rel="noopener">some</a> <a href="https://youtu.be/a4PaJWurmFw" target="_blank" rel="noopener">talks</a> <a href="https://youtu.be/JZ3cuz1Z0Ho" target="_blank" rel="noopener">because</a> <a href="https://youtu.be/Xd9g6h9Pdqw" target="_blank" rel="noopener">
547of</a> <a href="https://youtu.be/EFGt8HeDDKY" target="_blank" rel="noopener">their</a> <a href="https://youtu.be/R3WL1Y69myc" target="_blank" rel="noopener">titles</a>, <a href="https://youtu.be/ZeYxF9pkIs8" target="_blank" rel="noopener">some</a> <a href="https://youtu.be/2kt21jGdXMA" target="_blank" rel="noopener">because</a> <a href="https://youtu.be/IIJHfPtdZIc" target="_blank" rel="noopener">of</a> <a href="https://youtu.be/4S5SEkA4FSA" target="_blank" rel="noopener">their</a> <a href="https://youtu.be/e4zf76xVwYk" target="_blank" rel="noopener">speakers</a>. <a href="https://youtu.be/_k4K2q8aAYU" target="_blank" rel="noopener">Both</a> <a href="https://youtu.be/-yDO1E2369w" target="_blank" rel="noopener">of</a> <a href="https://youtu.be/H1QX-V90DwA" target="_blank" rel="noopener">these</a> <a href="https://youtu.be/6KgWiIy9vro" target="_blank" rel="noopener">types</a> didn’t disappoint. <a href="https://youtu.be/B6CX6i2QL7o" target="_blank" rel="noopener">Others</a> were <a href="https://youtu.be/LRWYFPT9B88" target="_blank" rel="noopener">hits</a> and <a href="https://youtu.be/Z6_ftKUJpkI" target="_blank" rel="noopener">misses</a>. Fourth, the constant bug in almost all of the talks was that typical questioner/commenter who hadn’t properly formulated (or even envisioned) the question before opening his (let’s be real, it’s definitely ‘his’ not ‘her’!) mouth, leading to mostly vague questions or tautological statements disguised as comments. That led in turn to more haw-ing, hmm-ing, “I mean(t)”’s, “you know, like”’s, “the the, so that thing”’s, “the usual thing, you know”’s, “so so, that’s what… the the, what I meant was”’s, etc. You get the idea why it bugged me, right? I wish Indian institutes had “graduate seminar courses” like I had at Stony Brook where professors actually gave you proper feedback on your talks. In addition, listening to fellow students in that ‘classroom’ setting made you aware of what to do and what not to do during talks, both yours and others. Some of the speakers &amp; listeners of this conference could have definitely used some exposure to such a course. Fifth, I didn’t listen to most of the discussion sessions, except one where the bugging became quite unbearable and I just had to leave. I had better things to do like watching “<a href="https://www.youtube.com/watch?v=Z2G_sgtdnDw" target="_blank" rel="noopener">Have I Got a Bit More News for You?</a>” or having dinner or anything else.</p>
548<p align="justify">Finally, we have to discuss what does all this mean. As in, what is the string theory research landscape in India like? Where is this research program headed in India? What are the differences compared to similar programs in other countries? And so on… I couldn’t care less about answering such boring questions. Indirect answers (or sketches thereof) to some of these questions are already available in the previous but one para. Even then, one ‘answer’ that just screams out of the data and charts above is: The ST research landscape is mostly covered by two (composite) peaks, those of black holes and (any kind of) holography. The two peaks from which I am as far away as possible given the plain areas of my research of the past decade. Let’s end this geological metaphor and get back on track of the un-punny sentences. My research falls majorly in the two minor categories of 3d (S)CFTs and other field theories (mostly dealing with 5d theories, as mentioned above) with some minor overlap with the major category of AdS/CFT &amp; holography. As we all know, 2021 was not a good year for many reasons for many people, and I can now add this one revelation to my personal list of such reasons. This might urge one to ask so would I be moving towards the peaks from the plains? Oh, Hell No! I am quite happy with the plains I prowl on.</p>
549<p align="justify">Wait, there’s one last thing to talk about: <a href="https://www.youtube.com/watch?v=DBgAjSAurwE&amp;ab_channel=IndianStringsMeeting2021" target="_blank" rel="noopener">Discussion session on gender imbalance in STEM</a>. I guess one of the reasons we have this problem is we talk about it at the end. Ok, enough with the jokes. This was a serious discussion… I learnt a lot with some precise language and data fleshing out the vague ideas I have had about the problem. The panelists in this discu
549ssion session represented ‘all walks of academic life’ who shared their experiences, talked about steps taken, being taken, steps in the pipeline, etc. Even general ideas encompassing treatment of minorities and amendment of systemic procedures to be more inclusive got an airing out. I encourage everyone to listen to this session, even re-listen, so that things (start to) change for the better.</p>
550<p align="left">Happy New Year! 🎉</p>
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559<article id="post-162" class="post-162 post type-post status-publish format-standard has-post-thumbnail hentry category-supersymmetry category-timepass tag-feynman-graphs tag-super-yang-mills-theory tag-superspace tag-supersymmetry">
560				<a href="https://jkmsmkj.fyi/2021/06/20/superspace/" title="Superspace"><img width="420" height="125" src="https://jkmsmkj.fyi/wp-content/uploads/2021/06/1l4ptgraphs.jpg?w=420" class="attachment-lovecraft-post-image size-lovecraft-post-image wp-post-image" alt="1-loop 4-point graph" decoding="async" loading="lazy" srcset="https://jkmsmkj.fyi/wp-content/uploads/2021/06/1l4ptgraphs.jpg 420w, https://jkmsmkj.fyi/wp-content/uploads/2021/06/1l4ptgraphs.jpg?w=150 150w, https://jkmsmkj.fyi/wp-content/uploads/2021/06/1l4ptgraphs.jpg?w=300 300w" sizes="auto, (max-width: 420px) 100vw, 420px" data-attachment-id="161" data-permalink="https://jkmsmkj.fyi/2021/06/20/superspace/1l4ptgraphs-jpg/" data-orig-file="https://jkmsmkj.fyi/wp-content/uploads/2021/06/1l4ptgraphs.jpg" data-orig-size="420,125" data-comments-opened="1" data-image-title="1L4PtGraphs.jpg" data-image-description="" data-image-caption="&lt;p&gt;1-loop 4-point graph&lt;/p&gt;
561" data-large-file="https://jkmsmkj.fyi/wp-content/uploads/2021/06/1l4ptgraphs.jpg?w=420" /></a>
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564			<h1 class="entry-title"><a href="https://jkmsmkj.fyi/2021/06/20/superspace/" rel="bookmark">Superspace</a></h1>
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566				<span class="posted-on">On <a href="https://jkmsmkj.fyi/2021/06/20/superspace/" rel="bookmark"><time class="entry-date published" datetime="2021-06-20T11:40:56+05:30">June 20, 2021</time><time class="updated" datetime="2025-06-06T18:12:41+05:30">June 6, 2025</time></a></span><span class="byline"> By <span class="author vcard"><a class="url fn n" href="https://jkmsmkj.fyi/author/jkmsmkj/">JKMSMKJ</a></span></span><span class="cat-links">In <a href="https://jkmsmkj.fyi/category/physics/supersymmetry/" rel="category tag">Supersymmetry</a>, <a href="https://jkmsmkj.fyi/category/timepass/" rel="category tag">TimePass</a></span><span class="comments-link"><a href="https://jkmsmkj.fyi/2021/06/20/superspace/#comments">2 Comments</a></span>			</div><!-- .entry-meta -->
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570			<p align="justify">It has been a while since I posted anything on this WordPress blog. Since I’ve just bought a domain name from WordPress, I thought I should post something, even if just to mention this update in the blog’s address. But that’s just some filler content that I post on my <a href="https://blog.jkmsmkj.fyi/">other blog</a>. So here I should at least talk about some Physics.</p>
571<p align="justify">Superspace – <a href="http://insti.physics.sunysb.edu/~siegel/parodies/stuperspace.html" target="_blank" rel="noopener">the greatest invention since the wheel</a> – is where I spent most of my graduate days. After graduating, I have spent less time there but I have regularly visited it over the years. Last year, I visited it to settle a 7 (or so) years old score with my collaborators by writing <a href="https://arxiv.org/abs/2007.13777" target="_blank" rel="noopener">a paper</a> on <strong>quantization of supersymmetric Yang-Mills theory in 4d <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=3" class="latex" /> harmonic superspace</strong> that is actually useful in computing <strong>loop graphs</strong>.</p>
572<p align="justify">Now, a popular physics blog would go to great lengths to try to explain to a layperson the various technical words that appear in the above paragraph, like quantization, SYM theory, <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=3" class="latex" />
572, harmonic superspace, and loop graphs. [Of course, 4d just refers to our (3+1)-dimensional spacetime: 3 space and 1 time directions.] As it turns out, <a href="https://4gravitons.com/" target="_blank" rel="noopener">4Gravitons</a> is just that sort of a blog and most of these terms have been explained there: <a href="https://4gravitons.com/whats-so-hard-about-quantum-field-theory-anyway/" target="_blank" rel="noopener">Quantization</a>, <a href="https://4gravitons.com/supersymmetry-to-the-rescue/" target="_blank" rel="noopener">Supersymmetric</a>, <a href="https://4gravitons.com/yang-mills-plays-well-with-itself/" target="_blank" rel="noopener">Yang-Mills theory</a>, <a href="https://4gravitons.com/n4-maximal-particles-for-maximal-fun/" target="_blank" rel="noopener"><img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=cb2018&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=cb2018&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=cb2018&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=3" class="latex" /></a> (well, <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=3" class="latex" /> is quite similar to <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D4&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D4&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D4&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=4" class="latex" />), <a href="https://4gravitons.com/a-wild-infinity-appears-or-renormalization/" target="_blank" rel="noopener">Loop graphs</a>.</p>
573<p align="justify">The most astute reader amongst you must have already realized that harmonic superspace is not explained there. Or maybe, it has been but I have deliberately not linked to that page. The reason for doing that is quite simple: I am supposed to be the (second-)last ‘expert’ on harmonic and projective superspaces in the whole world. By “(next-to-)last ‘expert’ in the whole world”, I don’t mean the pejorative phrase “world-famous in India”, I literally mean the former phrase and it does not have any connotation about fame at all in it. So, being one of the ‘experts’ in superspace, I can’t link everything to 4Gravitons (who is an <a href="https://4gravitons.com/about/" target="_blank" rel="noopener">expert in another subfield</a> of theoretical high energy physics) as I believe that one should undertake the undertaking about explaining something that one is an expert on to the lay-world.</p>
574<p align="justify">Here goes my feeble attempt then. [Disclaimer: Anyone expecting clarity and thoughtfulness in crafting the explanations, as found at 4Gravitons, will be disappointed.] First of all, we must agree with the basic fact that harmonic superspace is a kind of a superspace. Then, we need to define a superspace and later we will tag on the adjective ‘harmonic’ in a hand-waving way, which should be sufficient for the level of this post.</p>
575<p align="justify">A superspace is an extension of the usual spacetime <img src="https://s0.wp.com/latex.php?latex=x%5E%7B%5Cmu%7D%3D%28t%2Cx%2Cy%2Cz%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=x%5E%7B%5Cmu%7D%3D%28t%2Cx%2Cy%2Cz%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=x%5E%7B%5Cmu%7D%3D%28t%2Cx%2Cy%2Cz%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="x^{&#92;mu}=(t,x,y,z)" class="latex" /> with extra coordinates which have no basis in reality. (There, I said it; no beating around the bush.) These coordinates are such that (let’s call them <i>θ</i> (theta) to exude Greek sophistication) <img src="https://s0.wp.com/latex.php?latex=%5Ctheta%5E2%3D0&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Ctheta%5E2%3D0&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Ctheta%5E2%3D0&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;theta^2=0" class="latex" />. Now, a layperson would say: well, that just means <img src="https://s0.wp.com/latex.php?latex=%5Ctheta%3D0&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Ctheta%3D0&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Ctheta%3D0&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;theta=0" class="latex" />. That is, a layperson who has never heard of matrices. Anyway, hoping for this level of mathematical sophistication from our lay-audience, we can get past the fact that the vanishing square is not a problem and we can get on with building the most basic superspace, called <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=1" class="latex" /> superspace. This superspace has 4 <em>θ</em>’s [<img src="https://s0.wp.com/latex.php?latex=%5Ctheta_1%5Ctheta_2%3D-%5Ctheta_2%5Ctheta_1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Ctheta_1%5Ctheta_2%3D-%5Ctheta_2%5Ctheta_1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Ctheta_1%5Ctheta_2%3D-%5Ctheta_2%5Ctheta_1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;theta_1&#92;theta_2=-&#92;theta_2&#92;theta_1" class="latex" /> is satisfied by multiple <em>θ</em>’s. Recall that ordinary numbers satisfy <img src="https://s0.wp.com/latex.php?latex=x_1x_2%3D%2Bx_2x_1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=x_1x_2%3D%2Bx_2x_1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=x_1x_2%3D%2Bx_2x_1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="x_1x_2=+x_2x_1" class="latex" />] and can be used to describe <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=1" class="latex" /> (and higher) supersymmetric theories. This is useful because with this setup, supersymmetric theories can be described 
575in a compact notation with very concise expressions. Performing calculations in superspace is also quick and easy when compared to ordinary spacetime computations. Before superspace was invented, people found “miraculous cancellations” in their calculations, i.e., they would calculate a bunch of Feynman graphs and end up with an equivalent of &#8220;+1-1=0&#8221;. The problem is that they would have to compute the equivalents of &#8220;+1&#8221; and &#8220;-1&#8221; separately (which can be tedious) and find later that they summed to 0. Whereas in superspace, there are far fewer (super)Feynman graphs to compute and one directly sees the 0’s. In other words, nothing tedious to calculate and cancel later.</p>
576<p align="justify">Now, we come to another fact that there are <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=2" class="latex" /> supersymmetric theories and we should probably have <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=2" class="latex" /> superspace to describe them directly. So one may just append 4 more <em>θ</em>’s to <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=1" class="latex" /> superspace and call it a day. That approach is a valid one but doesn’t go far in achieving all the successes of <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=1" class="latex" /> superspace in simplifying computations for <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=2" class="latex" /> theories. (Because there are certain objects that do not exist in this naïve <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=2" class="latex" /> superspace, which turn out to be crucial for simplifying computations.) Thus, we have to invoke harmonic (or projective) variety of the <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D2&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=2" class="latex" /> superspace. Harmonic superspace contains the usual spacetime, only 4 <em>θ</em>’s (which are obtained as some combinations of the earlier 8 <em>θ</em>
576’s) and a 2d sphere (an ordinary sphere, nothing exotic like the <em>θ</em>’s but it is again not ‘real’; though, it has a very satisfying origin story in the mathematical structure of supersymmetric theories, whose discussion is beyond the scope of this post). It might still not be clear to a layperson what has ‘harmonic’ got to do with that extra 2d sphere. The punchline being that the study of functions on spheres is referred to as harmonic analysis in the mathematical literature. (The adjective ‘projective’ is also related to the 2d sphere, which can be treated as a <a href="https://en.wikipedia.org/wiki/Riemann_sphere#As_the_complex_projective_line" target="_blank" rel="noopener">complex projective line</a>. Actually, the word projective has a more satisfying origin in the way the coordinates transform in projective superspace. You guessed it right: they transform via <a href="https://jkmsmkj.wordpress.com/2013/07/27/projective-transformations/" target="_blank" rel="noopener">projective transformations</a>.) QED</p>
577<p align="justify">Just completing the loop, <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%3D3&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathcal{N}=3" class="latex" /> harmonic superspace consists of the usual spacetime, 8 <em>θ</em>’s (instead of the expected 12) and a 6d compact space (don’t really want to say anything more than that right now). Our 7-years-in-making paper studies the SYM theory in this superspace and derives Feynman rules to make computations of loop graphs easier. This problem has been unsolved for more than 3 decades. One proof-of-concept example we include in our paper slashes a 6-pages computation based on symmetry arguments to a ½-page direct computation of two very similar Feynman graphs.</p>
578<p><img loading="lazy" style="border:0 currentcolor;margin-right:auto;margin-left:auto;float:none;display:block;background-image:none" title="1 Loop 4 Point Graphs" src="https://jkmsmkj.fyi/wp-content/uploads/2021/06/1l4ptgraphs.jpg?w=420&#038;h=125" alt="1 Loop 4 Point Graphs" width="420" height="125" border="0" /></p>
579<p>So I guess that’s all for this post. Till next time, subscribe to 4Gravitons. You will learn a lot about <a href="https://4gravitons.com/2019/07/26/the-real-emc2/" target="_blank" rel="noopener">Theoretical High Energy Physic(icist)s</a>.</p>
580
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582					</div><!-- .entry-content -->
583	</div>
584</article><!-- #post-## -->
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588<article id="post-107" class="post-107 post type-post status-publish format-standard hentry category-latex category-physics category-timepass tag-arxiv-identifier tag-bibliography-styles tag-hyperlinks tag-latex">
589			<div class="entry-wrapper">
590		<header class="entry-header">
591			<h1 class="entry-title"><a href="https://jkmsmkj.fyi/2018/12/31/hephys-bibliography-style/" rel="bookmark">hephys Bibliography Style</a></h1>
592						<div class="entry-meta">
593				<span class="posted-on">On <a href="https://jkmsmkj.fyi/2018/12/31/hephys-bibliography-style/" rel="bookmark"><time class="entry-date published" datetime="2018-12-31T04:20:46+05:30">December 31, 2018</time><time class="updated" datetime="2026-03-21T20:19:20+05:30">March 21, 2026</time></a></span><span class="byline"> By <span class="author vcard"><a class="url fn n" href="https://jkmsmkj.fyi/author/jkmsmkj/">JKMSMKJ</a></span></span><span class="cat-links">In <a href="https://jkmsmkj.fyi/category/latex/" rel="category tag">LaTeX</a>, <a href="https://jkmsmkj.fyi/category/physics/" rel="category tag">Physics</a>, <a href="https://jkmsmkj.fyi/category/timepass/" rel="category tag">TimePass</a></span><span class="comments-link"><a href="https://jkmsmkj.fyi/2018/12/31/hephys-bibliography-style/#respond">Leave a comment</a></span>			</div><!-- .entry-meta -->
594					</header><!-- .entry-header -->
595
596		<div class="entry-content">
597			<p align="justify">[Start of ChangeLog: Changes noted here are reflected in the original text further below.</p>
598<p align="justify">Update(08/08/19):</p>
599<ol>
600<li>DOI hyperlinking for inCollection and inProceedings/Conference has been changed from &#8220;chapter, pages&#8221; to &#8220;title&#8221;.</li>
601<li>The keyword &#8220;chapter&#8221; has been made optional for inCollection.</li>
602</ol>
603<p>Update(15/11/21):</p>
604<ol>
605<li>DOI hyperlinking for inBook has been changed from &#8220;chapter, pages&#8221; to &#8220;title&#8221;.</li>
606<li>inBook is treated as inCollection if both authors and editors are present in the data entry! (This specific update is inspired by <a href="http://insti.physics.sunysb.edu/~siegel/plan.html" target="_blank" rel="noopener">Warren Siegel</a> citing <a href="https://inspirehep.net/literature/880993" target="_blank" rel="noopener">this paper</a> in <a href="https://arxiv.org/abs/2110.13010" target="_blank" rel="noopener">his paper</a>.)</li>
607<li>The keyword &#8220;pages&#8221; has been made optional for inCollection and inProceedings/Conference.</li>
608<li>A minor hyperlinking bug in Misc has been fixed.</li>
609</ol>
610<p>Update(21/03/26):</p>
611<ol>
612<li>The fixed string &#8220;arxiv&#8221; is changed to the keyword &#8220;archivePrefix&#8221;. (Just to accommodate viXra.org! 😂)</li>
613</ol>
614<p>End of ChangeLog.]</p>
615<p align="justify">You should have come from <a href="https://jkmsmkj.blogspot.com/2018/12/bibliography-style.html" target="_blank" rel="noopener">here</a>.</p>
616<p align="justify">Reiterating, I have created ‘hephys’ bibliography style which can be used to add references in hep-th papers. I was planning to compare hephys with “utphys” (my preferred style before I thought of creating my own) but that may leave a bad taste in some mouths including my own so let me just highlight some features of hephys, with no ulterior motive. hephys is optimized for InSpire&#8217;s BibTeX data so it’s mostly intended for people writing hep-th papers. In fact, utphys is quite general so anyone not writing hep-th papers should just stick to it.</p>
617<p align="justify">Some interesting facts about hephys:</p>
618<ol>
619<li>hephys tries to follow mostly LaTeX(EU) style typesetting of citations.</li>
620<li>hephys correctly typesets and links both old-style and new-style arXiv-identifiers.</li>
621<li>hephys treats url field as doi if doi field is not present.</li>
622<li>If doi field is present, then hephys appends url at the end of the citation <strong>only</strong> if it is a different link (it can’t be an arXiv link either; as I said before, very specialized to hep-th papers).</li>
623<li>hephys clubs number and volume together as in vol [num] for journal articles.</li>
624<li>After everything’s thought and written, there’s a possible deal-breaker: “Title”, vs. “Title,”. (No prizes for guessing which one hephys outputs!<img class="emoji" src="https://s0.wp.com/wp-content/mu-plugins/wpcom-smileys/twemoji/2/72x72/1f606.png?m=1781160205i" alt="😆" />)</li>
625</ol>
626<p align="justify">On to some history now: I used ‘makebst’ command (available in any LaTeX installation) to create a skeleton 
626style file ages ago. Why was it a skeleton, you ask? Because it had no (when I say no, I mean <strong>NO</strong>) support for hyperlinking or any idea for handling arXiv data properly. So where do we go from there? I don’t know about ‘we’ but I learnt that .bst files are written in an unnamed programming language using postfix notation! That brought back ‘great’ memories from class IX (nearly 2 decades ago) when we were taught some ‘computer theory’ along with BASIC (a basic programming language). Never thought that there would be a ‘serious’ language based on this notation. I have seen some discussion about prefix / infix / postfix notation in Mathematica somewhere but still as a curiosity rather than in any actual programming usage. But, here there was an opportunity to explore the thing that I once thought was &#8216;unexplore-able&#8217;. Considering myself a programming expert, I was able to edit a few things to make arXiv links appear as I wanted, after going through <a href="http://mirrors.ctan.org/biblio/bibtex/contrib/doc/btxhak.pdf" target="_blank" rel="noopener">this manual</a> half a dozen times.</p>
627<p align="justify">Then I remembered there’s something called DOI which can be used to hyperlink journal info! That was the time to open utphys.bst file and steal Jacques Distler’s code (basically the function “add.doi”). My expert senses, not to rest so easily, went on to give me sleepless nights till I fixed the behaviour of url field as highlighted in points 3 &amp; 4 above. Oh boy! writing if-else and for-loop in postfix notation is not child’s play (read: mind-numbingly frustrating) but wow, I’d be lying if I didn’t say It’s Quite Fulfilling! Also, I made very few keywords mandatory for a given citation style so not many errors pop up even if the BibTeX data is not up to the mark (i.e., cobbled up while in a half-awake state at midnight without any regards for health &amp; safety instructions). After that, to reduce the chances of hephys clashing with utphys too much, I removed a few things to make it unappealing to people writing non-hep-th papers. The following are the only citation styles (with the accepted / expected keywords; only bold ones are required) supported in hephys and their sample outputs (the blue colouring just means the expected hyperlink as explained in points 2, 3 &amp; 4 above):</p>
628<p align="justify">1. Article: <strong>author</strong>, collaboration, <strong>title</strong>, journal, volume, number, year, pages, doi, note, archivePrefix, eprint, primaryClass, url.</p>
629<p align="left">Samples:</p>
630<p align="justify">@article{art:1, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;article with both doi and url&#8221;, journal = &#8220;journal&#8221;, volume = &#8220;vol&#8221;, number = &#8220;num&#8221;, year = &#8220;year&#8221;, pages = &#8220;pages&#8221;, doi = &#8220;doi&#8221;, note = &#8220;note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;eprint&#8221;, primaryClass = &#8220;pClass&#8221;, url = &#8220;⋯.url.⋯&#8221;}</p>
631<p align="justify">[1] author (collaboration), &#8220;article with both doi and url&#8221;, <span style="color: #0000ff"><em>journal</em> <strong>vol[num]</strong> (year) pages</span>, note, <span style="color: #0000ff">arXiv:eprint [pClass]</span>, <span style="color: #0000ff">URL</span>.</p>
632<p align="justify">@article{art:2, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;article with both doi and url but duplicate url not used&#8221;, journal = &#8220;journal&#8221;, volume = &#8220;vol&#8221;, number = &#8220;num&#8221;, year = &#8220;year&#8221;, pages = &#8220;pages&#8221;, doi = &#8220;doi&#8221;, note = &#8220;note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;pClass/eprint&#8221;, primaryClass = &#8220;pClass&#8221;, url = &#8220;····://··.doi.···/10.1086⋯x&#8221;}</p>
633<p align="justify">[2] author (collaboration), &#8220;article with both doi and url but duplicate url not used&#8221;, <span style="color: #0000ff"><em>journal</em> <strong>vol[num]</strong> (year) pages</span>, note, <span style="color: #0000ff">arXiv:pClass/eprint</span>.</p>
634<p align="justify">@article{art:3, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;article with only url used as doi&#8221;, journal = &#8220;
634journal&#8221;, volume = &#8220;vol&#8221;, number = &#8220;num&#8221;, year = &#8220;year&#8221;, pages = &#8220;pages&#8221;, note = &#8220;note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;eprint&#8221;, primaryClass = &#8220;pClass&#8221;, url = &#8220;⋯.url.⋯&#8221;}</p>
635<p align="justify">[3] author (collaboration), &#8220;article with only url used as doi&#8221;, <span style="color: #0000ff"><em>journal</em> <strong>vol[num]</strong> (year) pages</span>, note, <span style="color: #0000ff">arXiv:eprint [pClass]</span>.</p>
636<p align="justify">@article{art:4, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;article with neither doi nor url&#8221;, journal = &#8220;journal&#8221;, volume = &#8220;vol&#8221;, number = &#8220;num&#8221;, year = &#8220;year&#8221;, pages = &#8220;pages&#8221;, note = &#8220;note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;pClass/eprint&#8221;, primaryClass = &#8220;pClass&#8221;}</p>
637<p align="justify">[4] author (collaboration), &#8220;article with neither doi nor url&#8221;, <em>journal</em> <strong>vol[num]</strong> (year) pages, note, <span style="color: #0000ff">arXiv:pClass/eprint</span>.</p>
638<p align="justify">@article{art:5, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;article with no journal&#8221;, year = &#8220;year&#8221;, pages = &#8220;pages&#8221;, note = &#8220;note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;pClass/eprint&#8221;, primaryClass = &#8220;pClass&#8221;}</p>
639<p align="justify">[5] author (collaboration), &#8220;article with no journal&#8221;, year, note, <span style="color: #0000ff">arXiv:pClass/eprint</span>.</p>
640<p align="justify">2. Book: <strong>author/editor</strong>, collaboration, <strong>title</strong>, doi, volume/number, series, edition, <strong>publisher</strong>, address, <strong>year</strong>, note, archivePrefix, eprint, primaryClass, url.</p>
641<p align="left">Samples:</p>
642<p align="justify">@book{boo:1, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;book&#8221;, doi = &#8220;doi&#8221;, number = &#8220;num&#8221;, series = &#8220;series&#8221;, edition = &#8220;edn&#8221;, publisher = &#8220;publisher&#8221;, address = &#8220;address&#8221;, year = &#8220;year&#8221;, note = &#8220;note&#8221;, archivePrefix = &#8220;arXiv&#8221;,eprint = &#8220;eprint&#8221;, primaryClass = &#8220;pClass&#8221;, url = &#8220;⋯.url.⋯&#8221;}</p>
643<p align="justify">[6] author (collaboration), <span style="color: #0000ff">book</span>, number num in series, edn edition, publisher, address year, note, <span style="color: #0000ff">arXiv:eprint [pClass]</span>, <span style="color: #0000ff">URL</span>.</p>
644<p align="justify">@book{boo:2, editor = &#8220;editor&#8221;, title = &#8220;book&#8221;, doi = &#8220;doi&#8221;, volume = &#8220;vol&#8221;, series = &#8220;series&#8221;, edition = &#8220;edn&#8221;, publisher = &#8220;publisher&#8221;, address = &#8220;address&#8221;, year = &#8220;year&#8221;, note = &#8220;note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;eprint&#8221;, primaryClass = &#8220;pClass&#8221;, url = &#8220;⋯.url.⋯&#8221;}</p>
645<p align="justify">[7] editor (Ed.) <span style="color: #0000ff">book</span>, volume vol of <i>series</i>, edn edition, publisher, address year, note, <span style="color: #0000ff">arXiv:eprint [pClass]</span>, <span style="color: #0000ff">URL</span>.</p>
646<p align="justify">3. Booklet: <strong>author</strong>, collaboration, <strong>title</strong>, howpublished, doi, address, year, note, archivePrefix, eprint, primaryClass, url.</p>
647<p align="left">Sample:</p>
648<p align="justify">@booklet{boo:3, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;booklet&#8221;, howpublished = &#8220;howpublished&#8221;, doi = &#8220;doi&#8221;, address = &#8220;address&#8221;, year = &#8220;year&#8221;, note = &#8220;note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;eprint&#8221;, primaryClass = &#8220;pClass&#8221;, url = &#8220;⋯.url.⋯&#8221;}</p>
649<p align="justify">[8] author (collaboration), &#8220;booklet&#8221;, <span style="color: #0000ff">howpublished</span>, address year, note, <span style="color: #0000ff">arXiv:eprint [pClass]</span>, <span style="color: #0000ff">URL</span>.</p>
650<p align="justify">4. inBook: <strong>author </strong>and/or<strong> editor</strong>, collaboration, <strong>title</strong>, doi, (<strong>booktitle</strong> if both author &amp; editor are present), volume/number, series, <strong>chapter</strong> and/or <strong>pages</strong>, type, edition, <strong>publisher</strong>, address, <strong>year</strong>, note, archivePrefix, eprint, primaryClass, url.</p>
651<p align="left">Sample:</p>
652<p align="justify">@inbook{inb:1, editor = &#8220;editor&#8221;, title = &#8220;inbook&#8221;, volume = &#8220;vol&#8221;, series = &#8220;series&#8221;, chapter = &#8220;ch&#8221;, pages = &#8220;pages&#8221;, doi = &#8220;doi&#8221;, edition = &#8220;edn&#8221;, publisher = &#8220;publisher&#8221;, address = &#8220;address&#8221;, year = &#8220;year&#8221;, note = &#8220;
652note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;eprint&#8221;, primaryClass = &#8220;pClass&#8221;, url = &#8220;⋯.url.⋯&#8221;}</p>
653<p align="justify">[9] editor (Ed.), <span style="color: #0000ff">inbook</span>, volume vol of <i>series</i>, <span style="color: #000000">chapter ch, p. pages</span>, edn edition, publisher, address year, note, <span style="color: #0000ff">arXiv:eprint [pClass]</span>, <span style="color: #0000ff">URL</span>.</p>
654<p align="justify">5. inCollection: <strong>author</strong>, collaboration, <strong>title</strong>, doi, <strong>booktitle</strong>, editor, edition, chapter, type, pages, volume/number, series, <strong>publisher</strong>, address, <strong>year</strong>, note, archivePrefix, eprint, primaryClass, url.</p>
655<p align="left">Sample:</p>
656<p align="justify">@incollection{inc:1, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;incollection&#8221;, doi = &#8220;doi&#8221;, booktitle = &#8220;booktitle&#8221;, editor = &#8220;editor&#8221;, edition = &#8220;edn&#8221;, chapter = &#8220;ch&#8221;, type = &#8220;Section&#8221;, pages = &#8220;pages&#8221;, number = &#8220;num&#8221;, series = &#8220;series&#8221;, publisher = &#8220;publisher&#8221;, address = &#8220;address&#8221;, year = &#8220;year&#8221;, note = &#8220;note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;eprint&#8221;, primaryClass = &#8220;pClass&#8221;, url = &#8220;⋯.url.⋯&#8221;}</p>
657<p align="justify">[10] author (collaboration), &#8220;<span style="color: #0000ff">incollection</span>&#8220;, <i>in</i> booktitle (edited by editor), edn edition, <span style="color: #000000">section ch, p. pages</span>, number num in series, publisher, address year, note, <span style="color: #0000ff">arXiv:eprint [pClass]</span>, <span style="color: #0000ff">URL</span>.</p>
658<p align="justify">6. inProceedings/Conference: <strong>author</strong>, collaboration, <strong>title</strong>, doi, <strong>booktitle</strong>, volume/number, series, editor, pages, organization, publisher, address, <strong>year</strong>, note, archivePrefix, eprint, primaryClass, url.</p>
659<p align="left">Sample:</p>
660<p align="justify">@inproceedings{inp:1, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;inproceedings&#8221;, doi = &#8220;doi&#8221;, booktitle = &#8220;booktitle&#8221;, volume = &#8220;vol&#8221;, series = &#8220;series&#8221;, editor = &#8220;editor&#8221;, pages = &#8220;pages&#8221;, organization = &#8220;organization&#8221;, publisher = &#8220;publisher&#8221;, address = &#8220;address&#8221;, year = &#8220;year&#8221;, note = &#8220;note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;eprint&#8221;, primaryClass = &#8220;pClass&#8221;, url = &#8220;⋯.url.⋯&#8221;}</p>
661<p align="justify">[11] author (collaboration), &#8220;<span style="color: #0000ff">inproceedings</span>&#8220;, <i>in</i> booktitle, volume vol of <i>series</i> (edited by editor), <span style="color: #000000">p. pages</span>, organization, publisher, address year, note, <span style="color: #0000ff">arXiv:eprint [pClass]</span>, <span style="color: #0000ff">URL</span>.</p>
662<p align="justify">7. MastersThesis: <strong>author</strong>, collaboration, <strong>title</strong>, doi, type, school, address, <strong>year</strong>, note, archivePrefix, eprint, primaryClass, url.</p>
663<p align="left">Sample:</p>
664<p align="justify">@mastersthesis{mas:1, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;mastersthesis&#8221;, doi = &#8220;doi&#8221;, school = &#8220;school&#8221;, address = &#8220;address&#8221;, year = &#8220;year&#8221;, note = &#8220;note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;eprint&#8221;, primaryClass = &#8220;pClass&#8221;, url = &#8220;⋯.url.⋯&#8221;}</p>
665<p align="justify">[12] author (collaboration), <span style="color: #0000ff">mastersthesis</span>, Master&#8217;s thesis, school, address year, note, <span style="color: #0000ff">arXiv:eprint [pClass]</span>, <span style="color: #0000ff">URL</span>.</p>
666<p align="justify">8. PhdThesis: <strong>author</strong>, collaboration, <strong>title</strong>, doi, type, school, address, <strong>year</strong>, note, archivePrefix, eprint, primaryClass, url.</p>
667<p align="left">Sample:</p>
668<p align="justify">@phdthesis{phd:1, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;phdthesis&#8221;, doi = &#8220;doi&#8221;, type = &#8220;{Ph.D.} dissertation&#8221;, school = &#8220;school&#8221;, address = &#8220;address&#8221;, year = &#8220;year&#8221;, note = &#8220;
668note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;eprint&#8221;, primaryClass = &#8220;pClass&#8221;, url = &#8220;⋯.url.⋯&#8221;}</p>
669<p align="justify">[13] author (collaboration), <span style="color: #0000ff">phdthesis</span>, Ph.D. dissertation, school, address year, note, <span style="color: #0000ff">arXiv:eprint [pClass]</span>, <span style="color: #0000ff">URL</span>.</p>
670<p align="justify">9. Proceedings: editor, <strong>title</strong>, doi, volume/number, series, organization, publisher, address, <strong>year</strong>, note, archivePrefix, eprint, primaryClass, url.</p>
671<p align="left">Sample:</p>
672<p align="justify">@proceedings{pro:1, editor = &#8220;editor&#8221;, title = &#8220;proceedings&#8221;, doi = &#8220;doi&#8221;, volume = &#8220;vol&#8221;, series = &#8220;series&#8221;, organization = &#8220;organization&#8221;, publisher = &#8220;publisher&#8221;, address = &#8220;address&#8221;, year = &#8220;year&#8221;, note = &#8220;note&#8221;, archivePrefix = &#8220;arXiv&#8221;, eprint = &#8220;eprint&#8221;, primaryClass = &#8220;pClass&#8221;, url = &#8220;⋯.url.⋯&#8221;}</p>
673<p align="justify">[14] editor (Ed.), <span style="color: #0000ff">proceedings</span>, volume vol of <i>series</i>, organization, publisher, address year, note, <span style="color: #0000ff">arXiv:eprint [pClass]</span>, <span style="color: #0000ff">URL</span>.</p>
674<p align="justify">10. Misc: author, collaboration, title, howpublished, doi, year, note, url.</p>
675<p align="left">Sample:</p>
676<p align="justify">@misc{mis:1, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;misc&#8221;, howpublished = &#8220;howpublished&#8221;, year = &#8220;year&#8221;, note = &#8220;note&#8221;, url = &#8220;⋯.url.⋯&#8221;}</p>
677<p align="justify">[15] author (collaboration), &#8220;misc&#8221;, <span style="color: #0000ff">howpublished</span> year, note.</p>
678<p align="justify">11. Unpublished: <strong>author</strong>, collaboration, title, year, <strong>note</strong>.</p>
679<p align="left">Sample:</p>
680<p align="justify">@unpublished{unp:1, author = &#8220;author&#8221;, collaboration = &#8220;collaboration&#8221;, title = &#8220;unpublished&#8221;, year = &#8220;year&#8221;, note = &#8220;note&#8221;}</p>
681<p align="justify">[16] author (collaboration), &#8220;unpublished&#8221;, year, note.</p>
682<p align="justify">So there you have it, the full ‘manual’ for hephys. If you find any bugs, do send me the offending citation’s BibTeX data &amp; expected output and I’ll look into it. There is one bug regarding citations having erratum/addendum information and two doi’s. It’s not handled correctly for now but I’m thinking of doing something about it in the new year. On second thought, there may not be a universal format for such data so I’m a bit hesitant to start looking into separating multiple doi’s and journal ref’s just yet. Anyway, if you (yes You, the zeroth reader of this post) are up for programming these and more fixes in your own ‘hephys’, do let me know. After all, the world can only rejoice with more choices to consider!‍<img class="emoji" src="https://s0.wp.com/wp-content/mu-plugins/wpcom-smileys/twemoji/2/72x72/1f926-200d-2642-fe0f.png?m=1593063836i" alt="🤦‍♂️" /></p>
683<p align="left"><a href="https://www.dropbox.com/s/mxmeptwvnh6li75/hephys.bst?dl=1">Download Link</a> for those who reached this point!</p>
684
685			
686					</div><!-- .entry-content -->
687	</div>
688</article><!-- #post-## -->
689
690			
691				
692<article id="post-89" class="post-89 post type-post status-publish format-standard has-post-thumbnail hentry category-linear-algebra category-mathematics category-timepass">
693				<a href="https://jkmsmkj.fyi/2013/07/27/projective-transformations/" title="Projective Transformations"><img width="420" height="315" src="https://jkmsmkj.fyi/wp-content/uploads/2013/07/wp_000066.jpg?w=420" class="attachment-lovecraft-post-image size-lovecraft-post-image wp-post-image" alt="Before keystone correction..." decoding="async" loading="lazy" srcset="https://jkmsmkj.fyi/wp-content/uploads/2013/07/wp_000066.jpg 420w, https://jkmsmkj.fyi/wp-content/uploads/2013/07/wp_000066.jpg?w=150 150w, https://jkmsmkj.fyi/wp-content/uploads/2013/07/wp_000066.jpg?w=300 300w" sizes="auto, (max-width: 420px) 100v
693w, 420px" data-attachment-id="85" data-permalink="https://jkmsmkj.fyi/2013/07/27/projective-transformations/wp_000066-jpg/" data-orig-file="https://jkmsmkj.fyi/wp-content/uploads/2013/07/wp_000066.jpg" data-orig-size="420,315" data-comments-opened="1" data-image-title="WP_000066.jpg" data-image-description="" data-image-caption="&lt;p&gt;Before keystone correction&amp;#8230;&lt;/p&gt;
694" data-large-file="https://jkmsmkj.fyi/wp-content/uploads/2013/07/wp_000066.jpg?w=420" /></a>
695		<div class="entry-wrapper">
696		<header class="entry-header">
697			<h1 class="entry-title"><a href="https://jkmsmkj.fyi/2013/07/27/projective-transformations/" rel="bookmark">Projective Transformations</a></h1>
698						<div class="entry-meta">
699				<span class="posted-on">On <a href="https://jkmsmkj.fyi/2013/07/27/projective-transformations/" rel="bookmark"><time class="entry-date published" datetime="2013-07-27T21:00:20+05:30">July 27, 2013</time><time class="updated" datetime="2025-06-06T18:10:08+05:30">June 6, 2025</time></a></span><span class="byline"> By <span class="author vcard"><a class="url fn n" href="https://jkmsmkj.fyi/author/jkmsmkj/">JKMSMKJ</a></span></span><span class="cat-links">In <a href="https://jkmsmkj.fyi/category/mathematics/linear-algebra/" rel="category tag">Linear Algebra</a>, <a href="https://jkmsmkj.fyi/category/mathematics/" rel="category tag">Mathematics</a>, <a href="https://jkmsmkj.fyi/category/timepass/" rel="category tag">TimePass</a></span><span class="comments-link"><a href="https://jkmsmkj.fyi/2013/07/27/projective-transformations/#comments">2 Comments</a></span>			</div><!-- .entry-meta -->
700					</header><!-- .entry-header -->
701
702		<div class="entry-content">
703			<p align="justify">Today’s problem is how to transform a given quadrilateral to another quadrilateral.</p>
704<p align="justify">Let me be more precise. The transformation is to be considered in the context of photography. For example, look at this ‘Scrabble’ board.</p>
705<p align="justify"><img loading="lazy" style="background-image: none;float: none;padding-top: 0;padding-left: 0;margin: 0 auto;display: block;padding-right: 0;border: 0" title="Quadrilateral" src="https://jkmsmkj.fyi/wp-content/uploads/2013/07/wp_000066.jpg?w=420&#038;h=315" alt="Quadrilateral" width="420" height="315" border="0" /></p>
706<p align="justify">As one might have expected for such a board to look like a square (instead of that unsightly trapezoid), it’s our job in this post to ‘correct’ the above image. That is, to transform a given quadrilateral to a ‘proper’ rectangle.</p>
707<p align="justify">The solution comes in the form of Projective transformation in 2D. Projective transformations are basically fractional linear transformations:</p>
708<p align="justify"><img src="https://s0.wp.com/latex.php?latex=x%5E%7B%5Cprime%7D%3D%5Cfrac%7Bax%2Bby%2Bc%7D%7Bgx%2Bhy%2Bi%7D%2C%5Cquad+y%5E%7B%5Cprime%7D%3D%5Cfrac%7Bdx%2Bey%2Bf%7D%7Bgx%2Bhy%2Bi%7D.&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=x%5E%7B%5Cprime%7D%3D%5Cfrac%7Bax%2Bby%2Bc%7D%7Bgx%2Bhy%2Bi%7D%2C%5Cquad+y%5E%7B%5Cprime%7D%3D%5Cfrac%7Bdx%2Bey%2Bf%7D%7Bgx%2Bhy%2Bi%7D.&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=x%5E%7B%5Cprime%7D%3D%5Cfrac%7Bax%2Bby%2Bc%7D%7Bgx%2Bhy%2Bi%7D%2C%5Cquad+y%5E%7B%5Cprime%7D%3D%5Cfrac%7Bdx%2Bey%2Bf%7D%7Bgx%2Bhy%2Bi%7D.&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="x^{&#92;prime}=&#92;frac{ax+by+c}{gx+hy+i},&#92;quad y^{&#92;prime}=&#92;frac{dx+ey+f}{gx+hy+i}." class="latex" /></p>
709<p align="justify">Why isn’t just a simple linear transformation enough for this job? Because as we know, a 2D matrix can only describe scaling, rotation and shear. It doesn’t have translations and it also can NOT make converging lines parallel! So, not only linear transformation is insufficient, even affine transformation won’t fit our purposes. The transformation which can make parallel lines converge (i.e. make ∞ come to a finite point) is what we are really after and the projective one does that job. Still all is not lost. All these 5 transformations can be made linear if we are willing to go up a bit from 2D to 3D.</p>
710<p align="justify">Let’s augment our 2D coordinates <img src="https://s0.wp.com/latex.php?latex=%28x%2Cy%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%28x%2Cy%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%28x%2Cy%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="(x,y)" class="latex" /> by <img src="https://s0.wp.com/latex.php?latex=w&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=w&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=w&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="w" class="latex" /> and consider the following 3D linear transformation:</p>
711<p align="justify"><img src="https://s0.wp.com/latex.php?latex=%5Cbegin%7Bpmatrix%7Dx%5E%7B%5Cprime%7D+%5C%5C+y%5E%7B%5Cprime%7D+%5C%5C+w%5E%7B%5Cprime%7D%5Cend%7Bpmatrix%7D%3D%5Cbegin%7Bpmatrix%7Da+%26+b+%26+c+%5C%5C+d+%26+e+%26+f+%5C%5C+g+%26+h+%26+i%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7Dx+%5C%5C+y+%5C%5C+w%5Cend%7Bpmatrix%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cbegin%7Bpmatrix%7Dx%5E%7B%5Cprime%7D+%5C%5C+y%5E%7B%5Cprime%7D+%5C%5C+w%5E%7B%5Cprime%7D%5Cend%7Bpmatrix%7D%3D%5Cbegin%7Bpmatrix%7Da+%26+b+%26+c+%5C%5C+d+%26+e+%26+f+%5C%5C+g+%26+h+%26+i%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7Dx+%5C%5C+y+%5C%5C+w%5Cend%7Bpmatrix%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cbegin%7Bpmatrix%7Dx%5E%7B%5Cprime%7D+%5C%5C+y%5E%7B%5Cprime%7D+%5C%5C+w%5E%7B%5Cprime%7D%5Cend%7Bpmatrix%7D%3D%5Cbegin%7Bpmatrix%7Da+%26+b+%26+c+%5C%5C+d+%26+e+%26+f+%5C%5C+g+%26+h+%26+i%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7Dx+%5C%5C+y+%5C%5C+w%5Cend%7Bpmatrix%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;begin{pmatrix}x^{&#92;prime} &#92;&#92; y^{&#92;prime} &#92;&#92; w^{&#92;prime}&#92;end{pmatrix}=&#92;begin{pmatrix}a &amp; b &amp; c &#92;&#92; d &amp; e &amp; f &#92;&#92; g &amp; h &amp; i&#92;end{pmatrix}&#92;begin{pmatrix}x &#92;&#92; y &#92;&#92; w&#92;end{pmatrix}" class="latex" /></p>
712<p align="justify">To get the 2D coordinates, we have to project back from 3D by dividing <img src="https://s0.wp.com/latex.php?latex=%28x%2C+y%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%28x%2C+y%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%28x%2C+y%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="(x, y)" class="latex" /> by <img src="https://s0.wp.com/latex.php?latex=w&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=w&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=w&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="w" class="latex" />. Since that division is a bit cumbersome, we will choose <img src="https://s0.wp.com/latex.php?latex=w%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=w%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=w%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="w=1" class="latex" />. (The division makes sure <img src="https://s0.wp.com/latex.php?latex=w%5E%7B%5Cprime%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=w%5E%7B%5Cprime%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=w%5E%7B%5Cprime%7D%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="w^{&#92;prime}=1" class="latex" /> too.) Let’s do a sanity check: <img src="https://s0.wp.com/latex.php?latex=a%2C+b%2C+d%2C+e&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=a%2C+b%2C+d%2C+e&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=a%2C+b%2C+d%2C+e&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="a, b, d, e" class="latex" /> give the usual linear (2D) transformations, <img src="https://s0.wp.com/latex.php?latex=c%2C+f&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=c%2C+f&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=c%2C+f&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="c, f" class="latex" /> give the translations, <img src="https://s0.wp.com/latex.php?latex=g%2C+h&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=g%2C+h&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=g%2C+h&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="g, h" class="latex" /> give the projective transformations, and <em>i</em> is a global scaling. This last transformation is redundant for our purposes so we can set <img src="https://s0.wp.com/latex.php?latex=i%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=i%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=i%3D1&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="i=1" class="latex" />.</p>
713<p align="justify">Now, we have the right transformation tool. Why, you may ask? The answer is because this transformation matrix has 8 (unknown) parameters and given the 4 corners of the (source) quadrilateral &amp; (destination) rectangle, we can write down 8 equations relating them. So 8 equations and 8 unknowns → any respectable linear algebra package should be able to do the ‘Maths’! (<strong>Caveat:</strong> You – How do you know the destination corners? Me – Well, that’s ‘beyond the scope’ of this post. You – $$$##%%^#)</p>
714<p align="justify">Actually, we can do better than that. If one stares at those 8 equations long enough, one realizes that those equations can be solved analytically if the transformation is from a unit square to a given quadrilateral! You might be thinking that’s not too helpful;
714 we want the reverse at the very least. Not quite! Because in image transformation business, if you think about it, “you don’t PUT the pixel, you GET the pixel”. (That’s my quote and you can fearlessly attribute it to me from now on. – Thanks.) <em>For more discussion on this revelation, read <a href="http://jkmsmkj.blogspot.com/2011/02/splitting-and-joining.html" target="_blank" rel="noopener">this post</a>.</em></p>
715<p align="justify">So what we have to do now is simple:</p>
716<ol>
717<li>
718<div align="justify">Get the 4 coordinates of the corners of the quadrilateral in the source image: <img src="https://s0.wp.com/latex.php?latex=X_q&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=X_q&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=X_q&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="X_q" class="latex" /> .</div>
719</li>
720<li>
721<div align="justify">Find the 8 parameters (<img src="https://s0.wp.com/latex.php?latex=a%2C+%5Ccdots%2C+h&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=a%2C+%5Ccdots%2C+h&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=a%2C+%5Ccdots%2C+h&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="a, &#92;cdots, h" class="latex" />) in the transformation matrix from the analytical solution for the coordinates <img src="https://s0.wp.com/latex.php?latex=X_q&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=X_q&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=X_q&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="X_q" class="latex" />: <img src="https://s0.wp.com/latex.php?latex=T_s&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=T_s&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=T_s&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="T_s" class="latex" /> .</div>
722</li>
723<li>
724<div align="justify">Figure out the corners of the destination rectangle: <img src="https://s0.wp.com/latex.php?latex=X_r&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=X_r&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=X_r&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="X_r" class="latex" /> .</div>
725</li>
726<li>
727<div align="justify">Scale the transformation matrix in <img src="https://s0.wp.com/latex.php?latex=%28x%2Cy%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%28x%2Cy%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%28x%2Cy%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="(x,y)" class="latex" />
727 direction appropriately so that the square scales to the required rectangle: <img src="https://s0.wp.com/latex.php?latex=T_r%3DT_s+%5Ccdot%5Ctext%7Bdiag%7D%5Cleft%28%5Ctfrac%7B1%7D%7BW%7D%2C+%5Ctfrac%7B1%7D%7BH%7D%2C+1%5Cright%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=T_r%3DT_s+%5Ccdot%5Ctext%7Bdiag%7D%5Cleft%28%5Ctfrac%7B1%7D%7BW%7D%2C+%5Ctfrac%7B1%7D%7BH%7D%2C+1%5Cright%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=T_r%3DT_s+%5Ccdot%5Ctext%7Bdiag%7D%5Cleft%28%5Ctfrac%7B1%7D%7BW%7D%2C+%5Ctfrac%7B1%7D%7BH%7D%2C+1%5Cright%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="T_r=T_s &#92;cdot&#92;text{diag}&#92;left(&#92;tfrac{1}{W}, &#92;tfrac{1}{H}, 1&#92;right)" class="latex" /> . (Width &amp; Height are figured out from <img src="https://s0.wp.com/latex.php?latex=X_r&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=X_r&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=X_r&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="X_r" class="latex" />.)</div>
728</li>
729<li>
730<div align="justify">Do the final transformation with translated coordinates: <img src="https://s0.wp.com/latex.php?latex=X_%7B%5Ctext%7Bsrc%7D%7D%3DT_r+%5Ccdot+%28X_%7B%5Ctext%7Bdest%7D%7D-X_%7B%5Ctext%7Btrans%7D%7D%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=X_%7B%5Ctext%7Bsrc%7D%7D%3DT_r+%5Ccdot+%28X_%7B%5Ctext%7Bdest%7D%7D-X_%7B%5Ctext%7Btrans%7D%7D%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=X_%7B%5Ctext%7Bsrc%7D%7D%3DT_r+%5Ccdot+%28X_%7B%5Ctext%7Bdest%7D%7D-X_%7B%5Ctext%7Btrans%7D%7D%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="X_{&#92;text{src}}=T_r &#92;cdot (X_{&#92;text{dest}}-X_{&#92;text{trans}})" class="latex" /> . (I find that incorporating the translations in the matrix is not that straightforward. Maybe it can be done, but translating the coordinates and then transforming them is simple enough. Also remember: “GET the Pixels”.)</div>
731</li>
732<li>
733<div align="justify">Crop the relevant portion of the transformed image.</div>
734</li>
735</ol>
736<p align="justify">After implementing the above algorithm, we can end up with something like this (I think the second last point becomes clear too):</p>
737<p align="justify"><a href="https://jkmsmkj.fyi/wp-content/uploads/2013/07/image.png" target="_blank" rel="noopener"><img loading="lazy" style="background-image: none;float: none;padding-top: 0;padding-left: 0;margin: 0 auto;display: block;padding-right: 0;border: 0" title="Intermediary" src="https://jkmsmkj.fyi/wp-content/uploads/2013/07/image_thumb.png?w=500&#038;h=312" alt="Intermediary" width="500" height="312" border="0" /></a></p>
738<p align="justify">Or to put it more bluntly, this:</p>
739<p align="justify"><img loading="lazy" style="background-image: none;float: none;padding-top: 0;padding-left: 0;margin: 0 auto;display: block;padding-right: 0;border: 0" title="Square" src="https://jkmsmkj.fyi/wp-content/uploads/2013/07/wp66s.jpg?w=420&#038;h=420" alt="Square" width="420" height="420" border="0" /></p>
740<p align="justify">If you’re starting to think, I thought of all this… You’re giving me too much credit. Here’s the paper from where I learnt about this solution / algorithm (though, I think the solution given there may have some typos. I say this because that solution didn’t work ‘out of the box’):</p>
741<p align="justify"><a href="http://graphics.cs.cmu.edu/courses/15-463/2008_fall/Papers/proj.pdf">Source</a></p>
742
743			
744					</div><!-- .entry-content -->
745	</div>
746</article><!-- #post-## -->
747
748			
749				
750<article id="post-72" class="post-72 post type-post status-publish format-standard hentry category-physics category-thermodynamics category-timepass">
751			<div class="entry-wrapper">
752		<header class="entry-header">
753			<h1 class="entry-title"><a href="https://jkmsmkj.fyi/2011/05/14/a-difficult-problem/" rel="bookmark">A Difficult Problem?</a></h1>
754						<div class="entry-meta">
755				<span class="posted-on">On <a href="https://jkmsmkj.fyi/2011/05/14/a-difficult-problem/" rel="bookmark"><time class="entry-date published" datetime="2011-05-14T16:30:57+05:30">May 14, 2011</time><time class="updated" datetime="2021-06-20T13:18:11+05:30">June 20, 2021</time></a></span><span class="byline"> By <span class="author vcard"><a class="url fn n" href="https://jkmsmkj.fyi/author/jkmsmkj/">JKMSMKJ</a></span></span><span class="cat-links">In <a href="https://jkmsmkj.fyi/category/physics/" rel="category tag">Physics</a>, <a href="https://jkmsmkj.fyi/category/physics/thermodynamics/" rel="category tag">Thermodynamics</a>, <a href="https://jkmsmkj.fyi/category/timepass/" rel="category tag">TimePass</a></span><span class="comments-link"><a href="https://jkmsmkj.fyi/2011/05/14/a-difficult-problem/#respond">Leave a comment</a></span>			</div><!-- .entry-meta -->
756					</header><!-- .entry-header -->
757
758		<div class="entry-content">
759			<p align="justify">Many people land on <a href="http://jkmsmkj.blogspot.com/2008/07/illusions-misconceptions.html" target="_blank" rel="noopener">this page</a> of my other blog in search of an answer for this problem:</p>
760<blockquote>
761<p align="justify">A circular lake 1.0 km in diameter is 10 m deep. Solar energy is incident on the lake at an average rate of 200 W/m². If the lake absorbs all this energy and does not exchange heat with its surroundings, how long will it take to warm from 10°C  to 20°C ?</p>
762</blockquote>
763<p align="justify">as seen in the following snapshot:</p>
764<p align="justify"><img loading="lazy" style="background-image:none;padding-left:0;padding-right:0;display:block;float:none;padding-top:0;border-width:0;margin:0 auto;" title="Problem" src="https://jkmsmkj.fyi/wp-content/uploads/2011/05/problem.jpg?w=480&#038;h=82" alt="Problem" width="480" height="82" border="0" /></p>
765<p align="justify">Now, I do offer a pathway to the solution there but I feel that not providing the final answer is tantamount to cheating on my part. So, this being a ‘Physics blog’, I provide the entire solution here (I don’t know or care what good does it do to those ‘searchers’!):</p>
766<p><img src="https://s0.wp.com/latex.php?latex=t%3D%5Cfrac%7BQ%7D%7BS.A%7D%3D%5Cfrac%7Bm.C.%5CDelta+T%7D%7BS.A%7D%3D%5Cfrac%7B%28%5Crho+.V%29C.%5CDelta+T%7D%7BS.A%7D%3D%5Cfrac%7B%5Crho%28A.h%29C.%5CDelta+T%7D%7BS.A%7D%3D%5Cfrac%7B%5Crho+.h.C.%5CDelta+T%7D%7BS%7D&#038;bg=ffffff&#038;fg=111111&#038;s=1&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=t%3D%5Cfrac%7BQ%7D%7BS.A%7D%3D%5Cfrac%7Bm.C.%5CDelta+T%7D%7BS.A%7D%3D%5Cfrac%7B%28%5Crho+.V%29C.%5CDelta+T%7D%7BS.A%7D%3D%5Cfrac%7B%5Crho%28A.h%29C.%5CDelta+T%7D%7BS.A%7D%3D%5Cfrac%7B%5Crho+.h.C.%5CDelta+T%7D%7BS%7D&#038;bg=ffffff&#038;fg=111111&#038;s=1&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=t%3D%5Cfrac%7BQ%7D%7BS.A%7D%3D%5Cfrac%7Bm.C.%5CDelta+T%7D%7BS.A%7D%3D%5Cfrac%7B%28%5Crho+.V%29C.%5CDelta+T%7D%7BS.A%7D%3D%5Cfrac%7B%5Crho%28A.h%29C.%5CDelta+T%7D%7BS.A%7D%3D%5Cfrac%7B%5Crho+.h.C.%5CDelta+T%7D%7BS%7D&#038;bg=ffffff&#038;fg=111111&#038;s=1&#038;c=20201002&#038;zoom=4.5 4x" alt="t=&#92;frac{Q}{S.A}=&#92;frac{m.C.&#92;Delta T}{S.A}=&#92;frac{(&#92;rho .V)C.&#92;Delta T}{S.A}=&#92;frac{&#92;rho(A.h)C.&#92;Delta T}{S.A}=&#92;frac{&#92;rho .h.C.&#92;Delta T}{S}" class="latex" /></p>
767<p><img src="https://s0.wp.com/latex.php?latex=%5CRightarrow+t%3D%5Cfrac%7B1000%5Ctimes+10%5Ctimes+4200%5Ctimes+%2820-10%29%7D%7B200%7D%3D+2.1%5Ctimes+10%5E6s%3D24d%7E7h%7E20m.&#038;bg=ffffff&#038;fg=111111&#038;s=1&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5CRightarrow+t%3D%5Cfrac%7B1000%5Ctimes+10%5Ctimes+4200%5Ctimes+%2820-10%29%7D%7B200%7D%3D+2.1%5Ctimes+10%5E6s%3D24d%7E7h%7E20m.&#038;bg=ffffff&#038;fg=111111&#038;s=1&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5CRightarrow+t%3D%5Cfrac%7B1000%5Ctimes+10%5Ctimes+4200%5Ctimes+%2820-10%29%7D%7B200%7D%3D+2.1%5Ctimes+10%5E6s%3D24d%7E7h%7E20m.&#038;bg=ffffff&#038;fg=111111&#038;s=1&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;Rightarrow t=&#92;frac{1000&#92;times 10&#92;times 4200&#92;times (20-10)}{200}= 2.1&#92;times 10^6s=24d~7h~20m." class="latex" /></p>
768
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776<article id="post-55" class="post-55 post type-post status-publish format-standard hentry category-linear-algebra category-mathematics category-physics category-timepass">
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778		<header class="entry-header">
779			<h1 class="entry-title"><a href="https://jkmsmkj.fyi/2010/06/19/3d-rotations/" rel="bookmark">3D Rotations</a></h1>
780						<div class="entry-meta">
781				<span class="posted-on">On <a href="https://jkmsmkj.fyi/2010/06/19/3d-rotations/" rel="bookmark"><time class="entry-date published" datetime="2010-06-19T15:46:42+05:30">June 19, 2010</time><time class="updated" datetime="2022-07-25T08:11:59+05:30">July 25, 2022</time></a></span><span class="byline"> By <span class="author vcard"><a class="url fn n" href="https://jkmsmkj.fyi/author/jkmsmkj/">JKMSMKJ</a></span></span><span class="cat-links">In <a href="https://jkmsmkj.fyi/category/mathematics/linear-algebra/" rel="category tag">Linear Algebra</a>, <a href="https://jkmsmkj.fyi/category/mathematics/" rel="category tag">Mathematics</a>, <a href="https://jkmsmkj.fyi/category/physics/" rel="category tag">Physics</a>, <a href="https://jkmsmkj.fyi/category/timepass/" rel="category tag">TimePass</a></span><span class="comments-link"><a href="https://jkmsmkj.fyi/2010/06/19/3d-rotations/#respond">Leave a comment</a></span>			</div><!-- .entry-meta -->
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783
784		<div class="entry-content">
785			<p align="justify">You should have come from <a href="http://jkmsmkj.blogspot.com/2010/06/how-many-triangles_19.html">here</a>!</p>
786<p align="justify">Here’s a quickie: What are the eigenvalues of a 2D rotation matrix?</p>
787<p align="justify">Here’s a problem: For a bunch of rotations performed one after another on a 3D object, find an equivalent single rotation which w
787ould give the same result.</p>
788<p align="justify">Here’s a solution: First of all, multiply all the rotation matrices (obviously!) such that <img src="https://s0.wp.com/latex.php?latex=R_%7Bs%7D%3DR_%7Bn%7DR_%7Bn-1%7D%5Ccdots+R_%7B2%7DR_%7B1%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=R_%7Bs%7D%3DR_%7Bn%7DR_%7Bn-1%7D%5Ccdots+R_%7B2%7DR_%7B1%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=R_%7Bs%7D%3DR_%7Bn%7DR_%7Bn-1%7D%5Ccdots+R_%7B2%7DR_%7B1%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="R_{s}=R_{n}R_{n-1}&#92;cdots R_{2}R_{1}" class="latex" />. A single rotation is characterized by an axis and an angle of rotation. Let’s get the angle first. Find the eigenvalues of <img src="https://s0.wp.com/latex.php?latex=R_%7Bs%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=R_%7Bs%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=R_%7Bs%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="R_{s}" class="latex" />. Two of them will be of the form: <img src="https://s0.wp.com/latex.php?latex=e%5E%7B%5Cdot%7B%5Ciota%7D%5Ctheta%7D%5C%2C%5C%26%5C%2Ce%5E%7B-%5Cdot%7B%5Ciota%7D%5Ctheta%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%5E%7B%5Cdot%7B%5Ciota%7D%5Ctheta%7D%5C%2C%5C%26%5C%2Ce%5E%7B-%5Cdot%7B%5Ciota%7D%5Ctheta%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=e%5E%7B%5Cdot%7B%5Ciota%7D%5Ctheta%7D%5C%2C%5C%26%5C%2Ce%5E%7B-%5Cdot%7B%5Ciota%7D%5Ctheta%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="e^{&#92;dot{&#92;iota}&#92;theta}&#92;,&#92;&amp;&#92;,e^{-&#92;dot{&#92;iota}&#92;theta}" class="latex" />. This <em>θ</em> is the required angle. Let’s get the axis now. So what would be the third eigenvalue? Remember <img src="https://s0.wp.com/latex.php?latex=R_%7Bs%7D+%5Cin+SO%283%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=R_%7Bs%7D+%5Cin+SO%283%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=R_%7Bs%7D+%5Cin+SO%283%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="R_{s} &#92;in SO(3)" class="latex" /> which means the third one is 1! Find the eigenvector corresponding to this eigenvalue which is the required axis!</p>
789<p align="justify">Here&#8217;s an example: Let’s do the problem mentioned in <em>Gravitation</em>.</p>
790<p align="center"><img src="https://s0.wp.com/latex.php?latex=R_%7B1%7D%3D%5Cbegin%7Bpmatrix%7D+0+%26+1+%26+0+%5C%5C+-1+%26+0+%26+0+%5C%5C+0+%26+0+%26+1+%5Cend%7Bpmatrix%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=R_%7B1%7D%3D%5Cbegin%7Bpmatrix%7D+0+%26+1+%26+0+%5C%5C+-1+%26+0+%26+0+%5C%5C+0+%26+0+%26+1+%5Cend%7Bpmatrix%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=R_%7B1%7D%3D%5Cbegin%7Bpmatrix%7D+0+%26+1+%26+0+%5C%5C+-1+%26+0+%26+0+%5C%5C+0+%26+0+%26+1+%5Cend%7Bpmatrix%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="R_{1}=&#92;begin{pmatrix} 0 &amp; 1 &amp; 0 &#92;&#92; -1 &amp; 0 &amp; 0 &#92;&#92; 0 &amp; 0 &amp; 1 &#92;end{pmatrix}" class="latex" />;</p>
791<p align="center"><img src="https://s0.wp.com/latex.php?latex=R_%7B2%7D%3D%5Cbegin%7Bpmatrix%7D+0+%26+0+%26+1+%5C%5C+0+%26+1+%26+0+%5C%5C+-1+%26+0+%26+0+%5Cend%7Bpmatrix%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=R_%7B2%7D%3D%5Cbegin%7Bpmatrix%7D+0+%26+0+%26+1+%5C%5C+0+%26+1+%26+0+%5C%5C+-1+%26+0+%26+0+%5Cend%7Bpmatrix%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=R_%7B2%7D%3D%5Cbegin%7Bpmatrix%7D+0+%26+0+%26+1+%5C%5C+0+%26+1+%26+0+%5C%5C+-1+%26+0+%26+0+%5Cend%7Bpmatrix%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="R_{2}=&#92;begin{pmatrix} 0 &amp; 0 &amp; 1 &#92;&#92; 0 &amp; 1 &amp; 0 &#92;&#92; -1 &amp; 0 &amp; 0 &#92;end{pmatrix}" class="latex" />.</p>
792<p align="center"><img src="https://s0.wp.com/latex.php?latex=%5CRightarrow+R_%7Bs%7D%3DR_%7B2%7DR_%7B1%7D%3D%5Cbegin%7Bpmatrix%7D+0+%26+0+%26+1+%5C%5C+-1+%26+0+%26+0+%
7925C%5C+0+%26+-1+%26+0+%5Cend%7Bpmatrix%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5CRightarrow+R_%7Bs%7D%3DR_%7B2%7DR_%7B1%7D%3D%5Cbegin%7Bpmatrix%7D+0+%26+0+%26+1+%5C%5C+-1+%26+0+%26+0+%5C%5C+0+%26+-1+%26+0+%5Cend%7Bpmatrix%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5CRightarrow+R_%7Bs%7D%3DR_%7B2%7DR_%7B1%7D%3D%5Cbegin%7Bpmatrix%7D+0+%26+0+%26+1+%5C%5C+-1+%26+0+%26+0+%5C%5C+0+%26+-1+%26+0+%5Cend%7Bpmatrix%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;Rightarrow R_{s}=R_{2}R_{1}=&#92;begin{pmatrix} 0 &amp; 0 &amp; 1 &#92;&#92; -1 &amp; 0 &amp; 0 &#92;&#92; 0 &amp; -1 &amp; 0 &#92;end{pmatrix}" class="latex" />.</p>
793<p align="justify">Relevant eigenvalues of <img src="https://s0.wp.com/latex.php?latex=R_%7Bs%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=R_%7Bs%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=R_%7Bs%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="R_{s}" class="latex" /> are <img src="https://s0.wp.com/latex.php?latex=-%5Cfrac%7B1%7D%7B2%7D%5Cpm%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Cdot%7B%5Ciota%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=-%5Cfrac%7B1%7D%7B2%7D%5Cpm%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Cdot%7B%5Ciota%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=-%5Cfrac%7B1%7D%7B2%7D%5Cpm%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Cdot%7B%5Ciota%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="-&#92;frac{1}{2}&#92;pm&#92;frac{&#92;sqrt{3}}{2}&#92;dot{&#92;iota}" class="latex" /> which give us the angle: <img src="https://s0.wp.com/latex.php?latex=%5Cmathrm%7Btan%7D%5E%7B-1%7D%5Cleft%28-%5Csqrt%7B3%7D%5Cright%29%3D120%5E%7B%5Ccirc%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathrm%7Btan%7D%5E%7B-1%7D%5Cleft%28-%5Csqrt%7B3%7D%5Cright%29%3D120%5E%7B%5Ccirc%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathrm%7Btan%7D%5E%7B-1%7D%5Cleft%28-%5Csqrt%7B3%7D%5Cright%29%3D120%5E%7B%5Ccirc%7D&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;mathrm{tan}^{-1}&#92;left(-&#92;sqrt{3}&#92;right)=120^{&#92;circ}" class="latex" />. The eigenvector corresponding to 1 is <img src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D%281%2C-1%2C1%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D%281%2C-1%2C1%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D%281%2C-1%2C1%29&#038;bg=ffffff&#038;fg=111111&#038;s=0&#038;c=20201002&#038;zoom=4.5 4x" alt="&#92;frac{1}{&#92;sqrt{3}}(1,-1,1)" class="latex" /> so the axis is one of the diagonals! This solution agrees with the one given in the book.</p>
794
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1033window.addEventListener( "DOMContentLoaded", function () {
1034				var link = document.createElement( "link" );
1035				link.href = "\/wp-content\/mu-plugins\/jetpack-plugin\/sun\/jetpack_vendor\/automattic\/jetpack-newsletter\/src\/..\/build\/action-bar.css?ver=35dd33285eb46dc308dd";
1036				link.rel = "stylesheet";
1037				document.head.appendChild( link );
1038
1039				var script = document.createElement( "script" );
1040				script.src = "\/wp-content\/mu-plugins\/jetpack-plugin\/sun\/jetpack_vendor\/automattic\/jetpack-newsletter\/src\/..\/build\/action-bar.js?ver=35dd33285eb46dc308dd";
1041				document.body.appendChild( script );
1042			} );
1043</script>
1043
1044<link rel="prefetch" href="/wp-content/mu-plugins/jetpack-plugin/sun/_inc/blocks/swiper.js" as="script" />
1045		<div id="jp-carousel-loading-overlay" style="display: none;">
1046			<div id="jp-carousel-loading-wrapper">
1047				<span id="jp-carousel-library-loading"><svg class="jetpack-spinner" width="40" height="40" viewBox="0 0 100 100" xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false"><circle cx="50" cy="50" r="46" fill="none" stroke="#ddd" stroke-width="8"/><path d="M 50 4 A 46 46 0 0 1 96 50" fill="none" stroke="currentColor" stroke-width="8" stroke-linecap="round"><animateTransform attributeName="transform" type="rotate" dur="1.4s" from="0 50 50" to="360 50 50" repeatCount="indefinite"/></path></svg></span>
1048			</div>
1049		</div>
1050		<div class="jp-carousel-overlay" style="display: none;">
1051
1052		<div class="jp-carousel-container">
1053			<!-- The Carousel Swiper -->
1054			<div
1055				class="jp-carousel-wrap swiper jp-carousel-swiper-container jp-carousel-transitions"
1056				itemscope
1057				itemtype="https://schema.org/ImageGallery">
1058				<div class="jp-carousel swiper-wrapper"></div>
1059				<div class="jp-swiper-button-prev swiper-button-prev">
1060					<svg width="25" height="24" viewBox="0 0 25 24" fill="none" xmlns="http://www.w3.org/2000/svg">
1061						<mask id="maskPrev" mask-type="alpha" maskUnits="userSpaceOnUse" x="8" y="6" width="9" height="12">
1062							<path d="M16.2072 16.59L11.6496 12L16.2072 7.41L14.8041 6L8.8335 12L14.8041 18L16.2072 16.59Z" fill="white"/>
1063						</mask>
1064						<g mask="url(#maskPrev)">
1065							<rect x="0.579102" width="23.8823" height="24" fill="#FFFFFF"/>
1066						</g>
1067					</svg>
1068				</div>
1069				<div class="jp-swiper-button-next swiper-button-next">
1070					<svg width="25" height="24" viewBox="0 0 25 24" fill="none" xmlns="http://www.w3.org/2000/svg">
1071						<mask id="maskNext" mask-type="alpha" maskUnits="userSpaceOnUse" x="8" y="6" width="8" height="12">
1072							<path d="M8.59814 16.59L13.1557 12L8.59814 7.41L10.0012 6L15.9718 12L10.0012 18L8.59814 16.59Z" fill="white"/>
1073						</mask>
1074						<g mask="url(#maskNext)">
1075							<rect x="0.34375" width="23.8822" height="24" fill="#FFFFFF"/>
1076						</g>
1077					</svg>
1078				</div>
1079			</div>
1080			<!-- The main close buton -->
1081			<div class="jp-carousel-close-hint">
1082				<svg width="25" height="24" viewBox="0 0 25 24" fill="none" xmlns="http://www.w3.org/2000/svg">
1083					<mask id="maskClose" mask-type="alpha" maskUnits="userSpaceOnUse" x="5" y="5" width="15" height="14">
1084						<path d="M19.3166 6.41L17.9135 5L12.3509 10.59L6.78834 5L5.38525 6.41L10.9478 12L5.38525 17.59L6.78834 19L12.3509 13.41L17.9135 19L19.3166 17.59L13.754 12L19.3166 6.41Z" fill="white"/>
1085					</mask>
1086					<g mask="url(#maskClose)">
1087						<rect x="0.409668" width="23.8823" height="24" fill="#FFFFFF"/>
1088					</g>
1089				</svg>
1090			</div>
1091			<!-- Image info, comments and meta -->
1092			<div class="jp-carousel-info">
1093				<div class="jp-carousel-info-footer">
1094					<div class="jp-carousel-pagination-container">
1095						<div class="jp-swiper-pagination swiper-pagination"></div>
1096						<div class="jp-carousel-pagination"></div>
1097					</div>
1098					<div class="jp-carousel-photo-title-container">
1099						<div class="jp-carousel-photo-caption"></div>
1100					</div>
1101					<div class="jp-carousel-photo-icons-container">
1102						<a href="#" class="jp-carousel-icon-btn jp-carousel-icon-info" aria-label="Toggle photo metadata visibility">
1103							<span class="jp-carousel-icon">
1104								<svg width="25" height="24" viewBox="0 0 25 24" fill="none" xmlns="http://www.w3.org/2000/svg">
1105									<mask id="maskInfo" mask-type="alpha" maskUnits="userSpaceOnUse" x="2" y="2" width="21" height="20">
1106										<path fill-rule="evenodd" clip-rule="evenodd" d="M12.7537 2C7.26076 2 2.80273 6.48 2.80273 12C2.80273 17.52 7.26076 22 12.7537 22C18.2466 22 22.7046 17.52 22.7046 12C22.7046 6.48 18.2466 2 12.7537 2ZM11.7586 7V9H13.7488V7H11.7586ZM11.7586 11V17H13.7488V11H11.7586ZM4.79292 12C4.79292 16.41 8.36531 20 12.7537 20C17.142 20 20.7144 16.41 20.7144 12C20.7144 7.59 17.142 4 12.7537 4C8.36531 4 4.79292 7.59 4.79292 12Z" fill="white"/>
1107									</mask>
1108									<g mask="url(#maskInfo)">
1109										<rect x="0.8125" width="23.8823" height="24" fill="#FFFFFF"/>
1110									</g>
1111								</svg>
1112							</span>
1113						</a>
1114												<a href="#" class="jp-carousel-icon-btn jp-carousel-icon-comments" aria-label="Toggle photo comments visibility">
1115							<span class="jp-carousel-icon">
1116								<svg width="25" height="24" viewBox="0 0 25 24" fill="none" xmlns="http://www.w3.org/2000/svg">
1117									<mask id="maskComments" mask-type="alpha" maskUnits="userSpaceOnUse" x="2" y="2" width="21" height="20">
1118										<path fill-rule="evenodd" clip-rule="evenodd" d="M4.3271 2H20.2486C21.3432 2 22.2388 2.9 22.2388 4V16C22.2388 17.1 21.3432 18 20.2486 18H6.31729L2.33691 22V4C2.33691 2.9 3.2325 2 4.3271 2ZM6.31729 16H20.2486V4H4.3271V18L6.31729 16Z" fill="white"/>
1119									</mask>
1120									<g mask="url(#maskComments)">
1121										<rect x="0.34668" width="23.8823" height="24" fill="#FFFFFF"/>
1122									</g>
1123								</svg>
1124
1125								<span class="jp-carousel-has-comments-indicator" aria-label="This image has comments."></span>
1126							</span>
1127						</a>
1128											</div>
1129				</div>
1130				<div class="jp-carousel-info-extra">
1131					<div class="jp-carousel-info-content-wrapper">
1132						<div class="jp-carousel-photo-title-container">
1133							<div class="jp-carousel-photo-title"></div>
1134						</div>
1135						<div class="jp-carousel-comments-wrapper">
1136															<div id="jp-carousel-comments-loading">
1137									<span>Loading Comments...</span>
1138								</div>
1139								<div class="jp-carousel-comments"></div>
1140								<div id="jp-carousel-comment-form-container">
1141									<span id="jp-carousel-comment-form-spinner"><svg class="jetpack-spinner" width="20" height="20" viewBox="0 0 100 100" xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false"><circle cx="50" cy="50" r="46" fill="none" stroke="#ddd" stroke-width="8"/><path d="M 50 4 A 46 46 0 0 1 96 50" fill="none" stroke="currentColor" stroke-width="8" stroke-linecap="round"><animateTransform attributeName="transform" type="rotate" dur="1.4s" from="0 50 50" to="360 50 50" repeatCount="indefinite"/></path></svg></span>
1142									<div id="jp-carousel-comment-post-results"></div>
1143																														<div id="jp-carousel-comment-form-commenting-as">
1144												<p id="jp-carousel-commenting-as">
1145													You must be <a href="#" class="jp-carousel-comment-login">logged in</a> to post a comment.												</p>
1146											</div>
1147																											</div>
1148													</div>
1149						<div class="jp-carousel-image-meta">
1150							<div class="jp-carousel-title-and-caption">
1151								<div class="jp-carousel-photo-info">
1152									<div class="jp-carousel-caption" itemprop="caption description"></div>
1153								</div>
1154
1155								<div class="jp-carousel-photo-description"></div>
1156							</div>
1157							<a class="jp-carousel-image-download" href="#" aria-label="Download image" target="_blank" style="display: none;">
1158								<svg width="25" height="24" viewBox="0 0 25 24" fill="none" xmlns="http://www.w3.org/2000/svg">
1159									<mask id="mask0" mask-type="alpha" maskUnits="userSpaceOnUse" x="3" y="3" width="19" height="18">
1160										<path fill-rule="evenodd" clip-rule="evenodd" d="M5.84615 5V19H19.7775V12H21.7677V19C21.7677 20.1 20.8721 21 19.7775 21H5.84615C4.74159 21 3.85596 20.1 3.85596 19V5C3.85596 3.9 4.74159 3 5.84615 3H12.8118V5H5.84615ZM14.802 5V3H21.7677V10H19.7775V6.41L9.99569 16.24L8.59261 14.83L18.3744 5H14.802Z" fill="white"/>
1161									</mask>
1162									<g mask="url(#mask0)">
1163										<rect x="0.870605" width="23.8823" height="24" fill="#FFFFFF"/>
1164									</g>
1165								</svg>
1166								<span class="jp-carousel-download-text"></span>
1167							</a>
1168							<div class="jp-carousel-image-map" style="display: none;"></div>
1169						</div>
1170					</div>
1171				</div>
1172			</div>
1173		</div>
1174
1175		</div>
1176		
1177	
1177<script type="text/javascript">
1178		(function () {
1179			var wpcom_reblog = {
1180				source: 'toolbar',
1181
1182				toggle_reblog_box_flair: function (obj_id, post_id) {
1183
1184					// Go to site selector. This will redirect to their blog if they only have one.
1185					const postEndpoint = `https://wordpress.com/post`;
1186
1187					// Ideally we would use the permalink here, but fortunately this will be replaced with the 
1188					// post permalink in the editor.
1189					const originalURL = `${ document.location.href }?page_id=${ post_id }`; 
1190					
1191					const url =
1192						postEndpoint +
1193						'?url=' +
1194						encodeURIComponent( originalURL ) +
1195						'&is_post_share=true' +
1196						'&v=5';
1197
1198					const redirect = function () {
1199						if (
1200							! window.open( url, '_blank' )
1201						) {
1202							location.href = url;
1203						}
1204					};
1205
1206					if ( /Firefox/.test( navigator.userAgent ) ) {
1207						setTimeout( redirect, 0 );
1208					} else {
1209						redirect();
1210					}
1211				},
1212			};
1213
1214			window.wpcom_reblog = wpcom_reblog;
1215		})();
1216	</script>
vendor: 1 bytes, line 1216
1216
1217<script id="jetpack-carousel-js-extra">
1218var jetpackSwiperLibraryPath = {"url":"/wp-content/mu-plugins/jetpack-plugin/sun/_inc/blocks/swiper.js"};
1219var jetpackCarouselStrings = {"widths":[370,700,1000,1200,1400,2000],"is_logged_in":"","lang":"en","ajaxurl":"https://jkmsmkj.fyi/wp-admin/admin-ajax.php","nonce":"61e17ecf2e","display_exif":"1","display_comments":"1","single_image_gallery":"1","single_image_gallery_media_file":"","background_color":"black","comment":"Comment","post_comment":"Post Comment","write_comment":"Write a Comment...","loading_comments":"Loading Comments...","image_label":"Open image in full-screen.","download_original":"View full size \u003Cspan class=\"photo-size\"\u003E{0}\u003Cspan class=\"photo-size-times\"\u003E\u00d7\u003C/span\u003E{1}\u003C/span\u003E","no_comment_text":"Please be sure to submit some text with your comment.","no_comment_email":"Please provide an email address to comment.","no_comment_author":"Please provide your name to comment.","comment_post_error":"Sorry, but there was an error posting your comment. Please try again later.","comment_approved":"Your comment was approved.","comment_unapproved":"Your comment is in moderation.","camera":"Camera","aperture":"Aperture","shutter_speed":"Shutter Speed","focal_length":"Focal Length","copyright":"Copyright","comment_registration":"1","require_name_email":"1","login_url":"https://jkmsmkj.wordpress.com/wp-login.php?redirect_to=https%3A%2F%2Fjkmsmkj.fyi%2F2010%2F06%2F19%2F3d-rotations%2F","blog_id":"9059097","meta_data":["camera","aperture","shutter_speed","focal_length","copyright"],"stats_query_args":"blog=9059097&v=wpcom&tz=5&user_id=0&arch_home=1&subd=jkmsmkj","is_public":"1"};
1220//# sourceURL=jetpack-carousel-js-extra
1221</script>
vendor: 1 bytes, line 1221
1221
1222<script crossorigin='anonymous' type='text/javascript'  src='/_static/??-eJyNj8EKwjAQRH/IdCsWigfxUyRdk7rpZhO6SdW/99KKeOrxDW8GBp7ZYJLipECsJnMdSRSCK9nitDJoFbiRIAyV+A4knoSKM4pzYt743USSJugBfkbLw0WnkOsAnBaHs/UFgoLYhUZbKO1ujJwGy3ttnSgbJpmMT1jVeHr9V3f/RTunqo6/2hash6/xcuzPbdu3p64LH1oGfpI='></script>
vendor: 1 bytes, line 1222
1222
1223<script data-wp-strategy="defer" defer id="tiled-gallery-js" src="/wp-content/mu-plugins/jetpack-plugin/sun/_inc/build/tiled-gallery/tiled-gallery/tiled-gallery.min.js?m=1755011788i&#038;ver=16.4-a.0"></script>
vendor: 1 bytes, line 1223
1223
1224<script crossorigin='anonymous' type='text/javascript'  src='/wp-content/mu-plugins/carousel-wpcom/carousel-wpcom.js?m=1761899756i'></script>
vendor: 1 bytes, line 1224
1224
1225<script id="wp-emoji-settings" type="application/json">
1226{"baseUrl":"https://s0.wp.com/wp-content/mu-plugins/wpcom-smileys/twemoji/2/72x72/","ext":".png","svgUrl":"https://s0.wp.com/wp-content/mu-plugins/wpcom-smileys/twemoji/2/svg/","svgExt":".svg","source":{"concatemoji":"/wp-includes/js/wp-emoji-release.min.js?m=1783090810i&ver=7.2-alpha-63973"}}
1227</script>
vendor: 1 bytes, line 1227
1227
1228<script type="module">
1229/*! This file is auto-generated */
1230var e="script#wp-emoji-settings",t=document.querySelector(e);if(!(t instanceof HTMLScriptElement))throw new Error("Element missing: "+e);const r=JSON.parse(t.text),s=(window._wpemojiSettings=r,"wpEmojiSettingsSupports"),o=["flag","emoji"];function i(e){try{var t={supportTests:e,timestamp:(new Date).valueOf()};sessionStorage.setItem(s,JSON.stringify(t))}catch(e){}}function c(e,t,n){e.clearRect(0,0,e.canvas.width,e.canvas.height),e.fillText(t,0,0);t=new Uint32Array(e.getImageData(0,0,e.canvas.width,e.canvas.height).data);e.clearRect(0,0,e.canvas.width,e.canvas.height),e.fillText(n,0,0);const r=new Uint32Array(e.getImageData(0,0,e.canvas.width,e.canvas.height).data);return t.every((e,t)=>e===r[t])}function p(e,t){e.clearRect(0,0,e.canvas.width,e.canvas.height),e.fillText(t,0,0);var n=e.getImageData(16,16,1,1);for(let e=0;e<n.data.length;e++)if(0!==n.data[e])return!1;return!0}function u(e,t,n,r){switch(t){case"flag":return n(e,"\ud83c\udff3\ufe0f\u200d\u26a7\ufe0f","\ud83c\udff3\ufe0f\u200b\u26a7\ufe0f")?!1:!n(e,"\ud83c\udde8\ud83c\uddf6","\ud83c\udde8\u200b\ud83c\uddf6")&&!n(e,"\ud83c\udff4\udb40\udc67\udb40\udc62\udb40\udc65\udb40\udc6e\udb40\udc67\udb40\udc7f","\ud83c\udff4\u200b\udb40\udc67\u200b\udb40\udc62\u200b\udb40\udc65\u200b\udb40\udc6e\u200b\udb40\udc67\u200b\udb40\udc7f");case"emoji":return!r(e,"\ud83e\udec8")}return!1}function f(e,t,n,r){let a;const s=(a="undefined"!=typeof WorkerGlobalScope&&self instanceof WorkerGlobalScope?new OffscreenCanvas(300,150):document.createElement("canvas")).getContext("2d",{willReadFrequently:!0}),o=(s.textBaseline="top",s.font="600 32px Arial",{});return e.forEach(e=>{o[e]=t(s,e,n,r)}),o}function a(e){var t=document.createElement("script");t.src=e,t.defer=!0,document.head.appendChild(t)}r.supports={everything:!0,everythingExceptFlag:!0},new Promise(t=>{let n=function(){try{var e=JSON.parse(sessionStorage.getItem(s));if("object"==typeof e&&"number"==typeof e.timestamp&&(new Date).valueOf()<e.timestamp+604800&&"object"==typeof e.supportTests)return e.supportTests}catch(e){}return null}();if(!n){if("undefined"!=typeof Worker&&"undefined"!=typeof OffscreenCanvas&&"undefined"!=typeof URL&&URL.createObjectURL&&"undefined"!=typeof Blob)try{var e="postMessage("+f.toString()+"("+[JSON.stringify(o),u.toString(),c.toString(),p.toString()].join(",")+"));",r=new Blob([e],{type:"text/javascript"});const a=new Worker(URL.createObjectURL(r),{name:"wpTestEmojiSupports"});return void(a.onmessage=e=>{i(n=e.data),a.terminate(),t(n)})}catch(e){}i(n=f(o,u,c,p))}t(n)}).then(e=>{for(const n in e)r.supports[n]=e[n],r.supports.everything=r.supports.everything&&r.supports[n],"flag"!==n&&(r.supports.everythingExceptFlag=r.supports.everythingExceptFlag&&r.supports[n]);var t;r.supports.everythingExceptFlag=r.supports.everythingExceptFlag&&!r.supports.flag,r.supports.everything||((t=r.source||{}).concatemoji?a(t.concatemoji):t.wpemoji&&t.twemoji&&(a(t.twemoji),a(t.wpemoji)))});
1231//# sourceURL=/wp-includes/js/wp-emoji-loader.min.js
1232</script>
1232
1233		
1233<script type="text/javascript">
1234			(function() {
1235				var extend = function(out) {
1236					out = out || {};
1237
1238					for (var i = 1; i < arguments.length; i++) {
1239						if (!arguments[i])
1240						continue;
1241
1242						for (var key in arguments[i]) {
1243						if (arguments[i].hasOwnProperty(key))
1244							out[key] = arguments[i][key];
1245						}
1246					}
1247
1248					return out;
1249				};
1250				extend( window.infiniteScroll.settings.scripts, ["jetpack-mu-wpcom-settings","rlt-proxy","wpcom-actionbar-placeholder","grofiles-cards","wpgroho","the-neverending-homepage","lovecraft-navigation","lovecraft-global","lovecraft-skip-link-focus-fix","jetpack-carousel","tiled-gallery","carousel-wpcom"] );
1251				extend( window.infiniteScroll.settings.styles, ["the-neverending-homepage","coblocks-frontend","wpcom-core-compat-playlist-styles","wpcom-text-widget-styles","wpcom-bbpress2-staff-css","lovecraft-style","genericons","lovecraft-wpcom","reblogging","geo-location-flair","a8c-global-print","h4-global","wp-img-auto-sizes-contain","wp-emoji-styles","wp-block-library","classic-theme-styles","wp-block-styles-placeholder","coblocks-inline-format","wp-global-styles-placeholder","lovecraft-fonts","jetpack-global-styles-frontend-style","wp-block-paragraph","global-styles","jetpack-top-posts-widget","jetpack-swiper-library","jetpack-carousel","tiled-gallery","core-block-supports-duotone"] );
1252			})();
1253		</script>
1253
1254				<span id="infinite-aria" aria-live="polite"></span>
1255		
1255<script src="//stats.wp.com/w.js?69" defer></script>
1255 
1255<script type="text/javascript">
1256_tkq = window._tkq || [];
1257_stq = window._stq || [];
1258_tkq.push(['storeContext', {'blog_id':'9059097','blog_tz':'5','user_lang':'en','blog_lang':'en','user_id':'0'}]);
1259		// Prevent sending pageview tracking from WP-Admin pages.
1260		_stq.push(['view', {'blog':'9059097','v':'wpcom','tz':'5','user_id':'0','arch_home':'1','subd':'jkmsmkj'}]);
1261		_stq.push(['extra', {'crypt':'UE5tW3cvZGRkOERlal9Yd291RzZXcmdOZVVpJVVkJTNPRktUU0lrQW9PVUpPZzRnc2JsLnZITS0taTNaV3FRWjFXVTNtVi42UWRvb19jc0hOLkRwVmtofnZsQVZ2OWUlVzBYN1o/b3dmUnJUSiZneDkxZ1dxSUYzNGNhMi8vfGt5UF0sQUF5SUFSMDA1NyZocT8/clA3RGhbVi83akpDbDJHSmYlbTB+W1dFNFNCeS4sMDRVZFN3Ll02Nj8/Vnx0REM0MjRrRUpQJWVkdlNsZXIuK3I='}]);
1262_stq.push([ 'clickTrackerInit', '9059097', '0' ]);
1263</script>
1263
1264<noscript><img src="https://pixel.wp.com/b.gif?v=noscript" style="height:1px;width:1px;overflow:hidden;position:absolute;bottom:1px;" alt="" /></noscript>
1265<meta id="bilmur" property="bilmur:data" content="" data-provider="wordpress.com" data-service="simple" data-site-tz="Asia/Kolkata" data-custom-props="{&quot;logged_in&quot;:&quot;0&quot;,&quot;wptheme&quot;:&quot;pub\/lovecraft&quot;,&quot;wptheme_is_block&quot;:&quot;0&quot;}"   >
1266		
1266<script defer src="/wp-content/js/bilmur.min.js?i=19&amp;m=202641"></script>
1266 	
1267</body>
1268</html>

Line numbers count LF bytes from the start of the resource, as the search results do. Vendor segments are library code the classifier recognised; they are stored but not indexed. Bytes are shown as Latin1 characters, one per byte.