Tiled With Pentagons

(Epistemic status: I mentored the SPAR stream that did all of this, pitched the project in the first place, and contributed project management and much of the understanding of the theory, which makes me exactly as impartial as that sentence implies; audit accordingly. This is meant as a companion post and reference explainer for the stream's talk Friday morning as delivered by FL; read it before or after, or refer to it during - whichever suits you best. I've deliberately left out much of the formal math here, saving that for the talk slides and more formal posts and papers that give a more in-depth treatment of the subject; there's pointers at the end for anyone who wants to go digging. For my entire SPAR stream, all fourteen of my first-coauthors: MA, FA, SMA, AB ,  SSMJ, AL, CL, AL (a different one), FL (obviously), RR, PR (a different one), ER (who is at the conference!), NT, and  MPdV . I am unspeakably proud of all of you, and I could never have made Project Newcomb f...

(Epistemic status: Philosophy of observation and memories of physics. The math is real and the metaphysics are probably reasonably solid too. As usual, morally correct, skipping over some of the finer details. Dedicated with bittersweet fondness for the Integrated Sciences Curriculum program. Even still, everything is a spring.)   Plucked guitar strings and the quantized modes of trumpet blasts. Carbon dioxide's vibrational modes. Cyclical stock market trends, despite the efficient market hypothesis. The length of a day over the course of a year. These and many other phenomensta should make you ask one simple question: why do so many physical systems like to vibrate? Why are so many such systems well-modelled by reducing them to simple or damped harmonic oscillators? That is: Why is everything a spring? After years of chewing it over in quiet moments, I've eventually come to a conclusion, and I'm not sure whether it's profound, trivial, both, or some secret fourth thing...

(This will make much more sense if you read "11. Why the First "High Dimension" is Six or Maybe Five" first. As before, epistemic status: morally correct, in the mathematician’s sense; here to give flavor and intuition without too much rigor. Frankly, I'm going to move even faster and more cursorily than last time. With thanks to IL, and to everyone I've tried to shill 4D Golf to.) After talking with IL a bit more and doing some thinking for myself, I realized with creeping delight that the long heuristic argument that I gave for why we should consider the first qualitatively high dimension to be six or maybe five is not remotely the only one. In fact, a shocking number of strong heuristic arguments all converge on the same figure: very specifically, "six or maybe five". That is: five is marginal, and six is definite. To give a very quick recap of the heuristic argument from the original post, the idea is that we can operationalize what it means fo...

(Epistemic status: A creased, stained map to what were once my favorite hunting grounds. Accessible to anyone who can support substantial abstraction; prior math knowledge is not necessary. In particular, ignorance of calculus is not an obstruction here, but total ignorance of geometry or like, arithmetic or logic, will be. Extremely dense and probably won’t get you there, but at least you’ll ask better questions. Partially dedicated to DG, JM, and PR.)   The path to my old hunting grounds (as pictured above) is long and twisty and winds through a lot of necessary math along the way. You'll be best served by carefully going through the parentheticals and answering the questions I ask, so as to keep track of the blazes and to keep your footing. You'll still get something out of this if you move more quickly, but you might end up lost further down the line. You might find the pace a little slow if you already know the territory - test yourself by answering those sa...

(With thanks to TT and IL for prompting me to think about this. Happy Pi Day.) Picture this: you're at a convention. You've got multiple badges to juggle - an attendee badge, two or three beautiful art badges to display, and maybe even an access or volunteer badge - so you've put them all on the same lanyard. In the morning, you carefully arrange them all facing the right way, put the lanyard on, and go out to the con. By noon, at least one badge will already have flipped backwards. By evening, another two have flipped backwards, and you could swear that you briefly saw one of them upside down. Over the course of the con, you fight a losing battle: you notice a backwards badge and flip it and it flips back on its own; you take off your lanyard altogether and carefully reorient all the badges correctly, and within an hour, half the badges are backwards and chaos reigns once more. It's long been a frustration of many who wear badges - the rectangular plastic kind that mig...