16| Gábor Domokos — The Gömböc, a shape at the limit of possibility
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The Gomboc is a curious shape. So curious many mathematicians thought it could not exist. And even to the untrained eye, it looks alien: neither the product of human or natural processes. This week Gábor Domokos relates his decade-long quest to prove the existence of a (convex, homogenous) shape with only two balance points.  The Gömböc is not just a mathematical curio, its discovery led to a theory of how "things fall apart", of the processes of abrasion that — whether on Earth, mars, or deep space — ineluctably reduce the number of balance points of objects.  The Gomboc is the shape all pebbles want to be, but can never reach. Show notes at multiverses.xyz [https://www.multiverses.xyz/podcast/mv16-gomboc-a-shape-at-the-limits-of-possibility-gabor-domokos/] (00:00) Intro (2:40) Start of conversation — what is a Gomboc? (4:30) The Gomboc is the "ultimate shape" it has only two balance points (5:30) The four vertex theorem: why a 2D shape must have 4 balance points (6:30) (almost) nobody thought a Gomboc existed (8:30) Vladimir Ilych Arnold's conjecture (9:00) Hamburg 1995, the beginning of a quest (10:30) "Mathematics is a part of physics where experiments are cheap" (11:50) A hungry scholar sits next to a mathematical superstar (13:00) Ten years of searching (15:00) Domokos and Varkonyi's gift for Arnold (15:30) Arnold's response: "good, but now do something serious" (16:50) We cannot easily speak about shapes. (18:00) A system for naming shapes (21:00) "The evolution of shapes is imprinted in these numbers" (21:50) Pebbles evolve towards the Gomboc, but never get there (24:50) How to find the balance points of shapes by hand (30:00) Physical intuition and empirical exploration can inform mathematics (30:30) A beach holiday (and a marital bifurcation point) (34:00) "No this was not fun, it was a markov process" (36:40) Working with NASA to understand the age of martian pebbles (38:20) An asteroid, or a spaceship? (43:00) The mechanisms of abrasion (45:50) The isoperimetric ratio — does not evolve monotonically … (47:50) … But the drift to less balance points is monotonic (49:00) The process of abrasion is a process of simplifying (50:00)  We can name the shape of Oumuamua because it is so simple (51:00)  Relationship between Gomboc and (one way of thinking about) entropy (55:00)  Abrasion and the heat equation — curvature is "like" heat and gets smoothed out (58:00)  The soap bar model — why pointy bits become smooth (1:00:00) Richard Hamilton, the Poincaré conjecture and pebbles (1:04:00) The connection between the Ricci flow and pebble evolution (1:09:00) Turning the lights on in a darkened labyrinth (1:12:00) The importance of geometric objects in physics (string theory) (1:13:30) Another way of naming natural shapes: the average number of faces and vertices (1:15:00) "Earth is made of cubes" — it turns out Plato was right (1:16:30) Could Plato's claim have been empirically inspired? (1:17:50) "Everything happens between 20 and 6" (1:18:30) The Cube and the Gomboc are the bookends of natural shapes (1:19:30) The Obelisk in 2001 — an unnatural, but almost natural shape (1:22:00) Poincaré on dreaming: genius taps the subconscious
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