James

The Gömböc is a peculiar shape. It’s too balanced and poised to appear natural but too clunky to suggest it is the fruit of human design. It seems alien or supernatural. This is appropriate, for the Gömböc is a shape that many thought impossible.

View the interactive version.

The Gömböc is the first known homogenous shape with the properties of having just one stable and one unstable balance point. Its weight is evenly distributed yet, however you put it, it will tumble and roll until resting always in the same position. Unless you pose it most delicately on its unstable point.

For convex, homogenous shapes the problem of finding balance points collapses to that of finding local maximal and minimal points from the centre of mass to the surface. For each minimum, there is a stable balance point, for each maximum an unstable one, and for those shapes that have a maximum and a minimum at the same point (appropriately called saddle points, for they are at the top and bottom of “hills”, like a saddle) — it is stable along one direction and unstable along others.

For many years, many mathematicians believed such a shape to be impossible, fruitlessly trying to produce a theorem to prove their intuition. Their hypothesis was informed by two pieces of evidence:

• For two-dimensional shapes, it can be shown that 4 balance points are the minimal possible. This is proved by the four-vertex theorem: any closed curve other than the circle must have four vertices — four points that are either local maxima or minima.
• No shape had been seen in nature.

Evidence can be misleading.

This week Gábor Domokos relates his decade-long quest to find the Gömböc. A tale of mathematical intuition, persistence, and a dose of luck.

But the story of this shape is far more than a mathematical curio.

Its discovery developed ways of thinking and led to a research program that has revolutionized our understanding of how things fall apart. Indeed, the Gömböc and the cube can be understood as two bookends of the evolutionary process by which material disintegrates.

Whether rocks on Mars, pebbles in Pisa, or asteroids flying through space, the Gömböc and the cube delimit the forms that things can take. At the one extreme, the cube sets an upper limit on (or, rather, just above) the number of balance points natural objects take.

Take a large boulder, or block of ice, and fracture it with a pickaxe, the resulting pieces will tend to have a similar number of faces as the cube. Just as randomly drawing lines on paper will produce, on average, shapes with four sides — see the image below, or animation here. As rocks weather, the areas of the highest curvature (the pointiest bits) weather fastest. This will tend to reduce the number of balance points as faces disappear and points of maximal and minimal distance elide.

The Markovian march towars the Gömböc from Pebbles, shapes, and equilibria — Gabor Domokos, András Árpád Sipos, Tímea Szabó & Péter Várkonyi

These straightforward techniques for categorizing forms in terms of balance points, average number of faces (and also of vertices, and faces per vertex) have proved incredibly productive. The Platonic pantheon of shapes (tetrahedra, cubes …) and the few others we learn of at school (cones, cylinders … ) are but a few citizens of a world that comes into view. A jostling populace that ineluctably marches towards the Gömböc.

The extraordinary consequence of this is that holding a pebble in your hand you can feel its age, not just in its sea-worn smoothness, but in its geometrical simplicity. In the loss of its stable points.

Things fall apart

View animated full screen version

Reading

• Scientists Uncover the Universal Geometry of Geology — a delightful Quanta article on the world’s cubism
• The Simple Geometry That Predicts Molecular Mosaics — another piece from Quanta on Gábor and collaborators’ more recent work bringing their geometric techniques (e.g. counting vertices) to molecular physics
• Plato’s Cube and the Natural Geometry of Fragmentation — journal article, as elegant as the title promises
• Static Equilibria of Rigid Bodies: Dice, Pebbles, and the Poincaré-Hopf Theorem — introducing a language for talking about convex shapes
• Pebbles, shapes, and equilibria — contains beautiful illustrations of shapes having differing numbers of balance points

Moments

(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