Seven short movements on group theory, from the six symmetries of a triangle to the Monster. Nothing here was recorded: every frame is computed in your browser as it plays, drawn for your screen at your display’s rate.
Runtime
4:00
Chapters
Seven, with a prologue and a coda
Resolution
Your screen’s, up to 4.4 MP
Frame rate
Your display’s
Frames recorded
None
Languages
English · 日本語 · 中文
Try it yourself
The pictures in this film are live. These open them as playgrounds.
The film never shows a symmetry as a property. It shows a move: something you do to a shape that leaves it looking as it did. An equilateral triangle has exactly six such moves, three rotations and three reflections, and the corner labels are the only way to tell them apart.
The table is a Latin square
Composing the six moves fills a six-by-six table. Because every move can be undone, each row and each column contains every move exactly once. Colour the rotations warm and the reflections cool, and the table turns out to be a two-by-two table in disguise: the triangle’s group, divided by its rotations, is the group of order two.
Cayley graphs
Draw one point per group element and one arrow per generator, and the group becomes a shape. The symmetric group on four letters, generated by adjacent swaps, is the permutohedron; the sixty rotations of the icosahedron, generated by a turn of order five and one of order two, are the truncated icosahedron. The relations of the group are the faces: a⁵ gives the pentagons, (ab)³ the hexagons.
Kaleidoscopes
Chapter V folds every pixel of the screen back into one small triangle, the way a kaleidoscope does, and counts how many reflections that took. The pattern grows outward in order of that count, which is the length of the group element that moved the triangle there. Bend the lines so they are no longer mirror images and the reflections vanish, leaving only the rotations.
The classification
Every finite group is assembled from simple groups, as every whole number is a product of primes. The list of all finite simple groups, finished in the 2000s after roughly fifty years and tens of thousands of pages, has eighteen infinite families and twenty-six sporadic exceptions. The largest is the Monster, whose order is shown in full in Chapter VII, together with the coincidence that began monstrous moonshine: 196,884 = 196,883 + 1.