Notes
A symmetry is an action
The film treats a symmetry not as a property but as an action: you move the whole shape, without stretching or bending it, and it lands exactly on its own outline. An equilateral triangle has exactly six such moves: three rotations (doing nothing counts as a turn by zero) and three reflections. Afterwards the triangle looks the same every time; only the markers the film adds (the corner labels, the small arrow, the shading after a flip), which are not part of the shape, tell the moves apart.
The group’s table: no repeats in any row or column
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; a table like this is called a Latin square. Colour the rotations warm and the reflections cool, and it turns out to be a two-by-two table in disguise: two rotations or two reflections make a rotation, and a rotation with a reflection makes a reflection. In the language of groups, the triangle’s group divided by its rotations is the group with just two elements.
Cayley graphs: a group as points and arrows
Draw one point for each element of the group and, from every point, one arrow for each generator, each generator in its own colour, and the group takes on a shape. The 24 ways to arrange four things (the symmetric group S₄), generated by swapping neighbours, give the permutohedron; the sixty rotations of the icosahedron, generated by a fifth of a turn, a, and a half turn, b, give the truncated icosahedron, the shape of a classic football. The basic relations of the group, which the film calls rules, show up as faces: every pentagon is a⁵ = e, every hexagon (ab)³ = e. Choose different generators and the shape changes.
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 carried the triangle there: the fewest reflections that will do it. Bend the lines so the pattern no longer matches its own mirror image, and the reflections vanish, leaving the rotations and translations.
Noether’s theorem
The symmetries in the second half of Chapter VI belong to the laws of physics. Strictly, what stays the same under a continuous family of changes is a quantity physicists call the action, from which a system’s equations of motion follow. In 1918 Emmy Noether proved that each independent direction of such a family comes with a quantity conserved as the system moves: shifts in time give energy, the three shifts in space give the three components of momentum, the three independent rotations give those of angular momentum. Her theorem says nothing about symmetries that come only in steps, like a crystal’s repeating lattice; local symmetries, which depend on arbitrary functions, fall under her second theorem, which gives identities among the equations of motion instead.
Classifying the finite simple groups
Every finite group can be taken apart, layer by layer, into simple groups, as molecules are made of atoms; and as in chemistry, the same atoms can make different molecules (C₄ and C₂ × C₂ are built from the same two pieces). The complete list of finite simple groups has eighteen infinite families and twenty-six exceptions, the sporadic groups. Proving that nothing is missing from it took roughly fifty years and tens of thousands of pages, and was finished in the 2000s. The largest exception is the Monster, whose order (its number of elements) is written out in full in Chapter VII, together with the observation that began ‘monstrous moonshine’: 196,883 + 1 = 196,884, and 196,884 is a coefficient of the j-function from number theory. In 1992 Richard Borcherds proved that this is no coincidence.