Today's "What's Happening in Fine Hall" lecture was given by John Conway on crystallographic space groups. An attempt at summary:

• The talk is motivated by the completion of a ten-year collaborative effort with Heidi Burgiel and Chaim Goodman-Strauss to write the book "The symmetries of things".
• A subset of the book is described in the talk: those pertaining to 3D space groups and 2D planar groups.
• The classification of 3D space groups and 2D plane groups are more or less a done subject by the beginning of twentieth century, through the separate efforts of three teams. The teams are led by Fedorov, Schoenflies, and Barlow.
• The crystallographer nomenclature puts the number of plane groups at 17 and number of space groups at 230. Conway claims that the "right way" of looking at things should give only 219 of the latter.
• The "right way" is to follow Thurston's commandment: one should only study geometric groups through its orbitfold. 11 of the classical groups are meta-chiral: not symmetric under reflection. The crystallographers counts both it and its chirality-dual as two groups, whereas the orbitfold notation does not distinguish chirality.
• Conway's method of drawing the orbitfold for planar groups: take a symmetry pattern. First draw all the mirror lines of the pattern in red (red for reflection). This should divide the plane pattern down to unit cells from which the entire pattern can be recovered by repeated reflection. Then pick out centers of rotation symmetry of the unit cell and draw it in blue (blue for true, but I don't remember what true is a reference to). One can think of the pattern as produced by a Kaleidoscope with mirrors on all the lines of reflection. Around each of the blue points we pinch the unit cell under the rotation symmetry (i.e., identify the points that are identical under the rotation). This is now an orbitfold. For most of the unit cell, a small neighborhood around it is planar: a circle around the point has 2π radians. For a point on the mirror, a neighborhood only has a semi-circle. For a point on the corner between two mirrors, a neighborhood only has an arc, whose length is π/n for some integer. For the blue point, the neighborhood as a circle, but the circle only has 2π/n radians for some integer n. An example: take the square as the unit cell, and consider a rotation about the center of the square of π radians. This identifies the top edge of the square with the bottom, and the left with the right. The orbitfold is actually then a cone: there is a cone-point with π radians around it near the top, and a boundary that has 2 corners, each turning π/2 radians.
• The idea is that all planar groups can be expressed as a bunch of cone points with a certain integer order of rotation, plus a bunch of kaleidoscope mirrors, which, under the rotation of the cone points, has a certain number of unique corners each rotating a certain angle (technically there is a little bit of something else, but those almost never show up). A topological argument places a certain weight on each of those objects (different weights for cone points of different orders and for corners of different angles); in other words, each descriptor of the orbitfold "costs" some money. The topological argument also states that for planar groups, you only have two dollars to spend, and you must spend exactly those two dollars. So as it turns out, given the different prices of different objects, there are only 17 collection of goodies that you can buy with exactly two dollars. Those 17 collections describe exactly the 17 planar groups.
• (For examples, Conway used the pattern of a brickwall first. Okounkov commented on the ubiquity of brick walls in the common room. Conway said: "I'm well aware of the brickwalls all around us, but I am not going to take out a piece and turn it around." The second example was the basket weave on the floor. And he actually drew on the floor...)
• (Conway also said that he knows quite a lot about brickwalls, actually, since he gave a lecture, on a bet that he couldn't make it interesting, titled "Staring at Brick Walls.")
• If one looks at, instead, groups on the surface of a two-sphere, the number of groups is infinite. On the otherhand, mod out by some integral parameters, we see that there are 7 "prime" groups which are true two dimensional groups, and 7 "composite" families of groups (parametrized by the integers). The composite families all are of the form that one axis of the sphere is fixed under all action of the group. which means that if you looked down along the axis, you recover a group as a composition of reflection along the axis, and a rotation about the axis: the composition of two 1D groups.
• The word "prime" should be taken to mean juicy, interesting, as in "USDA grade A prime beef".
• The distinction of prime and composite carries over to 3D. The 3D crystallographic groups can be categorized into 35 prime and 184 composite ones. The composite ones are "easy", they are compositions of 1D reflections of an axis with a planar group. The amazing thing is that the 35 prime groups can all be studied from one diagram!
• Claim: there must exist crystal groups with order 3 symmetry. One can obtain this result by deducing using Sylow theorem (non-standard... since the groups are infinite, one has to use non-standard arithmetic in a different formal logic system to apply Sylow theorem to the crystal group) plus representation theory. Or one can just be normal and use Thurston's geometrical argument.
• Take the cubic lattice. Color the alternate corners crimson and green (crimson for one, green for three). Let p denote the group of all rotations about body-diagonals. This group fixes the crimsons and greens. (Okay, I am not sure that I remember this first part right... should check Conway's book, which it is out, for details.) Now, take the body centers of the cubic cells, connecting them we get another cubic lattice. color the corners alternate yellow and blue (yellow for zero, blue for two). Now connect the yellow, green, crimson, and blue points. They form a quasiregular honeycomb for space by tetrahedral elements. Label the adjacent elements by black and white. It turns out that any space group can be described as a subgroup of P, the permutation group for the 0,1,2,3 and black and white. And P/p is C₂ x D₈. Therefore the heirachy (similar to the galois one for finite fields) between P and p describes, in fact, 27 of the prime space groups. (The other 8 is also hidden in the picture, just not from what we have considered).
• Scottish bubbles: the conjectured solution by Lord Kelvin to his namesake problem on the most efficient tiling of space by bubbles. (Even though Kelvin was born Irish...); Irish bubbles: the Weaire-Phelan structure that were found to actually solve the Kelvin problem, which is different from the Scottish bubbles. Both are contained as realizations of particular symmetry groups.
• Nonstandard arithmetic. Consider the formal logic system where one introduces a number c that satisfies all the properties which are satisfied by all integers, and let c be not an integer. The system is consistent in the sense of Godel completeness: it is a model extension of the integer system. Adding c to Z formally provides another additive group that is almost, but not quite, like Z. Replace the infinite part of the space group (the translation part), which is normally Z, by this new group. By considering primes p greater than c, we can formally reduce the infinite group down to a finite group, and apply Sylow's theorem. The problem really arises when one wants to say that the order 3 element is still order 3 in real world. (Heuristically speaking, he is saying that Z mod p is a finite group. So letting p tend to infinity, Z mod infinity should still be a finite group, and so Sylow's theorem applies...)