Reflections on the Letters r,s,t (Groups II)

Continuing the discussion from last week, let’s consider the 3-letter alphabet {r,s,t}. We are allowed to form all possible words in these letters (and their inverses, if you want to), but we agree that rr=ss=tt=1 and (rs)^2=(st)^3=(tr)^6=1. This defines the Coxeter group G(2,3,6). Last time we saw that the very similar group G(2,3,3) is finite, today we will see that G(2,3,6) is infinite. Below is the beginning of its Cayley graph.

Cayley236 01

We travel from one word to the next by appending r,s, or t. This looks much more complicated than what we saw for G(2,3,3), but things become clearer when we look at another group. Consider a yellow triangle with 30, 60, 90 degree angles (writing this as π/2, π/3,π/6 makes 2-3-6 reappear), and let ρ, σ, τ be the reflections at the lines extending its edges.

Reflectiongroup 01

These three reflections generate a group Γ(2,3,6) of Euclidean symmetries which has the yellow triangle as its fundamental domain. The clue is that Γ(2,3,6) = G(2,3,6). We can easily map G(2,3,6) to Γ(2,3,6) by sending R to ρ, s to &sigma, t to τ. This works because ρ, σ, τ satisfy the same relations as r, s, t. It doesn’t work as easily the other way because ρ, σ, &tau could also satisfy other, hidden relations.

Reflpath 01

Let’s look at the word tsrtst. Reading it from left to right gives us a path on the Cayley graph from the initial triangle to a target triangle. Translating from Latin tsrtst to Greek τσρτστ gives a composition of reflections that takes the initial triangle to the same target triangle. This is not completely trivial, you prove it by induction. Remember that the composition is applied from the right to the left, so we also change reading direction.

This observation can be used to show that the translation map G(2,3,6) to Γ(2,3,6) is injective. If a word in G is the identity in $γ, its path in the Cayley graph must be a closed loop. As the Euclidean plane where the tiling lives is simply connected, we can homotope it to a constant path, using elementary operations: Backtracking an edge, or shrinking a loop around a vertex to a point. The former is the accomplished using the relations rr=ss=tt=1, the latter using the other relations. This shows that the geometric homotopy can be realized using the relations of the group, and thus we can reduce the word to the trivial word 1.

This is essentially the proof of a famous theorem by Walther Franz Anton von Dyck: The group G(a,b,c) is finite if and only if 1/a+1/b+1/c>1. We have seen the relevant examples in the case
1/a+1/b+1/c>1 and 1/a+1/b+1/c=1. If 1/a+1/b+1/c <1, we need hyoperbolic geometry. Above is a picture of the Cayley graph of G(2,3,7) within the tiling of the hyperbolic plane by (π/2,π/3,π/7)-triangles.


Do and Undo (Groups I)

Groups are mathematical games being played with letters. In the simplest version, we use just one letter (say a), and are allowed to add it to or to remove it from a word. This is the free group of one generator, or the infinite cyclic group.

Cyclic 1 01

Clearly, this game of create and destroy needs more rules. A simple rule is to make it truly cyclic and finite by insisting that after using the letter a say 7 times, we are back where we started. This means aaaaaaa=1, which is a relief, but still not very interesting.

Cyclic 2 01

With two letters a and b, our game expands.


This Cayley graph is the dual graph of the tiling of the hyperbolic plane by ideal squares, and not accidentally so.

Again we can restrict the rules of the game. Let’s play with the three letters r, s, and t, and insist that rr=ss=tt=1 to avoid any repetitions, and also that rsrs=ststst=trtrtr=1. The result is the Coxeter group G(2,3,3), and after a while playing around with the words, you find its Cayley graph below, neatly laid out.


This is dual to a tiling of the sphere by spherical triangles with angles π/2,π/3,π/3, and this is also not accidentally so.
We’ll see more about this next week.


The Gyroids (Algorithmic Geometry III)

When we use squares bent by 90 degrees about one diagonal and extend by the rotate-about-edges rule, we get Petrie’s triply periodic skew polyhedron {4,6|4} which has six squares about each vertex. The two tunnel systems it divides space into are another crude approximation of the primitive surface of Schwarz.


Coxeter observed that this polyhedron can be used to construct Laves’ remarkable chiral triply periodic graph as follows. Choose any diagonal of any of the squares of {4,6|4}. Take an end point of the diagonal, adjacent to which are six squares. Look at the six diagonals of the squares that share the end point as a vertex, and take every other of them, starting with the already chosen diagonal. Keep extending the emerging graph like this.


You obtain the 3-valent Laves graph. At each vertex, the edges meet 120 degree angles. It turns out a mirror symmetric copy fits onto the {4,6|4} without intersections. These two graphs are the skeletons of the two components of the Gyroid, a triply periodic minimal surface discovered by Alan Schoen. You can read all about the discovery at his Geometry Garret.


The Laves graph also lies on the dual skeleton of the tiling of space of rhombic dodecahedra. That means that you can get a solid neighborhood of the Laves graph consisting of rhombic dodecahedra:


This can be done both for the Laves graph and its mirror still leaving a gap in which one can fit the gyroid. Alan Schoen also discovered a uniform polyhedral approximation of the gyroid, consisting of squares and star hexagons. To build it, take a star, attach a square to every other edge, bending the squares alternatingly up and down. Then attach six more stars to the free edges of the first star, fitting them to one free edge of one of the squares each:


Two copies of this piece (without the downward pointing stars and and squares) make a translational fundamental piece of the uniform gyroid.


Images of larger portions are hard to parse, but it makes a wonderful model.