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.

Reflections (Spheres I)


Large scale mirrors like the surface of a lake are awe inspiring. They simultaneously create complexity and
order. The order comes from the inherent symmetry, and the complexity from subtle differences between original and
mirror image.

Things get considerably more complicated when the mirrors are curved. The Cloud Gate sculpture by Anish Kapoor (the Bean) in the Millennium Park in Chicago is a popular example. The multiple reflections create an immediately surprising chaotic richness of the reflection: Taking one step to the side changes the appearance of the reflection dramatically. But the sculpture also extends and therefore enriches the architecture.

DSC 6142

Motivated by this, I began to experiment with the spherical mirrors, spheres being the simplest curved shapes.
For multiple spheres touching each other there is a surprising phenomenon that is best understood when we begin with seven spheres of equal size, one at the center, and the remaining six surrounding the central sphere symmetrically. Complete this configuration by adding two planes that touch all seven spheres. Now pretend that the two planes are in fact also gigantic spheres. Than these two and the central sphere all touch the remaining six spheres, which in turn form a chain where consecutive spheres touch.


It turns out that this picture is not just an approximation that only works in the ideal situation shown above where the big spheres are planes, but in fact works for spheres of any size. This is the content of Soddy’s theorem.


To turn this into some sort of virtual sculpture, it is best to make just one of the spheres a plane. Then place two spheres onto the plane so that they touch. If you continue placing more spheres onto the plane so that they also touch the two initial spheres and the previously placed sphere, they will form a chain of six spheres of which the last again touches the first.


Now imagine these being really large, reflective, slightly translucent, and illuminated with colored light sources. You might see something like this:


This is the first of a series of images featuring ray traced spheres.