The n-Category Café

Week 270 = 2 * 3^3 * 5 is delightful. Even through the stench of brimstone…

Suppose that physical space is actually a noncommutative geometry. What primitive aliens in an exotic environment could figure that out? I await the Connes-Egan collaboration…

I’m confused by your description of the (5,6,12) Steiner system. If I apply the transformation z -> 1/z to the block (0,1,3,4,5,9), I get (infty,1,4,3,9,5), which has a 5 element overlap with the original block. My guess is that I am supposed to only apply transformations z -> (az+b)/(cz+d) where ad-bc=1. Is this right?

John B. wrote:

People often say that Penrose tilings arise from lattices in 4d space. Maybe I’m finally starting to understand how! The $A_4$ lattice has a bunch of 4-simplices in it — but when we project these onto the plane correctly, they give pentagrams. I’d be very happy if this were the key.

It is the key! In fact Greg Egan has an applet based on this idea! It draws quasiperiodic tilings with 5-fold symmetry formed by slicing the $A_4$ lattice with a cleverly chosen plane.

Even better, it draws quasiperiodic tilings with $n$-fold symmetry formed by slicing the $A_{n-1}$ lattice with a cleverly chosen plane… when $n$ is odd. For $n$ even, it uses a slightly different trick… but I think we’re seeing a general relation between the noncrystallographic Coxeter group

  n
o---o 

and $A_{n-1}$, which generalizes the $n = 5$ case I talked about.

Even better, we can indeed get 3d quasicrystals with dodecahedral symmetry from slicing a $D_6$ lattice in 6 dimensions…

… and we can get 4d quasicrystals with the symmetry of the 120-cell from slicing an $E_8$ lattice in 8 dimensions!

• J. F. Sadoc and R. Mosseri, The $E_8$ lattice and quasicrystals, J. Non-crystalline Solids 153,154 (1993) 247–252.
• R. V. Moody and J. Patera, Quasicrystals and icosians, J. Phys. A. 26 (1993), 2829–2853.

The latter paper is tremendously fun; I haven’t looked at the former yet.

I will write more about this later. I was always scared of quasicrystals, but now I’m not, because I see how they connect to things I like!