Note the following slightly unintuitive fact: "There exist full-rank sublattices L of the cubic lattice Z^d such that the point-fixing automorphism group of L is greater than that of Z^d." For instance, in Z^4 we have that: L := {(a, b, c, d) : a + b + c + d is even} has an origin-fixing automorphism group of order 1152 (compared with 384 for the lattice Z^4). Similarly, in Z^8 we can take a scaled E_8 lattice: L := {(a_0, a_1, ..., a_7) : all coordinates have the same parity and a_0 + a_1 + ... + a_7 is divisible by 4} which has an automorphism group of 696729600, compared with 10321920 for the cubic lattice Z^8. The reason this phenomenon is slightly unintuitive is that it doesn't occur in dimensions d <= 3 so defies our geometric intuition. Best wishes, Adam P. Goucher
Sent: Saturday, January 09, 2016 at 6:18 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Voronoi tilings from grids with periodic deleted points
I would be very surprised if all the wallpaper groups could be realized. Surely the symmetry group of the Voronoi tiling is the same as that of the underlying collection of points -- work with that instead. I'll bet almost anything that, say, *236 can't be realized.
On Sat, Jan 9, 2016 at 12:03 PM, Thane Plambeck <tplambeck@gmail.com> wrote:
Make a 23x23 square array of points, numbered from 1 to 529 row-wise. Remove all the points whose number is congruent to 0, 1, or 2 modulo 5. Then compute the Voronoi cells of that arrangement of points inside a bounding rectangle.
This is the picture you get (use VoronoiMesh[] in Mathematica)
https://www.flickr.com/photos/thane/24191215291/in/dateposted-public/
Here's another one with different values of 23, 0, 1, 2, 5, so to speak (unfortunately I forgot to write them down in my enthusiasm)
https://www.flickr.com/photos/thane/24191270151/in/dateposted-public/
Anyway, this seems to be a simple way to make lots of interesting looking tilings on the cheap.
(1) I can't possibly be the first person to do this...I'd welcome references (2) Can all the wallpaper groups be realized this way?
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun