# Message #3273

From: Roice Nelson <roice3@gmail.com>

Subject: Császár and Szilassi polyhedra

Date: Sat, 12 Dec 2015 17:14:50 -0600

Yesterday I learned about the Császár polyhedron

<http://www.futilitycloset.com/2015/12/10/the-csaszar-polyhedron> on

Google+.

plus.google.com/u/0/+DavidJoyner/posts/HEBGDgqLgdG

It is the only known polyhedron besides the tetrahedron that has no

diagonals - all 7 vertices connect to every other. With 21 edges and 14

faces, its genus is 1. You can think of it as the complete graph

<https://en.wikipedia.org/wiki/Complete_graph> K_7 embedded on the torus.

It also has a dual, the Szilassi polyhedron

<https://en.wikipedia.org/wiki/Szilassi_polyhedron>. Both relate to

the Heawood

graph <http://blogs.ams.org/visualinsight/2015/08/01/heawood-graph/>.

Turns out I already had the latter configured in MagicTile (the {6,3}

7-Color), but I didn’t have the former, so I just added it. Here are some

pictures of the tilings.

https://goo.gl/photos/K1vYapeTqqYteGx58

https://goo.gl/photos/kQMxQCtbCqsL2Wj88

Both are in the Euclidean/Torus section of MagicTile.

www.gravitation3d.com/magictile

Enjoy!

Roice