# Message #3276

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] Császár and Szilassi polyhedra

Date: Thu, 17 Dec 2015 14:50:01 -0600

Looking at those, some differences I see are:

- The Császár has solid polygonal faces (topological disks), whereas a

holyhedron must have faces with a hole cut out of them (topological annuli).

- Steffen’s polyhedron has a few extra vertices (9), which I bet helps

allow its flexibility, and is genus-0 instead of genus-1. When following

your link, I read it has been proved that it is the simplest flexible

polyhedron with triangular faces.

So they appear different in some respects, though I wouldn’t be surprised

to find out there are connections! Also… a Visual Insight post

coincidentally showed up this week that mentions Steffen’s polyhedron:

blogs.ams.org/visualinsight/2015/12/15/kaleidocycle/

Cheers,

Roice

On Mon, Dec 14, 2015 at 4:53 PM, Melinda Green melinda@superliminal.com

[4D_Cubing] <4D_Cubing@yahoogroups.com> wrote:

>

>

> Interesting! Is it at all related to the holyhedron

> <https://en.wikipedia.org/wiki/Holyhedron>or the flexible Steffen model

> <http://mathworld.wolfram.com/FlexiblePolyhedron.html>? It looks a lot

> like the Steffen model which also happens to contains 14 triangular faces.

>

> -Melinda

>

>

> On 12/12/2015 3:14 PM, Roice Nelson roice3@gmail.com [4D_Cubing] wrote:

>

> 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

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