Thread: "=?UTF-8?Q?Re=3A_=5BMC4D=5D_Cs=C3=A1sz=C3=A1r_and_Szilassi_polyhedra?="

--001a11c38854a455ab05271e2a94
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: quoted-printable

Looking at those, some differences I see are:

- The Cs=C3=A1sz=C3=A1r has solid polygonal faces (topological disks), w=
hereas a
holyhedron must have faces with a hole cut out of them (topological annu=
li).


- 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 followin=
g
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
> or the flexible Steffen model
> ? 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 Cs=C3=A1sz=C3=A1r 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
> K_7 embedded on the
> torus. It also has a dual, the Szilassi polyhedron
> . Both relate to the =
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 som=
e
> 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
>
>
>
>
>
>=20
>

--001a11c38854a455ab05271e2a94
Content-Type: text/html; charset=UTF-8
Content-Transfer-Encoding: quoted-printable