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

From: Roice Nelson <roice3@gmail.com>
Date: Thu, 17 Dec 2015 14:50:01 -0600
Subject: =?UTF-8?Q?Re=3A_=5BMC4D=5D_Cs=C3=A1sz=C3=A1r_and_Szilassi_polyhedra?=



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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
>

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Looking at those, some differences I see are:
iv>
So they appear different in some respects, though I =
wouldn't be surprised to find out there are connections!=C2=A0 Also... =
a Visual Insight post coincidentally showed up this week that mentions Stef=
fen's polyhedron:=C2=A015/12/15/kaleidocycle/" target=3D"_blank">blogs.ams.org/visualinsight/2015/=
12/15/kaleidocycle/

Cheers,
Ro=
ice


quote">On Mon, Dec 14, 2015 at 4:53 PM, Melinda Green inda@superliminal.com" target=3D"_blank">melinda@superliminal.com [4D_C=
ubing] <arget=3D"_blank">4D_Cubing@yahoogroups.com> wrote:
uote class=3D"gmail_quote" style=3D"margin:0px 0px 0px 0.8ex;border-left-wi=
dth:1px;border-left-color:rgb(204,204,204);border-left-style:solid;padding-=
left:1ex">






=20=20=20=20=20=20=20=20

=20=20
=20=20=20=20
=20=20













Interesting! Is it at all related to the a.org/wiki/Holyhedron" target=3D"_blank">holyhedron or
the flexible .html" target=3D"_blank">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.co=
m
[4D_Cubing] wrote:



=20=20=20=20=20=20
=20=20=20=20=20=20









It is the only known polyhedron besides the tetrahedron
that has no diagonals - all 7 vertices connect to every
other.=C2=A0 With 21 edges and 14 faces, its genus is 1.=C2=A0 Yo=
u can
think of it as the ete_graph" target=3D"_blank">complete
graph
K_7 embedded on the torus.=C2=A0 It also has a dual,
the target=3D"_blank">Szilassi
polyhedron
.=C2=A0 Both relate to the s.ams.org/visualinsight/2015/08/01/heawood-graph/" target=3D"_blank">Heawoo=
d
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.=C2=A0 Here are some pictures of the tilings.








Both are in the Euclidean/Torus section of MagicTile.









Enjoy!

Roice







=20=20=20=20=20=20




=20=20





















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