Császár and Szilassi Polyhedra

[Nordehylop]

How would one get the Császár and Szilassi Polyhedra in stella? They are two polyhedra that I would like to build for my dad for Father's day. The Császár polyhedron is the only other polyhedron (besides the tetrahedron) without diagonals, and the Szilassi Polyhedron is the only other polyhedron (besides the tetrahedron) where every face touches every other face.

I have a model of the Szilassi Polyhedron already, from nets printed out from here:http://www.ac-noumea.nc/maths/amc/polyhedr/Csaszar_.htm (another question: how to put links in my posts!). But I would like to see the Császár polyhedron more clearly before I build a model of it, and also I would like to be able to size them the way I want them. Plus, they've got to have some pretty neat stellations...

Thanks!

[dysphras]

Here is the OFF contents for Szilassi and Csaszar polyhedra:

Szilassi:

14 7 21
12.0 0.0 12.0
-12.0 0.0 12.0
0.0 12.6 -12.0
0.0 -12.6 -12.0
2.0 -5.0 -8.0
-2.0 5.0 -8.0
3.75 3.75 -3.0
-3.75 -3.75 -3.0
4.5 -2.5 2.0
-4.5 2.5 2.0
7.0 0.0 2.0
-7.0 0.0 2.0
7.0 2.5 2.0
-7.0 -2.5 2.0
6 0 1 13 8 7 4
6 0 4 3 2 10 12
6 0 12 9 6 5 1
6 11 3 4 7 6 9
6 11 9 12 10 8 13
6 11 13 1 5 2 3
6 2 5 6 7 8 10

Csaszar 1:

7 14 21
3.0 -3.0 -7.5
-3.0 3.0 -7.5
3.0 3.0 -6.5
-3.0 -3.0 -6.5
1.0 2.0 -4.5
-1.0 -2.0 -4.5
0.0 0.0 7.5
3 0 1 2
3 0 2 5
3 0 5 4
3 0 4 6
3 0 6 3
3 0 3 1
3 1 3 4
3 1 4 5
3 1 5 6
3 1 6 2
3 2 6 4
3 2 4 3
3 2 3 5
3 5 3 6

Csaszar 4:

7 14 21
12.0 0.0 -8.485281374238570292810
-12.0 0.0 -8.485281374238570292810
0.0 12.0 8.485281374238570292810
0.0 -12.0 8.485281374238570292810
-4.0 -3.0 0.707106781186547524400
4.0 3.0 0.707106781186547524400
0.0 0.0 3.771236166328253463471
3 0 1 2
3 0 2 5
3 0 5 4
3 0 4 6
3 0 6 3
3 0 3 1
3 1 3 4
3 1 4 5
3 1 5 6
3 1 6 2
3 2 6 4
3 2 4 3
3 2 3 5
3 5 3 6

[robertw]

Also, you'll find that both have already been provided with Great Stella and Stella4D! Look in Stella Library -> More Stewart Toroids. They're not really Stewart toroids, but they are toroids and do appear in his book. You'll find both Csaszar and Szilassi in there.

Rob.

[Alex Doskey]

I have a website with Stewart Toroids and other figures from his book, including Császár and Szilassi. They are at the bottom of the page for Chapter 20.
http://toroid.doskey.com/

The hard part was modeling them before you could import OFF files. It took quite some effort.

[guy]

ISTR that these two are duals of each other. So you need only load one and view its dual to get both.

[Ulrich]

The hard part was modeling them before you could import OFF files. It took quite some effort.

Alex,

do you still remember how you did it and could you explain it?

Ulrich

[Nordehylop]

ISTR that these two are duals of each other. So you need only load one and view its dual to get both.

Actually, not so. I really have no idea why, but the duals of these polyhedra on Great Stella look nothing like eachother. Perhaps the polyhedra are distorted in relation to eachother, making it look like they are unrelated polyhedra.

[robertw]

Yes, that's right, they are not geometric duals of each other, just topological duals. If you take the geometric dual you lose the property of non-intersecting faces.

Rob.

[Alex Doskey]

Ulrich,
It really wasn't a complicated method. I found both models had integer coordinates (I had to multiply the coordinates I had from B.M. Stewart's book by 20 to make them integers). After that, I just made a 3D skeleton using cubes, and faceted out the models using the "top-right" corner of each cube at the desired location. It would have been easier to ask Rob to make it for me, or to hack my own .stel file by hand (back then the file formats were even easier to work with). But I wanted to see if I could do it within the given interface.