New Torus

For general discussion of polyhedra, not necessarily Stella-specific.
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ndl
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New Torus

Post by ndl » Thu Nov 29, 2018 3:31 am

I found a new interesting equal-edged torus today by excavating two J92's out of an icosidodecahedron
ID-J92 Torus.jpg
ID-J92 Torus.jpg (56.76 KiB) Viewed 1928 times
Has anyone ever made this before?
I never read Stewart's toroid book, maybe it's in there?

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robertw
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Re: New Torus

Post by robertw » Thu Nov 29, 2018 7:21 am

Nice. I'm not familiar with it offhand. It's not immediately obvious that J92 should reach exactly halfway into the icosidodecahedron. Interesting.

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Ulrich
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Re: New Torus

Post by Ulrich » Thu Nov 29, 2018 10:22 am

You should have a look at Alex Doskey‘s website. He studied Stewart’s book thoroughly and he has lots of such toroids:

polyhedra.doskey.com/stewart00.html

Ulrich

ndl
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Re: New Torus

Post by ndl » Thu Nov 29, 2018 7:00 pm

Thanks Ulrich for your suggestion, I actually did find his website last night and there are numerous interesting things there, but I didn't see this example. Really I need to get a copy of Stewart's book, like Alex said on his website it's really most useful as a supplement with the book.

I actually thought of this excavated icosidodecahedron because of something I saw on http://eusebeia.dyndns.org/4d/J92, he shows there that the J92 reaches the halfway point of the icosidodecahedron.

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Re: New Torus

Post by ndl » Wed Dec 19, 2018 7:12 pm

I found another interesting one yesterday:
J32 Torus.JPG
J32 Torus.JPG (30.31 KiB) Viewed 1655 times
This is a J91+J63 Tunnel through a J32 (Pentagonal Orthocupolarontunda)

I realized then that J32 had a cross section of 2 of those irregular hexagons that Stewart talked about inside a Rhombicosidodecahedron.

ndl
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Re: New Torus

Post by ndl » Wed Dec 19, 2018 7:20 pm

So I created a new toroid that has Stewart's quasi-convex property that doesn't seem to be mentioned in his book (I only have first edition, though):
J76 Torus.JPG
J76 Torus.JPG (44.46 KiB) Viewed 1652 times
This is J76 with a rotunda and two icosahedra removed for a genus of 2.

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robertw
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Re: New Torus

Post by robertw » Thu Dec 20, 2018 2:28 am

That one's nice. Seems unexpected that the icosahedron should align like that.

ndl
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Re: New Torus

Post by ndl » Thu Dec 20, 2018 2:35 am

Even better, I got one with a genus of 6 and retaining the original symmetry of J76 by cutting through with a Dodecahedron and 5 J11s
J76 Genus 6.JPG
J76 Genus 6.JPG (46 KiB) Viewed 1650 times
Still quasi-convex!

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robertw
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Re: New Torus

Post by robertw » Thu Dec 20, 2018 2:43 am

Yep very nice. J76 minus a dodecahedron and 5 diminished icosahedra.

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Re: New Torus

Post by ndl » Thu Dec 20, 2018 2:54 am

And an alternate version with antiprism and cupola replacing rotunda:
J76 Genus 6 alt.JPG
J76 Genus 6 alt.JPG (57.53 KiB) Viewed 1649 times

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