New Torus

[ndl]

I found a new interesting equal-edged torus today by excavating two J92's out of an icosidodecahedron

ID-J92 Torus.jpg (56.76 KiB) Viewed 70881 times

Has anyone ever made this before?
I never read Stewart's toroid book, maybe it's in there?

[robertw]

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

[Ulrich]

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]

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.

[ndl]

I found another interesting one yesterday:

J32 Torus.JPG (30.31 KiB) Viewed 70608 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]

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 (44.46 KiB) Viewed 70605 times

This is J76 with a rotunda and two icosahedra removed for a genus of 2.

[robertw]

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

[ndl]

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 (46 KiB) Viewed 70603 times

Still quasi-convex!

[robertw]

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

[ndl]

And an alternate version with antiprism and cupola replacing rotunda:

J76 Genus 6 alt.JPG (57.53 KiB) Viewed 70602 times