4D stellations

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ndl
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4D stellations

Post by ndl » Sun Aug 18, 2019 4:33 am

I wanted to create 3D stellation patterns for 4D regulars, and I figured out how to do it I'm just missing one step that maybe someone would be able to help me out with. The patterns just consist of sets of planes in a 3D realm and are easy to create with Stella. The problem I have is how do I calculate the radius of those stellation planes where the cell would hit the next group of cells. I was able to calculate some groups using known stellations of the regulars and setting the inradius to be the same. However for those groups that don't have any defined stellations I can't figure out how to calculate the radius. The orientation I can get by looking at the orientation of the vertices of the dual with a given chord, but the distance I have no idea. I would appreciate any help. Thanks

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robertw
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Re: 4D stellations

Post by robertw » Sun Aug 18, 2019 9:19 am

Interesting idea. People have asked about support for 4D stellations in Stella, but there would be a lot involved, and beyond the simplest polychora it would get crazy very quickly. However I'd never thought about just supporting the 3D stellation diagrams. Something I should think about sometime.

I'm not sure immediately how to help you with that right now though. It would probably require more time than I have to spare at the moment, but good luck with it! Let us know how you go.

ndl
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Re: 4D stellations

Post by ndl » Mon Aug 19, 2019 9:18 pm

So I figured out a convoluted way to do it:
First I used the arcsine of height of the dual from vertex first projection to another level of vertices divided by the chord length to that vertex. That gives me the angle to tangent realm. Then I used one over the cosine of that angle times half of that chord to give me distance from center of cell to face of stellation!

Here's some of them that I created:
Stellation cores.zip
(6.32 KiB) Downloaded 41 times
The model is just the cores and the stellations represent the 3D stellation cell for the 4D polychoron. Now you can't tell from this which ones are actually valid stellations, but it's a start.

When I was working on the 600-cell, after adding a bunch of these cores by using the blend from memory I ran into an error:

"Couldn't find exit edge"

What does that mean?

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robertw
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Re: 4D stellations

Post by robertw » Thu Aug 22, 2019 2:35 pm

ndl wrote:
Mon Aug 19, 2019 9:18 pm
When I was working on the 600-cell, after adding a bunch of these cores by using the blend from memory I ran into an error:

"Couldn't find exit edge"

What does that mean?
It means something went wrong when adding lines to the stellation diagram. Could you send me a .stel file of the first model that gives that error? I'll see if I can find why.

Thanks.

ndl
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Location: Queens, NY

Re: 4D stellations

Post by ndl » Fri Aug 23, 2019 3:43 am

I used a more exact calculation for the inradius and it solved the problem. I think the pieces that were supposed to line up didn't because it wasn't exact enough.

ndl
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Re: 4D stellations

Post by ndl » Wed Aug 28, 2019 5:46 pm

I've had some success with these stellations, I just have one more question (for now): Is there an easy way to reverse a polyhedron's orientation in a given symmetry (tetrahedral for example) to correctly blend it with another model. What I have been doing (which only works sometimes) is to export it to .off and then change the signs on the coordinates. Is there a better way? Thanks

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robertw
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Re: 4D stellations

Post by robertw » Thu Aug 29, 2019 3:41 am

Use "Poly->Create Mirror Image". Only available when the base model doesn't have mirror symmetry already.

Roiger
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Re: 4D stellations

Post by Roiger » Mon Dec 02, 2019 3:46 pm

Most base models should have mirror symmetry already though, right?

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robertw
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Re: 4D stellations

Post by robertw » Tue Dec 03, 2019 1:30 am

Roiger wrote:
Mon Dec 02, 2019 3:46 pm
Most base models should have mirror symmetry already though, right?
Yes many do, and many don't. Models like the snub cube, or compound of five tetrahedra have lots of rotational symmetry, but no mirror symmetry.

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