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  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Mon Aug 24, 2015 1:07 pm   Subject: How does Stella4D determine the dual?
Adrian,

Nice job finding the equation to calculate the centroid of the Chestahedron. The vast majority of polytopes require an algorithm to find the centroid, but for special cases of symmetry, thi ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Mon Aug 24, 2015 1:02 pm   Subject: How does Stella4D determine the dual?
Thx Adrian, & Guy. Interesting replies, both:) Such a fascinating topic. Just read up on abstract polytopes. FYI, duals seem so much easier in that context. It's when you try to realize a special ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 9:45 pm   Subject: How does Stella4D determine the dual?
On a coincidental side note, stumbled onto this while perusing the math/art gallery from this year's recent Bridges conference. (Perhaps you attended?)...

http://www.ams.org/mathimagery/albums/user ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 8:53 pm   Subject: How does Stella4D determine the dual?
P.P.S. Finished reading your paper, "Vertex figures" (for the first time, at least). Very interesting. And, quite well written - in line with your ability of expression demonstrated in this ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 8:52 pm   Subject: How does Stella4D determine the dual?
P.S. I have acquired the book, 'Dual Models' (MWenninger), and lectures, 'Lectures on Polytopes' (GunterMZiegler). That should keep me busy for a while! Very Happy ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 7:47 pm   Subject: How does Stella4D determine the dual?
I see. (And thx for the correction on the misattributed quotation.)

So, the new edge does not have to be perpendicular to the original, and, in fact, is only so for certain specific midsurfaces and ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 7:19 pm   Subject: How does Stella4D determine the dual?
Hi Guy,

As to your third, recent reply: "I see what adrian means,..."

The centroid, for triangle side length of 1, is ~0.419 (according to Mathematica), but ~0.544 (according to Antipr ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 7:00 pm   Subject: How does Stella4D determine the dual?
Hi Guy,

As to your second, recent reply: "Going back to the canonical construction of the solid and its dual..."

First, by simple inspection, I did not imagine the Chestahedron to be s ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 5:42 pm   Subject: How does Stella4D determine the dual?
FYI, I have started to read your fine paper, "Vertex figures". The intro addresses much of what I have encounter, and intimated. Looking forward to reading in its entirety. I'm sure it will ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 5:31 pm   Subject: How does Stella4D determine the dual?
Hi Guy,

Thx for the great replies. I have read all three. I will address them in turn, in time. As for the first one (on duality/projective geom)...

Yes, I have come to understand that "dua ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 9:08 am   Subject: How does Stella4D determine the dual?
Hi Guy,

Again, very well stated. Thx. I believe I understand. But, although I have studied projective geometry, I am not as comfortable with it as I'd like to be. I will reread your insight, and co ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 5:55 am   Subject: How does Stella4D determine the dual?
Wow, Guy! Great stuff. Thx. Much of what you shared I have been discovering along the way, or at least getting a hint of. Thx for so eloquently clarifying things. A fascinating journey it has been:)
...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 5:38 am   Subject: How does Stella4D determine the dual?
Several duals using Antiprism/pol_recip: top to bottom...

Green: -c v (vert/face balanced)
Blue: -c a (v/f/e balanced)
Red: -c e (edge balanced)
Orange: -c M (mid-sphere (approximate))
https:// ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 5:23 am   Subject: How does Stella4D determine the dual?
Hi, I'm back:)...

https://farm6.staticflickr.com/5789/20184466073_a198395aa2_c.jpg

The graphic above shows one version of a dual for the Chestahedron. This "polar reciprocation" method ...
  Topic: How does Stella4D determine the dual?
Swami Polytope

Replies: 30
Views: 40268

PostForum: Stella Forum   Posted: Fri Aug 21, 2015 5:47 pm   Subject: How does Stella4D determine the dual?
Thx again, Adrian. I appreciate your knowledge in this area.

Started to write more of a reply, but realized I need to consider things more, and do some testing.

So, will get back to you, but wan ...
 
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