Search found 20 matches
- Mon Aug 24, 2015 1:07 pm
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
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, this aspect can be used to intuite a formula. Oh, and thx for clarifying the distinction of the volume cen...
- Mon Aug 24, 2015 1:02 pm
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
- Sun Aug 23, 2015 9:45 pm
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
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/userpics/10002/jmm15-nelson-segerman.jpg ""Hyperbolic Catacombs," by Roice Nelson (Austin, TX) and Henry Se...
- Sun Aug 23, 2015 8:53 pm
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
- Sun Aug 23, 2015 8:52 pm
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
- Sun Aug 23, 2015 7:47 pm
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
- Sun Aug 23, 2015 7:19 pm
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
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 Antiprism). Not sure what I've done differently, or am missing. Anyhow, neither of these is the midsphere center, or midel...
- Sun Aug 23, 2015 7:00 pm
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
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 self-dual if reciprocated in the ellipsoid. You wrote: "If you scale the chesterhedron model to make the edge-tangent...
- Sun Aug 23, 2015 5:42 pm
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
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 be helpful in my understanding in these matters. Thx. I, of course, in my research, encountered the vertex fig...
- Sun Aug 23, 2015 5:31 pm
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
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 "duality" is a many faceted thing;):) And, I have been aware that projective geometry is a beast unto itself. None ...
- Sun Aug 23, 2015 9:08 am
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
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 continue to advance in my study of projective geometry. I am very much enjoying this adventure:)...
Swami
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 continue to advance in my study of projective geometry. I am very much enjoying this adventure:)...
Swami
- Sun Aug 23, 2015 5:55 am
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
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:) As for the Chestahedron and the midellipsoid I discovered for it - yes, I got "lucky":) And, as noted...
- Sun Aug 23, 2015 5:38 am
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
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)) [-c: reciprocation centre] Notes: - E, A, & V centres (i.e. with inversion) gave same results as without - at least...
- Sun Aug 23, 2015 5:23 am
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065
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 can produce similar, but slightly different results, depending on how the "center of reciprocation" is calculated. ...
- Fri Aug 21, 2015 5:47 pm
- Forum: Stella Forum
- Topic: How does Stella4D determine the dual?
- Replies: 31
- Views: 108065