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 ...

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 case in a classical sense/space that things get complicated.

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 ...

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 thread. Glad to know you:)

P.S. I have acquired the book, 'Dual Models' (MWenninger), and lectures, 'Lectures on Polytopes' (GunterMZiegler). That should keep me busy for a while! ...

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 symmetrical cases. The dual is a tricky little fellow!:)

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 ...

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 ...

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 ...

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 ...

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

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 ...

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 ...

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 ...

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 wanted to confirm receipt, and show appreciation.

Swami