New here; first post. Excited to explore Stella4D, & polytopes!:)...

(This post got a bit long, but I've endeavored to be as concise and clear as possible, while adequately presenting the matter. Read on, if you will... TIA.)

So, my current object of study is the "Chestahedron". (See interactive version, & dynamic.) Basically, it's a heptahedron where the 7 faces are of equal area (3 congruent equilateral triangles, & 3 congruent kites).

Among

*many*things, I'm interested in the dual of the Chestahedron. All references I have found give methods for a REGULAR solid, which the Chestahedron is

*not*. I have generated some interesting "duals" based on some of these methods, plus some variations I have concocted.

I have made a Chestahedron by importing an OFF of points only, then Stella4D made the convex hull, and then the dual. Here is the STEL file.

One correspondent claims the Chestahedron is self-dual, though I can graphically demonstrate it is not (at least if I'm correct in my understanding: the dual must fit inside its reciprocal and have each vert uniquely touch a face of its reciprocal).

Now, for a regular polyhedron, a simple method to get the dual is to locate a vert at the centroid of each face, then connect to form the new solid. But this clearly does not generate a proper dual for the Chestahedron (again, I can graphically demonstrate this).

Another method is to determine the midsphere (if such exists), then create new edges perpendicular to the tangent point at the original edge. Here's an interesting animated visualization of the idea. This is often called the Dorman-Luke construction, though in my research it is stated as appropriate for regular solids. I have considered implementing this method, but have opted for an easier one first...

Dual from the inELLIPSOID. The Chestahedron is not regular, nor does it admit a circumsphere, midsphere, or insphere. It's a bit elongated, so an ellipsoid seemed a reasonable alternative. In fact, I have found the circumellipsoid, and the inellipsoid. Using the in ellipsoid, I locate a unique point on each face (not a centroid). Connecting these yields a solid that at least conforms to the requirements for a proper dual. Also, it is like a (inverted) Chestahedron, but with different proportions.

So, here is the "dual" from the inellipsoid method...

(Note: This is very close to a Chestahderon with an areas ratio of 2:1 (in a family I call "k-Chestahedra", where k is a non-zero positive integer representing the ratio of areas). Since I did this graphically/experimentally, not analytically, it is inexact (though quite close), which might account for the numbers not showing it to be exactly a k-2 Chestahedron. So, this matter is still open.

FYI, in addition to the above investigation, and using Stella4D, I have attempted to use Macaulay2, but with limited success. It does find duals (the 'polar'), but is not a geometry app, per se, but for algebraic geometry/commutative algebras - and therein lies a snag:( The dual it returns

*almost*conforms as a proper dual, but it includes a "ray". Unfortunately, at this time, I do not fully understand what this is in the context of the realm of Macaulay2. It's a direction associated with a "cone". Anyhow, regardless of what it is, it prevents me from getting a proper solid to inspect - though I have looked at what it generated, and it does not look "correct".

In closing, I seek the dual of the Chestahedron. My understanding of a "dual" is decent, but not complete, and possibly in error in some aspects. I'm hoping to clear up all of this, and come to a properly informed presentation of the Chestahedron dual.

And again, I ask, How does Stella4D determine the dual? TIA.

Swami Polytope