Talk:Acrohedron

Heptagonal prism is uniform

"The smallest convex regular polygon not found in the uniform or Johnson solids is the heptagon." However the heptagonal prism is listed as a 7-4-4 acrohedron in the next sentence is a uniform Johnson solid. It would seem that for any n  then the n -gonal prism is a n -4-4 acrohedron. I think something else is meant. Sycamore916 (talk) 03:40, 26 December 2022 (UTC)

Is this a known 3-4-5 acrohedron?

Polyhedron in question.

I'm wondering if this 3-4-5 acrohedron is known (image on the right). It has 16 faces, so it's not setting any records, but it appears in some constructions of Stewart toroids. Sycamore916 (talk) 00:51, 16 January 2023 (UTC)

Stewart gives a gyrated version of this polyhedron in Adventures among the toroids, with the name $A_{5}''m^{-2}$. It appears several times in the section "Accordion polyhedra". That one is probably more useful for constructing toroids since its pentagonal faces are parallel. Sycamore916 (talk) 23:00, 28 January 2023 (UTC)

This polyhedron can be found here (it's "biaugQ2.5modeC"). So I think it was known to Stewart. I don't know it exactly, but I'm very familiar the units it's made up of, "biaugQ2.5modeC = half A" and " = half B" on the aforementioned site. To me they can be gateways into the art of Stewart toroids with irregular "cross-sections"/pseudofaces. (Imagine the 3D formations of Chapter 7, but made of the irregular-face-containing polyhedra briefly described in Chapter 20, in such a way that the polyhedra blend at the irregular faces and produce a (R) result. Some of them merely resemble bloated versions of other toroids, because their components are just expanded versions of known polyhedra, but others are more unique.) Oh, and one can also make a relatively compact (R) toroid of genus 1 out of 3 of this polyhedron and 3 J63, blending them at the pentagons. It has some coplanar triangular faces (but that could be avoided by inserting 6 P5 in between the other parts), an icosahedron-shaped dent on one side, and a dent reminiscent of W' (or trigyrate J92) on the other side. I'd upload a picture but I'm wary of using the wiki for social purposes. The New Kid (talk) 10:12, 2 April 2023 (UTC)