geometrypolyhedra
Does anyone recognize / know the name of the convex polyhedron depicted below
as the intersection of a
Cuboctahedron and a Rhombicdodecahedron?
Please note you have to interpret this picture and extract the intersection.
Its faces are squares, equilateral triangles, and rectangles.
The presence of rectangles takes it out of the realm of the Archimedean
and Johnson solids.
Thanks for any pointers!
Once you glue $S$ and $T$ along their equal perimeters, the result is a unique convex polyhedron, an amazing result of Alexandrov. This theorem is described in many places, including Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Chapter 23, Section 3. It is, however, quite difficult in general to construct the unique polyhedron realized. I wrote a short high-level description of the algorithm of Bobenko and Izmestiev here:
Because $S$ and $T$ have the same perimeter, there are an infinite number of ways those perimeters can be glued together, resulting in a continuum of polyhedra so realized. These continua have been studied in particular cases (cf. Geometric Folding Algorithms), but in general are not well-understood.
I believe it is a gyroelongated hexagonal bipyramid.
https://en.wikipedia.org/wiki/Gyroelongated_bipyramid
The example shown in the Wikipedia article uses equilateral triangles, which results in coplanar faces around the degree-6 vertices, but as you mention, the model uses isosceles triangles.
Best Answer
I believe this is what's referred to as a Cuboctahedron-Rhombic Dodecahedron Compound.
According to Mathworld on compound polyhedron:
See Mathworld on Cuboctahedron-Rhombic Dodecahedron Compound, and in particular, I believe the figure at very bottom, left: the Cuboctahedron-Rhombic Dodecahedron Compound- depicts the figure you've provided above. It's a compound of two (dual) Archimedean polyhedra: the cuboctahedron (pictured immediately below, to left) and the rhombic dodecahedron (image immediately below, to right).
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EDIT: and actually, I think the figure below (right) is what you're after: it is described as the "Cuboctahedron-Rhombic Dodecahedron solid which is common to both polyhedra"; i.e. their intersection. Each figure (above) is the dual of the other. No specific name is given to the convex figure of intersection (below, to right), save for referring to it as a "Cuboctahedron-Rhombic Dodecahedron convex solid."
The page linked is worth a visit, since you can see the figures below, rotate them, and the page provides additional information on the polyhedra, the surface area, volume, etc. (edges, vertices,...). There are also links to other polyhedral compounds.
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