I tried to make a symmetrohedron using regular octadecagons and regular decagons, but that combination forces the octadecagons to overlap, and that causes the would-be symmetrohedron to be non-convex.
I tried to augment these octadecagons with antiprisms, and then form the convex hull of the result. Here’s what I found:
I made these using Stella 4d, which you can try for free right here.
The first of these two polyhedra also includes isosceles triangles, two types of isosceles trapezoids, and twelve regular pentagons.
It is also possible to make a similar polyhedron where the twelve pentagons are replaced by regular decagons, but only by allowing the twenty octadecagons to overlap.
These polyhedra were constructed using Stella 4d, which can be found at http://www.software3d.com/Stella.php.
The octagonal design on each face appears in the last post here, and was made using both Geometer’s Sketchpad and MS-Paint. After cropping this image, I projected it onto the faces of this polyhedron, the rhombic triacontahedron, using Stella 4d, a program you can try for yourself at http://www.software3d.com/php.
RobertLovesPi.net 