It’s hard to get regular pentagons, regular star pentagons, regular decagons, and related polygons to tessellate the plane while maintaining radial symmetry. This is my latest attempt.
I made both of these while playing around with Stella 4d: Polyhedron Navigator, a program you can try out for free at http://www.software3d.com/Stella.php.
Symmetrohedra are polyhedra with some form of polyhedral symmetry, and many (not necessarily all) regular faces. The first two symmetrohedra here each include four regular enneagons as faces.
The next two symmetrohedra each include four regular dodecagons as faces.
All four of these were made using Stella 4d, which you can try out for free at http://www.software3d.com/Stella.php.
To make this polyhedron using Stella 4d (available here), I began with the dodecahedron, dropped the symmetry of the model from icosahedral to tetrahedral, and then stellated it thirteen times.
This stellated polyhedron has pyritohedral symmetry, but this is easier to see in its convex hull:
The eight blue triangles in this convex hull are equilateral, while the twelve yellow ones are golden isosceles triangles.
All of these polyhedra were made using Stella 4d: Polyhedron Navigator. If you’d like to try this program yourself, simply visit http://www.software3d.com/Stella.php, where a free trial download is available.
I created these with Stella 4d, which you may try for free at this website. To make a given polyhedral stellation appear larger, simply click on it.
Each of these has a tetrahedron hidden from view in the center.
These were made using Stella 4d, which you may try for yourself here.