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.
Any of these rotating polyhedra may be made larger with a click. I created them using Stella 4d, a program you may try (as a free trial download) at http://www.software3d.com/Stella.php.
Any of these rotating polyhedra may be made larger with a single click. All were created using Stella 4d, a program you may try for free at this website: http://www.software3d.com/Stella.php.
Mathematicians have discovered more than one set of rules for polyhedral stellation. The software I use for rapidly manipulating polyhedra (Stella 4d, available here, including as a free trial download) lets the user choose between different sets of stellation criteria, but I generally favor what are called the “fully supported” stellation rules.
For this exercise, I still used the fully supported stellation rules, but set Stella to view these polyhedra as having only tetrahedral symmetry, rather than icosidodecahedral (or “icosahedral”) symmetry. For the icosahedron, this tetrahedral symmetry can be seen in this coloring-pattern.
The next image shows what the icosahedron looks like after a single stellation, when performed through the “lens” of tetrahedral symmetry. This stellation extends the red triangles as kites, and hides the yellow triangles from view in the process.
The second such stellation produces this polyhedron — a pyritohedral dodecahedron — by further-extending the red faces, and obscuring the blue triangles in the process.
The third tetrahedral stellation of the icosahedron produces another pyritohedral figure, which further demonstrates that pyritohedral symmetry is related to both icosidodecahedral and tetrahedral symmetry.
The fourth such stellation produces a Platonic octahedron, but one where the coloring-scheme makes it plain that Stella is still viewing this figure as having tetrahedral symmetry. Given that the octahedron itself has cuboctahedral (or “octahedral”) symmetry, this is an increase in the number of polyhedral symmetry-types which have appeared, so far, in this brief survey.
Next, I looked at the fifth tetrahedral stellation of the icosahedron, and was surprised at what I found.
While I was curious about what would happen if I continued stellating this polyhedron, I also wanted to see this fifth stellation’s convex hull, since I could already tell it would have only hexagons and triangles as faces. Here is that convex hull:
For the last step in this survey, I performed one more tetrahedral stellation, this time on the convex hull I had just produced.