If I've tried this once, I've tried it at least fifty-five times….

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.

Four Symmetrohedra with Tetrahedral Symmetry

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.

A Pyritohedral, Stellated Polyhedron, and Its Convex Hull

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. 

Dodeca 13th tetstell.gif

This stellated polyhedron has pyritohedral symmetry, but this is easier to see in its convex hull:

Convex hull of the dodecahedron's 13th tetstell.gif

The eight blue triangles in this convex hull are equilateral, while the twelve yellow ones are golden isosceles triangles.

A Miscellany of Polyhedra with Icosidodecahedral Symmetry

20 at 5 and 60 at 5 and 20 at 6 and 60 at 7242 faces mostly quadsConvex hull bstarballAll quads 240 facesAugmented DisdyakistriacontaConvex hull with 12 reg icosagonsConvex hulldecagons enneagons bowtie trapezoids stellated several timesDual of Convex hullashgdfasfdual of polyhedron with 12 pentadecagonsDual of Unnamed DualFaceted Dodeca duAL OF FIRST STELLATION OF ICOSAFaceted DodecaFaceted ICOSIDodecaHEDRAL POLYHEDRONStellated Convex hull 2Stellated Convex hullstellated Dual of Convex hullashgdfasfStellated Poly 4Stellated Poly 3.gifZonohedrified Poly.gifUnnamed Dual nc.gifZonohedrified Icosa.gif

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.

Some Tetrahedral Stellations of the Truncated Cube

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.

Tetrahedrally-Symmetric Creatures with Polyhedral Legs

Each of these has a tetrahedron hidden from view in the center.

augmented-tetra

augmented-tetra-2

augmented-tetra-3

augmented-tetra-4

These were made using Stella 4d, which you may try for yourself here.

A Gallery of Two Dozen Polyhedra with Icosidodecahedral Symmetry, with a Few of Them Chiral

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.

A Gallery of Fourteen Polyhedra with Cuboctahedral Symmetry

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.

A Tetrahedral Exploration of the Icosahedron

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.

Icosa showing tet symm

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.

Icosa showing tet symm stellation 1

The second such stellation produces this polyhedron — a pyritohedral dodecahedron — by further-extending the red faces, and obscuring the blue triangles in the process.

Icosa showing tet symm stellation 2 pyritohedral dodecahedron

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.

Icosa showing tet symm stellation 3

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.

Icosa showing tet symm stellation 4 an octahedron with 2 face types

Next, I looked at the fifth tetrahedral stellation of the icosahedron, and was surprised at what I found.

Icosa showing tet symm stellation 5

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:

Icosa tet sym stellation 5's 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.

Icosa tet sym stellation 5's Convex hull ist stellation