The Pyritohedral Golden Icosahedron

Both the Platonic icosahedron and the golden icosahedron have twenty triangular faces. In the Platonic version, these faces are all equilateral triangles. The golden icosahedron has eight such triangles, but the other twelve are golden triangles, which have a leg-to-base ratio which is the golden ratio. These golden triangles appear in pairs, and the six pairs are arranged in such a way as to make this a solid with pyritohedral symmetry: the symmetry of a standard volleyball.

A net for the golden icosahedron appears below. Both images were made using a program called Stella 4d, which you can try for free right here.

A Non-convex, Pyritohedral Dodecahedron with Non-convex Pentagonal Faces

I created this using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php. Starting with the Platonic dodecahedron, I dropped the symmetry of the model down from icosahedral to tetrahedral, then stellated it six times. I also put the resulting polyhedron into “rainbow color mode” before making this .gif image.

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 Compound of an Octahedron and a Pyritohedral Dodecahedron

compound of a pyritohedral dodecahedron and an octahedron

I made this using Stella 4d: Polyhedron Navigator. You can try this program for free at http://www.software3d.com/Stella.php.

A Pyritohedral Polyhedron Featuring Only Hexagonal and Rhombic Faces

Zonohedrified Stellated Icosa face-based zonohedron starting with a pyritohedral dodecahedron

Software credit: I used Stella 4d to make this, and you can find that program at http://www.software3d.com/Stella.php, with a free trial download available there.

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

A Pyritohedral and Pentagon-Faced Polyhedron

pyritohedral 36 pentagons

Twelve of the faces of this polyhedron are pink, and the other twenty-four are blue. It has no faces which are not pentagons. I made it using Stella 4d: Polyhedron Navigator, which is avialable at http://www.software3d.com/Stella.php

A Large Collection of Polyhedra with Pyritohedral Symmetry, Some of Them Chiral

pyritohedral polyhedron featuring octagons

pyritohedral polyhedron featuring heptagons

14 hexagons of two types 12 heptagons 24 pentagons 50 faces in all and pyritohedral

hourglasses

hexagons and pyritohedral rectangles

Dual of Convex hull

idea pyrito

interesting pyritohedral convex hull from 8Apr15 blog post

kites and paired irregular pentagons pyritohedral

pyrito and interesting

Pyrito Compound of enantiomorphic pair

jsfasdkf

pyritic Convex hull

pyrito Dual of Convex hull

pyrito RCO

pyrito stellation of great Dodeca 2 actuaaly these are tetstells

pyrito stellation of great Dodeca

pyrito

pyritohedral and chiral 3

pyritohedral and chiral

idea stellated

pyritohedral Convex hull

pyritohedral double snub cube

pyritohedral dual

Pyritohedral Icosahedral with added octagons

pyritohedral triangles and heptagons again

pyritohedral triangles and heptagons

pyritol

pyritoonvex hull

pyritowrinkles by no of sides

Unmessnamed Dual

Faceted Dual pyritohedral and chiral

Final Stellation of Compound of Icosidodeca and Trunc Cube also pyritohedral

chiral Dual of Convex hull

Unnamed Dual

It’s quite an informal way to define it, but pyritohedral symmetry is the symmetry-type of a standard volleyball. These images of pyritohedral polyhedra were made using Stella 4d, software available at http://software3d.com/Stella.php.

A Pyritohedral Coloring-Scheme for the Truncated Icosahedron

pyritohedral coloring of the truncated icosahedron

While the polyhedron above, informally known as the “soccer ball,” has icosidodecahedral symmetry, its coloring-scheme does not. Instead, I colored the faces in such a way that the coloring-scheme has pyritohedral symmetry — the symmetry of a standard volleyball. This rotating image was made with Stella 4d, a program you can buy, or try for free, right here: http://www.software3d.com/Stella.php.