An Rhombic Dodecahedral Lattice, Made of Icosahedra

Augmented Icosa

I used Stella 4d: Polyhedron Navigator to make this. You can find this program at

Posted in Mathematics | Tagged , , , , | Leave a comment

Compound of Three Square Dipyramids

The reason I am not calling this a compound of three octahedra is that the faces of the dipyramids aren’t quite equilateral. They are, however, isosceles, with triangle-bases where  pyramid-bases of the same color meet each other.

compound of three square dipyramids

Posted in Uncategorized | Tagged , , , , , | Leave a comment

Happy Second Anniversary of Your Simulated Existence


The world ended on this day in 2012 — December 21 — when the Mayan calendar began a new cycle. We now secretly live in a computer simulation run by highly advanced ancient Mayan aliens. They have authorized me to wish you a happy second anniversary of the end of your previous existence.

[Image credit: within this simulation, you can find this picture at]

Posted in Life, Religion | Tagged , , , , , , , | Leave a comment

A Partially-Invisible Rhombicosidodecahedron, and One of Its Stellations


The polyhedron above originally had thirty yellow square faces, but I rendered them invisible so that the interior structure of this polyhedron could be seen.

When stellating such a partially-invisible figure, the new faces “inherited” from the “parent polyhedron” are either visible or invisible, depending on which type of face they are derived from. This makes for a very unusual look for some stellations, such as this, the rhombicosidodecahedron’s 50th:

Rhombicosidodeca w inv squares 50th stellations

I created these images using a program called Stella 4d: Polyhedron Navigator. You may try it for yourself at

Posted in Mathematics | Tagged , , , , , , | Leave a comment

The Seven Zonish Dodecahedra with Zones Added Based on Faces, Edges, and/or Vertices

If a zonish dodecahedron is created with zones based on the dodecahedron’s vertices, here is the result.

zonish dod v

If the same thing is done with edges, this is the result — an edge-distorted version of the great rhombicosidodecahedron.

zonish dodeca edges only

Another option is faces-only. Although I haven’t checked the bond-lengths, this one does have the general shape of the most-symmetrical 80-carbon-atom fullerene molecule. Also, this shape is sometimes called the “pseudo-truncated-icosahedron.”

zonish dodec faces only

The next zonish dodecahedron has had zones added based on the dodecahedron’s faces and edges, both.

zonish dodeca e & f

Here’s the one for vertices and edges.

zonish dodec v & e

Here’s the one for faces and vertices.

zonish dodec v & f

Finally, the last of this set of seven has had zones added based on all three: faces, vertices, and edges.

zonish dodec vfe

All seven of these were made with Stella 4d, which is available at

Posted in Mathematics | Tagged , , , , , , , | Leave a comment

Thirty-Three Polyhedra with Icosidodecahedral Symmetry

Note:  icosidodecahedral symmetry, a term coined (as far as I know) by George Hart, means exactly the same thing as icosahedral symmetry. I simply use the term I like better. Also, a few of these, but not many, are chiral.

15 reg decagons 30 reg hex 120 trapsl

15x5 20x61 30x62 120x5 182 total

20x9 12x5 and 60x6 and 60x5 total 152

360 triangles

362 faces 12x10 20x18 30x10' 60x7 60x3 and 120 tiny triangles

480 triangular faces

542 faces incl 30x16 20x12 60x6 60x6' 12x5 60x7 120x5 and 120 timy triangles


The images directly above and below show the shape of the most symmetrical 240-carbon-atom fullerene.


chiral convex hull Convex hull

compound five tet

The image above is of the compound of five tetrahedra. This compound is chiral, and the next image is the compound of the compound above, and its mirror-image.

Compound of enantiomorphic pair

Comvnvex hjsdgaull

Conhgvedsfasdfx hull

Convedsfasdfx hull

Convex hjsdgaull

Convex hulfsgl

Convex hullll

Dual of Cjhfonvex hull

Dual of Convex hull

Dual of Convex hullb

dual of kite-variant of snub dodec

Faceted Convex hull augmentation with length 5 prisms

Faceted Convex hull

features twenty reg dodecagons 12 reg pents 60 kites 60 rectangles

In the next two, I was experimenting with placing really big spheres at the vertices of polyhedra. The first one is the great dodecahedron, rendered in this unusual style, with the faces rendered invisible.

great dodec


icosa variant

kites and triangles

rhombi and octagons

Stellated Poly



I made these using Stella 4d: Polyhedron Navigator. You may try this program for free at

Posted in Mathematics | Tagged , , , , , , | Leave a comment

A Radial Tiling of Regular Decagons and Bowtie Hexagons

decagon and bowtie hexagons

This tiling could be continued indefinitely, while still maintaining its five-fold radial symmetry, giving it the overall appearance of a pentagon.

Posted in Art, Mathematics | Tagged , , , , , , , , , , , , , , , | Leave a comment