Two Polyhedral Compounds: the Icosidodecahedron with the Truncated Cube, and the Rhombic Triacontahedron with the Triakis Octahedron

Compound of Icosidodeca and Trunc Cube

These two compounds, above and below, are duals. Also, in each of them, one polyhedron with icosidodecahedral symmetry is combined with a second polyhedron with cuboctahedral symmetry to form a compound with pyritohedral symmetry: the symmetry of a standard volleyball.

Compound of RTC and Triakis octahedron also pyritohedral

A program called Stella 4d was used to make these compounds, and create these images. It may be purchased, or tried for free, at this website.

The Greatly Augmented Icosidodecahedron, and Its Dual

Augmented Icosidodeca

If a central polyhedron’s pentagonal and triangular faces are augmented by great dodecahedra and great icosahedra, I refer to it as a “greatly augmented” polyhedron. Here, this has been done with an icosidodecahedron. The same figure appears below, but in “rainbow color” mode.

Augmented Icosidodeca colored rainbow

In the next image, “color by face type,” based on symmetry, was used.

Augmented Icosidodeca colored by face type

The next image shows the dual of this polyhedral cluster, with face color chosen on the basis of number of sides.

Augmented Icosidodeca colored by whether sides have 5 or 16 sides

Here is another version of the dual, this one in “rainbow color” mode.

Augmented Icosidodeca colored rainbow DUAL

Finally, this image of the dual is colored based on face type.

Augmented Icosidodeca colored by face type DUAL

These six images were made with Stella 4d, which may be found here.

Combining Octahedral and Icosahedral Symmetry to Form Pyritohedral Symmetry

Compound of Octa and Icosa

Pyritohedral symmetry, seen by example both above and below, is most often described at the symmetry of a volleyball:


[Image of volleyball found here.]

To make the rotating polyhedral compound at the top, from an octahedron and an icosahedron, I simply combined these two polyhedra, using Stella 4d, which may be purchased (or tried for free) here.

In the process, I demonstrated that it is possible to combine a figure with octahedral (sometimes called cuboctahedral) symmetry, with a figure with icosahedral (sometimes called icosidodecahedral) symmetry, to produce a figure with pyritohedral symmetry.

Now I can continue with the rest of my day. No matter what happens, I’ll at least know I accomplished something.

Icosidodecahedral Stained Glass

icosidodecahedral stained glass

Polyhedra are one of the areas (there are at least a few others) where the fields of mathematics and art intersect. Stella 4d, the program I used to make this image, is a great tool for the exploration of this region of intersection. This software may be tried for free right here.

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

A Collection of Rotating Polyhedra with Icosidodecahedral Symmetry

I’ve received a request to slow down the rotational speed of the polyhedral models I make and post here, and am going to try to do exactly that. First, though, I need to empty my collection of already-made image files which haven’t yet been posted, so that I can start again, with models which rotate more slowly, after deleting all the “speedy” ones. From my backlog of polyhedral images to post, then, here are most of the ones with icosidodecahedral symmetry.

60 hexagons and 30 rhombi

60 rhombi and 120 trapezoids92 faces including 20 enneagons120 of traingle A and 120 of triangle B and 60 rhombi for 300 faces in all

The next one shown has 362 faces — the closest I have come, so far, to a polyhedron with a number of faces which matches the number of days in a year.

362 faces close to a year

big Convex hull

bowtie polyhedron with 20 enneagons and 12 decagons

Convex hhgdull


cool too

irregular pentagons and hexagons

The next one is a variant of the rhombic enneacontahedron, with that polyhedron’s wide rhombic faces replaced by kites, and its narrow rhombi replaced by pairs of isosceles triangles.

kite and triangle variant of the REC

multiple stellated pentagonal dokaiheptacontahedrongif


Stellated Convex hull

Stellated Convex hull 2

Stellated Convex hull 3I call this next one a “thrice-truncated rhombic triacontahedron.”

Thrice-truncated RTCIn the remaining polyhedral images in this post, some faces have been rendered invisible. I do this, on occasion, either so that the front and back of the polyhedra can be seen at the same time, or simply for aesthetic reasons.

CoGSHSnvex hhgdull

Expanded GRID shell

Stellated Convex hull 2b

Stellated Convex hull 3b

All of these images were created using Stella 4d:  Polyhedron Navigator. If you’d like to try this program for yourself, the website to visit for a free trial download is