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


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


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


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


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.

My Third Solution to the Zome Cryptocube Puzzle

The President of the Zometool Corporation, Carlos Neumann, gave me a challenge, not long ago: find a solution to the Zome Cryptocube puzzle which uses only B0s, which I call “tiny blue struts.” For the Cryptocube puzzle, though, these “blue” struts actually appear white. Carlos knows me well, and knows I cannot resist a challenge involving Zome. Here is what I came up with, before the removal of the black cube, which is what the Zome Cryptocube puzzle starts with.


In a “pure” Crypocube solution, the red Zomeballs would also be white — not just the “blue” struts. However, when Carlos issued this challenge, I was at home, with all the white Zomeballs I own located at the school where I teach — so I used red Zomeballs, instead, since I had them at home, and did not wish to wait.

Here’s what this Cryptocube solution looks like, without the black cube’s black struts. You can still “see” the black cube, though, for the black Zomeballs which are the eight corners of the black cube are still present. As is happens, this particular Cryptocube solution has pyritohedral symmetry — better known as the symmetry of a standard volleyball.


While the Cryptocube puzzle is not currently available on the Zome website, http://www.zometool.com, it should be there soon — hopefully, in time for this excellent Zome kit to be bought as a Christmas present. Once a child is old enough so that small parts present no choking hazard, that child is old enough to start playing with Zome — and it is my firm belief that such play stimulates the intellectual growth of both children and adults. As far as a maximum age where Zome is an appropriate Christmas gift, the answer to that is simple: there isn’t one.

Also: while I do openly advertise Zome, I do not get paid to do so. I do this unpaid advertising for one reason: I firmly believe that Zome is a fantastic product, especially for those interested in mathematics, or for those who wish to develop an interest in mathematics — especially geometry. Also, Zome is fun!

A Compound of the Octahedron, and a Pyritohedral Dodecahedron

Compound of octa and pyritohedral dodeca

This compound is the first I have seen which combines a Platonic solid (the blue octahedron) with a pyritohedral modification of a Platonic solid. Here’s what a pyritohedral dodecahedron looks like, by itself:

pyritohedral dodecahedron

Stella 4d: Polyhedron Navigator was used to make these — software you can try right here: http://www.software3d.com/Stella.php.

A Hybrid Polyhedron: The “Offspring” of Jessen’s Icosahedron and the Great Dodecahedron

I stumbled upon this interesting hybrid of two well-known polyhedra, while simply playing around with Stella 4d, the software I use to make these rotating polyhedral images (you can try a free trial download of it here).

Jessens icosa meets the great dodeca

The faces of the above polyhedron are twelve modified regular pentagons, each with a triangular piece removed which contained one of the pentagon’s edges. Therefore, it would also be correct to refer to these modified pentagons as non-convex hexagons. These modified pentagons interpenetrate, so all that can be seen are triangular “facelets” — the parts of the faces which are not hidden inside the polyhedron. Each of these facelets is a golden gnomon (an obtuse, isosceles triangle with a base:leg ratio which is the golden ratio), and these golden gnomons come in two sizes. The larger ones were “inherited” from Jessen’s icosahedron, and there are twelve of them. The smaller golden gnomons, on the other hand, were “inherited” from the great dodecahedron, and are twenty-four in number, in eight sets of three. Like Jessen’s icosahedron itself, but unlike the great dodecahedron, this hybrid has pyritohedral symmetry.

For more information about Jessen’s icosahedron, please visit this site at Wolfram Mathworld. Also, here is an image of Jessen’s icosahedron, one of the two “parents” of the hybrid above.

Jessens Icosa

While Jessen’s icosahedron is a relatively new discovery (Børge Jessen revealed it to the world in 1967), the hybrid’s other “parent,” the great dodecahedron, has been known for much longer; Louis Poinsot discovered it in 1809, according to this source. Here’s an image of the great dodecahedron.

Great Dodeca

As you can see, the smaller golden gnomons found in the hybrid above were “inherited” from the great dodecahedron, while the larger ones came from the six indented face-pairs found in Jessen’s icosahedron.

A well-known property of Jessen’s icosahedron is that it is “shaky,” unlike most polyhedra, which are rigid. A physical model of Jessen’s icosahedron, made from paper and tape, can, in fact, be collapsed to form an octahedron. While I suspect that a physical, paper-and-tape model of this newly-discovered hybrid polyhedron would share these properties (“shakiness,” and at least some degree of collapsibility), I have not (yet) tested this conjecture.

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.

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.

A Survey of Polyhedra with Pyritohedral Symmetry

The simplest way for many to understand pyritohedral symmetry is simply to realize that it is the symmetry of the seams in a volleyball. The first time I encountered this unusual symmetry-type was in the golden icosahedron I blogged about here, a figure which much resembles this pyritohedral icosahedron, except the dozen isosceles triangles in this one have a leg-to-base ratio which is not the golden ratio.

non golden pyritohedral icosahedron

Earlier today, I went on a search for polyhedra with pyritohedral symmetry. I found several, but the worthwhile findings from the search are far from exhausted. Here are some others I found, exploring and manipulating polyhedra using Stella 4d, which you can try at this website.

another pyritohedral version of an icosahedronIn the version of the pyritohedral icosahedron above, the twelve green triangles have become heptagons which use very short sides to approximate triangles. The one below is of a similar figure, but one in which truncations has happened, so I call it a truncated pyritohedral icosahedron.

pyritohedral version of a truncated icosahedron

There also exist many pyritohedral polyhedra based, more or less, on the cube. These are a few I have found:

pyritohedral cube

pyritohedral cube variant

another pyritohedral cube

Now, is this next one a pyritohedral cube, or a pyritohedral dodecahedron? A case could be made for either, so it inhabits a “gray zone” between varying categories.

pyritohedral dodecahedron

Here is a pyritohedral icosidodecahedron:

pyritohedral icosidoecahedronl

This one could probably be described in multiple ways, also, but it looks, to me, like a rhombic dodecahedron with its six four-valent vertices being double-truncated in a pyritohedral manner, with pairs of isosceles trapezoids appearing where the truncations took place.

Convex hull of icosahedron plus CO

One thing that this one, and the last, have in common is that the largest faces are heptagons. It appears to be a pyritohedral dodecahedron which has been only partially truncated.

12 helptagons and 8 trianlgesl

This survey could not have been performed without a program called Stella 4d, which I rely on heavily for polyhedral investigations. It may be purchased, or tried for free, at http://www.software3d.com/Stella.php.