Three Convex Polyhedra with Tetrahedral Symmetry, Each Featuring Four Regular Enneagons


In addition to the four regular enneagons, the polyhedron above also has rhombi and isosceles triangles as faces. The next one, however, adds equilateral triangles, instead, to the four regular enneagons, along with trapezoids and rectangles.


Only the last of these three truly deserves to be called a symmetrohedron, in my opinion, for both its hexagons and enneagons are regular. Only the “bowtie trapezoid” pairs are irregular.

four reg enneagons and four reg hexagons and six pairs of bowtie hexagons

All three of these polyhedra were created using software called Stella 4d: Polyhedron Navigator, which I use frequently for the blog-posts here. You can try it for free at this website.

Nine (2015) / Nine (2013)

First, the newer version I just made:

nine 2015

Next, the 2013 version, which I recently found, along with a bunch of other previously-lost stuff from around then. The two are simply color-inversions of each other, according to the rules for color-inversion used by MS-Paint.

nine 2013

A Polyhedron Featuring Enneagons and Two Types of Pentagon

only the blue pentagons are regular

Enneagons are nine-sided polygons, and some people prefer to call them “nonagons.” I try not to use the latter term because it mixes Greek and Latin word-parts, which the former term, derived purely from Greek, does not do.

This was made using Stella 4d, a program you may try for yourself here.

A Polyhedron Featuring Twelve Regular Pentadecagons, and Twenty Regular Enneagons

152 faces featuring regular enneagons and pentadecagons

In the last post here, there were two polyhedra shown, and the second one included faces with nine sides (enneagons, also known as nonagons), as well as fifteen sides (pentadecagons), but those faces were not regular.

The program I use to manipulate polyhedra, Stella 4d (available at,  has a “try to make faces regular” function included. When I applied it to that last polyhedron, in the post before this one, Stella was able to make the twenty enneagons and twelve pentadecagons regular. The quadrilaterals are still irregular, but only because squares simply won’t work to close the gaps of a polyhedron containing twenty regular enneagons and twelve regular pentadecagons. These quadrilaterals are grouped into thirty panels of four each, so there are (4)(30) = 120 of them. Added to the twelve pentadecagons and twenty enneagons, this gives a total of 152 faces for this polyhedron.

A Polyhedral Journey, Beginning with a Near-Miss Johnson Solid Featuring Enneagons

When Norman Johnson first found, and named, all the Johnson solids in the latter 1960s, he came across a number of “near-misses” — polyhedra which are almost Johnson solids. If you aren’t familiar with the Johnson solids, you can find a definition of them here. The “near-miss” which is most well-known features regular enneagons (nine-sided polygons):

ennneagonal-faced near-miss

This is the dual of the above polyhedron:

ennneagonal-faced near-miss dual

As with all polyhedra and their duals, a compound can be made of these two polyhedra, and here it is:

ennneagonal-faced near-miss base=dual compound

Finding this polyhedron interesting, I proceeded to use Stella 4d (polyhedron-manipulation software, available at to make its convex hull.

Convex hull of near-miss base-dual compound

Here, then, is the dual of this convex hull:

dual of Convex hull of near-miss base-dual compound

Stella 4d has a “try to make faces regular” function, and I next used it on the polyhedron immediately above. If this function cannot work, though — because making the faces regular is mathematically impossible — one sometimes gets completely unexpected, and interesting, results. Such was the case here.

attempt no make latest polyhedron have regular faces

Next, I found the dual of this latest polyhedron.

attempt no make latest polyhedron have regular faces's dual

The above polyhedron’s “wrinkled” appearance completely surprised me. The next thing I did to change it, once more, was to create this wrinkled polyhedron’s convex hull. A convex hull of a non-convex polyhedron is simply the smallest convex polyhedron which can contain the non-convex polyhedron, and this process often has interesting results.

Convex hull of wrinkled dual

Next, I created this latest polyhedron’s dual:

dual of Convex hull of wrinkled dual

I then attempted “try to make faces regular” again, and, once more, had unexpected and interesting results:

dual of latest polyhedron

The next step was to take the convex hull of this latest polyhedron. In the result, below, all of the faces are kites — two sets of twenty-four each.

convex hull of last polyhedron with two sets of two dozen kites each

I next stellated this kite-faced polyhedron 33 times, looking for an interesting result, and found this:

33rd stellation of latest polyhedron

This looked like a compound to me, so I told Stella 4d to color it as a compound, if possible, and, sure enough, it worked.

33rd stellation of latest polyhedron colored as a compound

The components of this compound looked like triakis tetrahedra to me. The triakis tetrahedron, shown below, is the dual of the truncated tetrahedron. However, I checked the angle measurement of a face, and the components of the above compound-dual are only close, but not quite, to being the same as the true triakis tetrahedron, which is shown below.

Triakistetra -- ANGLES AREN'T QUITE A MATCH for last polyhedron

This seemed like a logical place to end my latest journey through the world of polyhedra, so I did.

Pulsating Cuboctahedron, Featuring Enneagrammic Mandalas


Pulsating Cuboctahedron, Featuring Enneagrammic Mandalas

The enneagramic mandalas on the square faces of this cuboctahedron are from the last post, with inverted-color, smaller versions of the same image on the triangular faces. These mandalas were created using Geometer’s Sketchpad and MS-Paint. Projecting them onto the faces of the cuboctahedron, and then creating this rotating, pulsating .gif image, however, took a third program: Stella 4d, which you can buy, or try for free, at