I made this using *Stella 4d*, which you can find here.

# Monthly Archives: April 2015

# New “Near-Miss” Candidate?

As a proposed new “near-miss” to the Johnson solids, I created this polyhedron using *Stella 4d*, which can be found for purchase, or trial download, here. To make it, I started with a tetrahedron, augmented each face with icosidodecahedra, created the convex hull of the resulting cluster of polyhedra, and then used *Stella*‘s “try to make faces regular” function, which worked well. What you see is the result.

This polyhedron has no name as of yet (suggestions are welcome), but does have tetrahedral symmetry, and fifty faces. Of those faces, the eight blue triangles are regular, although the four dark blue triangles are ~2.3% larger by edge length, and ~4.6% larger by area, when compared to the four light blue triangles. The twelve yellow triangles are isosceles, with their bases (adjacent to the pink quadrilaterals) ~1.5% longer than their legs, which are each adjacent to one of the twelve red, regular pentagons. These yellow isosceles trapezoids have vertex angles measuring 61.0154º. The six pink quadrilaterals themselves are rectangles, but just barely, with their longer sides only ~0.3% longer than their shorter sides — the shorter sides being those adjacent to the green quadrilaterals.

The twelve green quadrilaterals are trapezoids, and are the most irregular of the faces in this near-miss candidate. These trapezoids have ~90.992º base angles next to the light blue triangles, and ~89.008º angles next to the pink triangles. Their shortest side is the base shared with light blue triangles. The legs of these trapezoids are ~2.3% longer than this short base, and the long base is ~3.5% longer than the short base.

If this has been found before, I don’t know about it — but, if you do, please let me know in a comment.

UPDATE: It turns out that this polyhedron has, in fact, been found before. It’s called the “tetrahedrally expanded tetrated dodecahedron,” and is the second polyhedron shown on this page. I still don’t know who discovered it, but at least I did gather more information about it — the statistics which appear above, as well as a method for constructing it with *Stella*.

# A Faceting of the Truncated Dodecahedron, Together with Its Dual

This faceting of the truncated dodecahedron, one of many, was made with *Stella 4d*, software you can buy, or try for free, here. Here is its dual, below.

# Albert Einstein, on Morality

# On Consistent and Inconsistent Combining of Chiralities, Using Polyhedral Augmentation

For any given chiral polyhedron, a way already exists to combine it with its own mirror-image — by creating a compound. However, using augmentation, rather than compounding, opens up new possibilities.

The most well-known chiral polyhedron is the snub cube. This reflection of it will be referred to here using the letter “A.”

To avoid unnecessary confusion, the same direction of rotation is used throughout this post. Apart from that, though, the image below, “Snub Cube B,” is the reflection of the first snub cube shown.

There are many ways to modify polyhedra, and one of them is augmentation. One way to augment a snub cube is to attach additional snub cubes to each square face of a central snub cube, creating a cluster of seven snub cubes. In the next image, all seven are of the “A” variety.

If one examines the reflection of this cluster of seven “A” snub cubes, all seven, in the reflection, are of the “B” variety, as shown here:

Even though one is the reflection of the other, both clusters of seven snub cubes above have something in common: consistent chirality. As the next image shows, inconsistent chirality is also possible.

The cluster shown immediately above has a central snub cube of the “A” variety, but is augmented with six “B”-variety snub cubes. It therefore exhibits inconsistent chirality, as does its reflection, a “B” snub cube augmented with “A” snub cubes:

With simple seven-part snub-cube clusters formed by augmentation of a central snub cube’s square faces by six snub cubes of identical chirality to each other, this exhausts the four possibilities. However, multiplying the possibilities would be easy, by adding more components, using other polyhedra, mixing chiralities within the set of polyhedra added during an augmentation, and/or mixing consistent and inconsistent chirality, at different stages of the growth of a polyhedral cluster formed via repeated augmentation.

All the images in this post were created using *Stella 4d*, which you can try for yourself at this website.

# Compound of the Great and Small Stellated Dodecahedra

In this compound, as shown above, the small stellated dodecahedron is yellow, while the red polyhedron is the great stellated dodecahedron. Below, the same compound is colored differently; each face has its own color, unless faces are in parallel planes, in which case they have the same color.

Making a physical model of this compound would have taken most of the day, if I did it using such things as posterboard or card stock, compass, ruler, tape, scissors, and pencils. For the first several years I built models of polyhedra, starting about nineteen years ago, that was how I built such models. The virtual polyhedra shown above, by contrast, took about ten minutes to make, using *Stella 4d: Polyhedron Navigator*, which you can try for free, or purchase, here.

There’s also a middle path: using *Stella* to print out nets on cardstock, cutting them out, and then taping or gluing these *Stella*-generated nets together to make physical models. I haven’t spent much time on this road myself, but I have several friends who have, including the creator of Stella. You can see some of his incredible models here, and some amazing photographs of other Stella users’ paper models, as well as some in other media, at this website.

# Two Different Forty-Part Polyhedral Compounds

The polyhedron above is a compound of twenty cubes and twenty octahedra, colored by symmetry-based face-type. If the same compound is viewed in “rainbow color mode,” it looks like this:

With this particular compound, though, there are two versions — without taking coloring into consideration at all. The other version simply has the twenty cubes and twenty octahedron in a different, but still symmetrical, arrangement:

The compound above uses this second arrangement, colored by face type, and the next image is the same (second) compound, but in “rainbow color mode.”

These rotating polyhedral images were made with *Stella 4d*, software you can try for yourself, right here.