Two Different Forty-Part Polyhedral Compounds

Cubes 20 A

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:

Cubes 20 Octa 20 A

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:

Cubes 20 Octa 20 B

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.”

Cubes 20 B

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

Three Polyhedra, Each Featuring One Dozen Regular Icosagons

Icosagons are polygons with twenty sides, and do not appear in any well-known polyhedra. The first of these three regular-icosagon-based polyhedra has 122 faces.

122 faces including 12 regular icosagons

The second of these polyhedra, each of which bears an overall resemblance to a dodecahedron, has 132 faces.

132 faces including 12 regular icosagons

Finally, the third of these polyhedra has a total of 152 faces.

152 faces ncluding 12 regular icosagonsI used Stella 4d to make each of these virtual polyhedron models — and you may try this program for free at

A Polyhedral Demonstration of the Fact That Twenty Times Four Is Eighty

20 times 4 is 80

The Platonic solid known as the icosahedron has twenty triangular faces. This polyhedron resembles the icosahedron, but with each of the icosahedron’s triangles replaced by a panel of four faces:  three isosceles trapezoids surrounding a central triangle. Since (20)(4) = 80, it is possible to know that this polyhedron has eighty faces — without actually counting them.

To let you see the interior structure of this figure, I next rendered its triangular faces invisible, to form “windows,” and then, just for fun, put the remaining figure in “rainbow color mode.”

20 times 4 is 80 version twoI perform these manipulations of polyhedra using software called Stella 4d. If you’d like to try this program for yourself, the website to visit for a free trial download is

Do Not Drink the Twenty Proof Gasoline!


We’ve all seen labels like this, stuck to gasoline pumps. While filling up my car’s gas tank earlier today, I felt compelled to take a picture of this familiar label — because I suddenly realized that what this small sign actually means is that the alcohol content of the gasoline being sold (in an area where liquor sales are illegal, no less) might be as much as twenty proof.

Twenty proof gasoline. Twenty proof gasoline! One never thinks of it this way, but it is both mathematically and chemically accurate. There are many different alcohols, but the one people drink for purposes of intoxication, and the one found in this gasoline, are the exact same molecule: C2H5OH. I then realized that the people who design these labels are being sneaky with the wording on purpose, for they don’t put “contains alcohol,” or anything like that, on these stickers found on gas pumps all over the place.

The reason for use of the official, less-familiar chemical term “ethanol” then became both obvious, and horrifying, all at once. Gas pumps must be labeled this way because there are people out there who are so incredibly stupid that they would actually drink gasoline if they knew it contained, well, booze.

What’s more, there is an unwritten assumption in play here, and I think (or at least hope) it is a valid one: anyone sufficiently educated to know that “ethanol” and the “the alcohol people drink to get drunk” are synonyms is also, presumably, smart enough to know better than to drink gasoline. Drinking gasoline would, of course, be dangerous in the extreme. Even inhaling gasoline fumes is hazardous, but drinking the stuff would be far worse. Consuming enough of this ethanol-containing gasoline to actually get drunk would, in fact, very likely be fatal, due to the mixture of toxic hydrocarbons present, in addition to the alcohol. The most toxic component of gasoline with which I am familiar is benzene, a potent carcinogen. Benzene is really nasty stuff, if it somehow makes it into a human body.

So, for the record, do not drink the up-to-twenty-proof gasoline — even though that is an accurate way to describe it.

Cluster of Twenty Snub Dodecahedra


Cluster of Twenty Snub Dodecahedra

This was made by the augmentation of an icosahedron, using snub dodecahedra on each of its twenty faces. I used software available at