Octahedra and Truncated Cubes Can Fill Space Without Leaving Any Gaps

Augmented Trunc Cube

I created this using Stella 4d, which you can try for free right here. It’s much like a tessellation, but in three dimensions instead of two.

A Faceted Version of the Truncated Cube

This is the truncated cube, which is one of the Archimedean solids.

Convex hull.gif

To make a faceted version of this solid, one must connect at least some of the vertices in different ways. Doing that creates new faces.

Faceted Trunc Cube 8 hexagons blue and yellow triangles.gif

This faceted version of the truncated cube includes eight blue equilateral triangles, eight larger, yellow equilateral triangles, and eight irregular, red hexagons. It’s easy to spot the yellow and blue triangles, but seeing the red hexagons is harder. In the final picture here, I have hidden all faces except for three of the hexagons, so that their positions can be more easily seen.

Faceted Trunc Cube 8 hexagons blue and yellow triangles some parts hidden.gif

I made all three of these images using Robert Webb’s program called Stella 4d: Polyhedron Navigator. It is available for purchase, or as a free trial download, at http://www.software3d.com/Stella.php.

Three Archimedean Solids Which Fill Space Together: The Great Rhombcuboctahedron, the Truncated Tetrahedron, and the Truncated Cube

To start building this space-filling honeycomb of three Archimedean solids, I begin with a great rhombcuboctahedron. This polyhedron is also called the great rhombicuboctahedron, as well as the truncated cuboctahedron.

Trunc Cubocta honeycomb core

Next, I augment the hexagonal faces with truncated tetrahedra.

Trunc Cubocta honeycomb core plus 1.gif

The next polyhedra to be added are truncated cubes.

Trunc Cubocta honeycomb core plus 2

Now it’s time for another layer of great rhombcuboctahedra.

Trunc Cubocta honeycomb core plus 3

Now more truncated tetrahedra are added.

Trunc Cubocta honeycomb core plus 4

Now it’s time for a few more great rhombcuboctahedra.

Trunc Cubocta honeycomb core plus 5

Next come more truncated cubes.

Trunc Cubocta honeycomb core plus 6

More great rhombcuboctahedra come next.

Trunc Cubocta honeycomb core plus 7

More augmentations using these three Archimedean solids can be continued, in this manner, indefinitely. The images above were created with Stella 4d: Polyhedron Navigator, a program you may try for yourself at http://www.software3d.com/Stella.php.

Three Stellations of the Truncated Cube

12TH STELLATION OF Trunc Cube.gif

The polyhedron above is the 12th stellation of the truncated cube. The one below is the 14th.

14th-stellation-of-trunc-cube

The next one shown is the 18th and final stellation. If stellated again, the result is an ordinary truncated cube.

18th-and-final-stellation-of-trunc-cube

These virtual models were made using Stella 4d, software you may try for yourself at http://www.software3d.com/Stella.php.

 

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.

A Space-Filling Arrangement of Polyhedra Using Truncated Cubes, Rhombcuboctahedra, Cubes, and Octagonal Prisms

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms

This image above has only one polyhedron-type hidden from view, in the center:  a red truncated cube. Next, more of this pattern I just found will be added.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 2

The next step will be to add another layer of blue octagonal prisms.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 3And now, more yellow cubes.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 4This was an accidental discovery I made, just messing around with Stella 4d, a program you may try for yourself at http://www.software3d.com/Stella.php. The next cells added will be red truncated cubes.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 5

Next up, I’ll add a set of pink rhombcuboctahedra.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 6The next set of polyhedra added: some yellow cubes, and blue octagonal prisms.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 7Now I’ll add more of the red truncated cubes.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 8At this point, more yellow cubes are needed.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 9The next polyhedra added will be pink rhombcuboctahedra.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 10

And now, more of the blue octagonal prisms.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 11

As long as this pattern is followed, this may be continued without limit, filling space, without leaving any gaps.

Five of the Thirteen Archimedean Solids Have Multiple English Names

Image

Four Archimedean Solids with Multiple English Names

I call the polyhedron above the rhombcuboctahedron. Other names for it are the rhombicuboctahedron (note the “i”), the small rhombcuboctahedron, and the small rhombicuboctahedron. Sometimes, the word “small,” when it appears, is put in parentheses. Of these multiple names, all of which I have seen in print, the second one given above is the most common, but I prefer to leave the “i” out, simply to make the word look and sound less like “rhombicosidodecahedron,” one of the polyhedra coming later in this post.

Trunc Cubocta

My preferred name for this polyhedron is the great rhombcuboctahedron, and it is also called the great rhombicuboctahedron. The only difference there is the “i,” and my reasoning for preferring the first name is the same as with its “little brother,” above. However, as with the first polyhedron in this post, the “i”-included version is more common than the name I prefer.

Unfortunately, this second polyhedron has another name, one I intensely dislike, but probably the most popular one of all — the truncated cuboctahedron. Johannes Kepler came up with this name, centuries ago, but there’s a big problem with it: if you truncate a cuboctahedron, you don’t get square faces where the truncated parts are removed. Instead, you get rectangles, and then have to deform the result to turn the rectangles into squares. Other names for this same polyhedron include the rhombitruncated cuboctahedron (given it by Magnus Wenninger) and the omnitruncated cube or cantitruncated cube (both of these names originated with Norman Johnson). My source for the named originators of these names is the Wikipedia article for this polyhedron, and, of course, the sources cited there.

Rhombicosidodeca

This third polyhedron (which, incidentally, is the one of the thirteen Archimedean solids I find most attractive) is most commonly called the rhombicosidodecahedron. To my knowledge, no one intentionally leaves out the “i” after “rhomb-” in this name, and, for once, the most popular name is also the one I prefer. However, it also has a “big brother,” just like the polyhedron at the top of this post. For that reason, this polyhedron is sometimes called the small rhombicosidodecahedron, or even the (small) rhombicosidodecahedron, parentheses included.

Trunc Icosidodeca

I call this polyhedron the great rhombicosidodecahedron, and many others do as well — that is its second-most-popular name, and identifies it as the “big brother” of the third polyhedron shown in this post. Less frequently, you will find it referred to as the rhombitruncated icosidodecahedron (coined by Wenninger) or the omnitruncated dodecahedron or icosahedron (names given it by Johnson). Again, Wikipedia, and the sources cited there, are my sources for these attributions.

While I don’t use Wenninger’s nor Johnson’s names for this polyhedron, their terms for it don’t bother me, either, for they represent attempts to reduce confusion, rather than increase it. As with the second polyhedron shown above, this confusion started with Kepler, who, in his finite wisdom, called this polyhedron the truncated icosidodecahedron — a name which has “stuck” through the centuries, and is still its most popular name. However, it’s a bad name, unlike the others given it by Wenninger and Johnson. Here’s why: if you truncate an icosidodecahedron (just as with the truncation of a cuboctahedron, described in the commentary about the second polyhedron pictured above), you don’t get the square faces you see here. Instead, the squares come out of the truncation as rectangles, and then edge lengths must be adjusted in order to make all the faces regular, once more. I see that as cheating, and that’s why I wish the name “truncated icosidodecahedron,” along with “truncated cuboctahedron” for the great rhombcuboctahedron, would simply go away.

Here’s the last of the Archimedean solids with more than one English name:

Trunc Cube

Most who recognize this shape, including myself, call it the truncated cube. A few people, though, are extreme purists when it comes to Greek-derived words — worse than me, and I take that pretty far sometimes — and they won’t even call an ordinary (Platonic) cube a cube, preferring “hexahedron,” instead. These same people, predictably, call this Archimedean solid the truncated hexahedron. They are, technically, correct, I must admit. However, with the cube being, easily, the polyhedron most familiar to the general public, almost none of whom know, let alone use, the word “hexahedron,” this alternate term for the truncated cube will, I am certain, never gain much popularity.

It is unfortunate that five of the thirteen Archimedean solids have multiple names, for learning to spell and pronounce just one name for each of them would be task enough. Unlike in the field of chemistry, however, geometricians have no equivalent to the IUPAC (International Union of Pure and Applied Chemists), the folks who, among other things, select official, permanent names and symbols for newly-synthesized elements. For this reason, the multiple-name problem for certain polyhedra isn’t going away, any time soon.

(Image credit:  a program called Stella 4d, available at www.software3d.com/Stella.php, was used to create all of the pictures in this post.)

Truncated Cube Decorated with Waves

Image

Truncated Cube Decorated with Waves

On the octagonal faces, the waves shown in blue, red, and purple are sine waves of varying wavelength and amplitude. The other three waves were each formed by combining two of these component waves by simple addition — wave interference, in other words. All three possible combinations of two component waves are included.

I needed three different types of software to create this. First, Geometer’s Sketchpad was used to create the multiple-wave image itself. Next, with MS-Paint, I cropped the image and converted it into a usable format. Finally, Stella 4d was used to place the image on the large faces of this truncated cube, and create this rotating .gif file.

Of these three programs, Stella 4d is easily my favorite, and it also happens to be written by a friend of mine. Trial version / purchase information for this program may be found at http://www.software3d.com/stella.php.