# The Icositetrachoron and the Truncated Icositetrachoron, Rotating in Hyperspace

There are six regular, convex four-dimensional polytopes. Five of them correspond on a 1:1 basis with the Platonic solids (as the tesseract corresponds to the cube), leaving one four-dimensional polytope without a three-dimensional analogue among the Platonics. That polychoron is the icositetrachoron, also named the 24-cell and made of 24 octahedral cells. It also happens to be self-dual.

If, in hyperspace, the corners are cut off just right, new cells are created with 24 cubic cells created at the corners, with 24 truncated octahedral cells remaining from the original polychoron. This is the truncated icositetrachoron:

4-dimensional polytopes have 3-dimensional nets. These nets are shown below — first for the icositetrachoron, and then for the truncated icositetrachoron.

I used Stella 4d to create these images. You can try Stella for free at http://www.software3d.com/Stella.php.

# The Dodecagonal Duoprism

There are objects in hyperspace known as duoprisms, which have prismatic cells. This one’s cells are 24 dodecagonal prisms. It was made using Stella 4d, available here.

# Slowly Rotating Hyperdodecahedron

This is the hyperdodecahedron, or 120-cell, one of the six four-dimensional analogs of the Platonic solids. It’s been shown on this blog before, but this image has one major change: a much slower rotational speed. It is my hope that this will help people, including myself, with the difficult task of understanding four-dimensional objects.

This image was created using Stella 4d, a program you can try, as a free trial download, at this website.

# 23-Tex: The Truncated 600-Cell

One of the regular 4-dimensional polytopes, or polychora, is called the 600-cell. If you truncate it, you get the figure above, called “23-tex” for short. Its unit cells are 600 truncated tetrahedra, as well as 120 icosahedra. As shown above, you see only the cells, with space between them so you can see them, and vertices and edges rendered invisible. If the vertices and edges are shown, though, 23-tex looks like this:

Since these are four-dimensional figures, it’s not possible to see what these really look like all at once, in hyperspace, because you, the observer, are trapped in a universe where only three spatial dimensions are easy to perceive. That’s one reason I’m showing these figures with them rotating — as they spin, in hyperspace, the “slice” of the shape which appears as a three-dimensional projection, in our universe, changes. Over time, then, you get to see it all — but, in hyperspace, its appearance would be different. If you try to imagine a living creature living inside a horizontal Euclidean plane, and this creature trying to picture a three-dimensional shape, that can help with understanding the nature of the problem that arises when three-dimensional beings try to picture anything involving a fourth spatial dimension.

Three-dimensional polyhedra can be unfolded, and shown as flat “nets” made out of polygons. Well, what does the “net” of a four-dimensional polytope look like? The answer is one dimension “down” from four: the net is a bunch of three-dimensional polyhedra, stuck together at their faces, just as the nets of polyhedra are made of polygons joined at their edges. Here’s one possible net for 23-tex:

The three images above were produced using Stella 4d: Polyhedron Navigator, software you may try for yourself, here.

## The Icositetrachoron, or 24-Cell: An Oddball In Hyperspace

### Image

In three dimensions, there are five regular, convex polyhedra. Similarly, in five dimensions, there are five regular, convex polytopes. There are also five of them in six, seven, eight dimensions . . . and so on, for as long as care to venture into higher-dimensional realms. However, in hyperspace — that is, four dimensions — there are, strangely, six.

The five Platonic solids have analogs among these six convex polychora, and then there’s one left over — the oddball among the regular, convex polychora. It’s the figure you see above, rotating in hyperspace: the 24-cell, also known as the icositetrachoron. Its twenty-four cells are octahedra.

Like the simplest regular convex polychoron, the 5-cell (analogous to the tetrahedron), the 24-cell is self-dual. No matter how many dimensions you are dealing with, it is always possible to make a compound of any polytope and its dual. Here, then, is the compound of two 24-cells (which may be enlarged by clicking on it):

Both of these moving pictures were generated using software called Stella 4d:  Polyhedron Navigator. You can buy it, or try a free trial version, right here:  www.software3d.com/Stella.php.

## The Hyperspace Analog of the Dodecahedron/Icosahedron Compound

### Image

The dodecahedron and the icosahedron are dual to each other, and can be combined to make this well-known compound.

In hyperspace, the analog to the dodecahedron is the hyperdodecahedron, also known as the 120-cell, as well as the hecatonicosachoron. Its dual is the 600-cell, or hexacosichoron, made of 600 tetrahedral cells. The image at the top is the compound of these two polychora, rotating in hyperspace.

These images were made using Stella 4d, available at http://www.software3d.com/Stella.php.

## The Hyperspace Analog of the Cube/Octahedron Compound

### Image

The cube and the octahedron are dual to each other, and can be combined to make this well-known compound (below; can be enlarged with a click).

In hyperspace, the analog to the cube is the tesseract, also known the 8-cell, the octachoron, and the hypercube. Its dual is the 16-cell, or hexadecachoron, made of 16 tetrahedral cells. The image at the top is the compound of these two polychora, rotating in hyperspace.

These images were made using Stella 4d, available at http://www.software3d.com/Stella.php.

## 20-Thex: A Four-Dimensional Polytope

### Image

In hyperspace, or four-space, there are six regular polychora, analogous to the Platonic Solids in three-space. Beyond the Platonics in the study of polyhedra comes, of course, the Archimedean Solids, which include many truncated forms of Platonic polyhedra.

In hyperspace, there are varieties of progressively-less regular polychora, also, and one of these, in a group called the truncates, is called 20-thex, or simply the “thex.” (Those are short names for this polychoron; it’s also called the truncated hexadecachoron, or truncated 16-cell.) What you see above is a (seemingly) three-dimensional projection of a thex, as it rotates in hyperspace.

Just as polyhedra have polygons as faces, polychora have polyhedra as unit cells. This is the net for the thex. As you can see, the thex is composed of both truncated tetrahedra and octahedra.

Both of these images were created using Stella 4d, which you can try for yourself at http://www.software3d.com/Stella.php.

## 165-Srix

### Image

In three-dimensional space, there are five Platonic and thirteen Archimedean polyhedra, plus numerous other shapes, in several categories. The whole collection can appear to be quite a confusing jumble — until, and unless, you start surveying four-dimensional polytopes, known as polychora.

There are six regular polychora, and they are analogous to the five Platonic solids. Each three-dimensional cell is regular, and all are identical, within a single one of these six. When peering beyond these six, however, things can get very confusing, very quickly.

The software I used to generate this image, Stella 4d, has a built-in library of polyhedra and polychora. You can examine it as a free trial download at http://www.software3d.com/stella.php. Today, motivated by curiosity, I went surveying, using this program, into the more complex polychora — beyond the six regular ones — which have different polyhedra as cells, looking for one I could (try to) understand, and which appealed to me aesthetically.

The one I settled on for this post is known as 165-Srix, as well as the small rhombated 600-cell, a/k/a the cantellated 600-cell. It has 600 cells which are cuboctahedra, shown here in yellow, 120 more which are icosidodecahedra, shown here in blue, and 720 cells which are regular pentagonal prisms.

I must admit this: I’m more than a little jealous of those who seem to be able to easily understand these four-dimensional shapes. I am definitely not one of them.