The Dodecagonal Duoprism

12-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.

Zome Hyperdodecahedron

This is one projection of the four-dimensional hyperdodecahedron, or 120-cell, rendered in Zome. All the part for this come in a single kit, and, if you want it for yourself, you can find it for sale at this website.

hyperdo

I did have student help with the construction of this, for which I am grateful. However, for legal and ethical reasons, I cannot credit the students by name.

Here’s a closer view, through the “core” of all-blue pentagons:

160114_0001

Zome is a great product. I recommend it strongly, and without reservation (and no, they aren’t paying me anything to write this).

23-Tex: The Truncated Hexacosichoron

23-Tex

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:

23-Tex with

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:

23-Tex net

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

Four Views of a Tesseract, Rotating in Hyperspace

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Four Views of a Tesseract, Rotating in Hyperspace

These tesseract views are all of the perspective projection-type, with the first one, above, being done in cell-first fashion.

The next one is projected edge-first.

persp edge 2-Tes, 8 -cell, Octachoron, Tesseract

The third one is projected vertex-first.

persp vertex 2-Tes, 8 -cell, Octachoron, Tesseract

Lastly, face-first:

persp face 2-Tes, 8 -cell, Octachoron, Tesseract

Although all of these are rotating in the same direction in hyperspace, the different projection-choices make the second and fourth images appear to be rotating in different directions. Why? I’m still trying to figure that out!

These animations were created with Stella 4d, software available at www.software3d.com/Stella.php.

20-Thex: A Four-Dimensional Polytope

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The 20-Thex:  A Four-Dimensional Polytope

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.

20-Thexnet

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

The Hyperspace Analogue of the Stella Octangula

The simplest polyhedron is the tetrahedron, and it is self-dual. The compound of two tetrahedra puts these duals together, and is most often called the Stella Octangula, a name Johannes Kepler gave it in the early 17th Century.

Image

In hyperspace, or 4-space, the simplest polychoron is the pentachoron, or 5-cell. Like the tetrahedron in 3-space, it is also self-dual. Here is the compound of two of them: hyperspace’s version of the Stella Octangula.

Compound of 1-Pen, 5-cell, Pentachoron and dual

Website to find the software used to make these images:  www.software3d.com/stella.php