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.

# Tag Archives: 4-d

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

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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

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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.