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

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

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):

4-Ico, 24-cell, Icositetrachoron with dual

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.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s