Two Views of the Truncated Tesseract

19 truncated tesseract orthogonal projection

The figure above, rotating in hyperspace, is an orthogonal projection of a four-dimensional polychoron known as the truncated tesseract. It is analogous to the truncated cube, one of the Archimedean solids. The image below is of the same figure, but is shown as a perspective projection.

19-Tat perspective projection

Both images were created using Stella 4d, software you can buy (with a free trial download available, first) at It’s great software, and a friend of mine wrote it — but no, he doesn’t pay me to give his program free advertising, as some have wondered.

Four Views of a Tesseract, Rotating in Hyperspace


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

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: