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

This Faceting of the Truncated Icosahedron is Also a Truncation of the Great Dodecahedron

This first version shows this polyhedron colored by face type.

In the next image, only parallel faces share a color. This is the traditional coloring-scheme for the great dodecahedron.

Both images were created with Stella 4d, which is available as a free trial download at this website. Also, the obvious change needed with this polyhedron — making its faces regular — is in the next post.

Selected Stellations of the Truncated Dodecahedron

This is the truncated dodecahedron. It is one of the Archimedean solids.

Trunc Dodeca

This polyhedron has a long stellation-series, from which I selected several on aesthetic grounds. The figure immediately below is the truncated dodecahedron’s 16th stellation.

16th stellation of Trunc Dodeca

Here is the 21st stellation.

21st stellation of Trunc Dodeca

It’s easy to stellate polyhedra rapidly, and make many other changes to them, with Stella 4d: Polyhedron Navigator. You can try it for free at

25th stellation of Trunc Dodeca.gif

The stellation shown immediately above is the 25th, and the one shown immediately below is the 27th.

27th stellation of Trunc Dodeca

Here is the next stellation: the 28th. Unlike the ones shown above, it is chiral.

28th stellation of Trunc Dodeca.gif

This is the truncated dodecahedron’s 31st stellation.

31st stellation of Trunc Dodeca.gif

This one is the 38th stellation.

38th stellation of Trunc Dodeca.gif

This one is the 44th.

44th stellation of Trunc Dodeca.gif

The last one shown here is called the truncated dodecahedron’s final stellation because, if it is stellated once more, it returns to the original truncated dodecahedron.

Final stellation of Trunc Dodeca

Some Tetrahedral Stellations of the Truncated Cube

I created these with Stella 4d, which you may try for free at this website. To make a given polyhedral stellation appear larger, simply click on it.

The Truncated Cube, with Two Variations Featuring Regular Dodecagons

This is the truncated cube, one of the thirteen Archimedean solids.


If the truncation-planes are shifted, and increased in number, in just the right way, this variation is produced. Its purple faces are regular dodecagons, and the orange faces are kites — two dozen, in eight sets of three.


Applying yet another truncation, of a specific type, produces the next polyhedron. Here, the regular dodecagons are blue, and the red triangles are equilateral. The yellow triangles are isosceles, with a vertex angle of ~41.4 degrees.


All three of these images were produced using Stella 4d, available at this website.

An Oblique Truncation of the Tetrahedron

kite-bounded tetrahedron

This polyhedron has sixteen faces: four equilateral triangles, and a dozen kites. It was created using Stella 4d, which may be found at

A Faceted Truncated Tetrahedron, or Seven Tetrahedra Joined at Their Edges — Your Choice

Faceted Trunc Tetra

I made this by faceting a truncated tetrahedron, giving it faces which are interpenetrating, red, equilateral triangles, as well as yellow crossed-edged hexagons. It can also be viewed as a central tetrahedron, with six more tetrahedra attached to its edges. This was made with Stella 4d, available at this website.

Partially Truncated Platonic and Rhombic Dodecahedra

chiral polyhedron featuring a dozen hexagons and four triangles

Each of these dodecahedra were modified by truncations  at exactly four of their three-valent vertices. As a result, each has four equilateral triangles as faces. In the one above, the Platonic dodecahedron’s pentagonal faces are modified into a dozen irregular hexagons by these truncations, while, in the one below, the rhombic dodecahedron’s faces are modified into twelve irregular pentagons.

dozen pents 4 triangles

Both of these polyhedra were created using Stella 4d, software you can try for yourself at this website.