This is the rhombic triacontahedron, one of the Catalan solids.
Now here’s the same polyhedron, but with the three-valent vertices truncated, exposing twenty triangular faces.
Here’s what it looks like with only the twelve five-valent vertices truncated.
Finally, here’s the fully truncated rhombic triacontahedron.
I created these polyhedra using Stella 4d, a program you can try for free right here.
This is the rhombic dodecahedron, one of the Catalan solids.
Now here’s the same polyhedron, but with the three-valent vertices truncated, exposing eight triangular faces.
Here’s what it looks like with only the six four-valent vertices truncated.
Finally, here’s the fully truncated rhombic dodecahedron.
I created these polyhedra using Stella 4d, a program you can try for free right here.
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 http://www.software3d.com/Stella.php.
To get from the last image posted to this one, I used Stella 4d‘s “try to make faces regular” function. (You can get a free trial download of this program right here.)
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.
This can also be called the compound of two truncated tetrahedra.
This image was created using Stella 4d, which you can try at this website.
This is the truncated dodecahedron. It is one of the Archimedean solids.
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.
Here is the 21st stellation.
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 http://www.software3d.com/Stella.php.
The stellation shown immediately above is the 25th, and the one shown immediately below is the 27th.
Here is the next stellation: the 28th. Unlike the ones shown above, it is chiral.
This is the truncated dodecahedron’s 31st stellation.
This one is the 38th stellation.
This one is the 44th.
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.
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.
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.
This polyhedron has sixteen faces: four equilateral triangles, and a dozen kites. It was created using Stella 4d, which may be found at http://www.software3d.com/Stella.php.
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