In addition to the sixty heptagons, there are twelve regular pentagons and sixty quadrilaterals in this polyhedron, along with 380 triangles of various shapes and sizes. That’s 512 faces in all.
I used Stella 4d: Polyhedron Navigator to make this. You may try this program, for free, at http://www.software3d.com/Stella.php.
In addition to the eighteen regular octagons, this polyhedron also has eight equiangular hexagons and twenty-four isosceles trapezoids among its fifty faces. I made it using Stella 4d, which you can try, for free, right here.
Here’s a single dodecahedron.
A new “cluster polyhedron” can be made by augmenting each pentagonal face with another dodecahedron.
If you can do it once, you can do it again, augmenting each pentagon with a new dodecahedron.
Once more.
I made these polyhedral clusters using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.
Here is this polyhedron’s dual.
I made these using Stella 4d, a program you can try for free at http://www.software3d.com/Stella.php.
I made this using Stella 4d, which you can try for free at this website.
I made this using Stella 4d, which you can try for free right here.
The components of this compound are eight-faced trapezohedra. Here’s what just one of them looks like:
I made these images using Stella 4d, which you can try for free right here.
I made this using Stella 4d, which you can try for free right here.
Twelve heptagonal antiprisms, meeting at triangular faces, can make a ring with only a small gap remaining. It’s a “near miss,” made using Stella 4d, which you can try for free right here.
This polyhedron contains, as faces, 12 regular decagons, 20 regular hexagons, 30 squares, and 120 irregular hexagons. I made it using Stella 4d, which you can try for free at this website.
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