In the previous post (here), I showed a great icosidodecahedron with its hexagonal faces replaced by inward-facing triangular cupolas. This one is similar, but the excavations take place on the decagonal faces, and take the shape of pentagonal cupolas.
Lastly, here’s a great rhombicosidecahedron with cupola-excavations performed on both its hexagonal and decagonal faces.
I made these images using Stella 4d, which you can try for free here.
This great rhombicosidodecahedron has had its hexagonal faces replaced by indentations, each the shape of a triangular cupola. I made it using Stella 4d, which you can try for yourself, free, at this website.
Each of these symmetrohedra has 302 faces. The one above was created by using the “morph duals by expansion” function, on the snub dodecahedron, in Stella 4d, the program I use to manipulate polyhedra (go here if you want to download a free trial of this software). It has twelve regular pentagons, sixty almost-square rectangles, and eighty equilateral triangles, along with ninety more obviously non-square rectangles, and sixty irregular pentagons.
I next used Stella’s “try to make faces regular” function, which produced this result:
This second polyhedron has 72 regular pentagons as faces, along with 20 equilateral triangles, 60 narrow isosceles triangles, and 150 irregular quadrilaterals. That’s 92 regular faces, in each of these two polyhedra.
I made this using Stella 4d, which you can for yourself, as a free trial download, right here: http://www.software3d.com/Stella.php.
This symmetrohedron has 122 faces. They are: (1) twenty-four blue, regular pentagons; (2) six green squares, (3) eight pink, equilateral triangles, (4) sixty red, irregular quadrilaterals, and (5) twenty-four yellow, scalene triangles. I made it, starting with the snub cube, using Stella 4d, a program you may try for free at this website: http://www.software3d.com/Stella.php.
I found this polyhedron by faceting the icosidodecahedron in such a way as to get twelve pentagons and twelve pentagrams as faces. Once I had done so, the result looked familiar, so I checked the uniform solids (a class of polyhedra I don’t yet know well), and found it there. Here is its dual, the medial rhombic triacontahedron.
I used Stella 4d to make these; you can try it for free at this website.
The faces of this polyhedron are twelve yellow regular star pentagons, twenty blue equilateral triangles, and thirty golden rectangles, shown in red. I used Stella 4d to make this, and you can try this program for free, at http://www.software3d.com/Stella.php.
I made this using Stella 4d, which you can try for yourself right here.
This non-convex polyhedron illustrates three related equations: 60 = (12)(5) — twelve sets of five yellow hexagons, 60 = (30)(2) — thirty pairs of red isosceles triangles, and 60 = (20)(3) — twenty sets of three yellow hexagons.
For a second look at this polyhedron, I made the triangles invisible, and then put the rest of the model in “rainbow color mode.”
I made both of these images using Stella 4d, which you can try for free at this website.
I made this using Stella 4d, which is available as a free trial download right here. It could also be viewed as a dodecahedron augmented with pentagonal frustrums.
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