A Polyhedron Featuring a Dozen Regular Heptagons and a Whole Mess of Quadrilaterals

There are four different shapes of quadrilaterals here, but they have one thing in common: there are twelve of each of them. Add those 48 quads to the twelve regular heptagons, and that gives a total of 60 faces. I found this polyhedron while playing with the snub dodecahedron, using Stella 4d: Polyhedron Navigator, a program you can try for free right here.

In the image below, each of the four types of quadrilateral appears with its own color.

The Third Stellation of the Rhombic Dodecahedron

The model above shows the rhombic dodecahedron colored with one color per face, unless those faces are parallel. The one below shows this solid in “rainbow color mode.”

I made both of these using Stella 4d: Polyhedron Navigator. You can try this program for free at http://www.software3d.com/Stella.php.

Playing Around With the Pentagonal Icositetrahedron

This, above, is the pentagonal icositetrahedron — one of the Catalan solids, and the dual of the snub cube. If faceting is used to remove square-based pyramids, the result is the solid shown below — a chiral polyhedron with six square faces, and 24 faces which are isosceles trapezoids.

I use a program called Stella 4d: Polyhedron Navigator (available here) to perform these polyhedral manipulations. This program has a “try to make faces regular” function, and the solid shown below is the result of using that function on the polyhedron shown above.

The “try to make faces regular” function appeared at first to leave the squares as squares (which it did), but turn the trapezoids into kites (which it did not; the yellow quadrilaterals’ longer sides are actually of two slightly different lengths). Lastly, I used Stella to create the compound of the polyhedron above and its dual. The result is shown below.

A Chiral Polyhedron Featuring Pentagons and Isosceles Trapezoids

I made this chiral polyhedron using Stella 4d, which you can try for yourself — free — at this website. It is a faceted form of the pentagonal hexecontahedron, which is the dual of the snub dodecahedron.