To make this, I stellated the dual of the tetrated dodecahedron over 200 times, and then applied Stella 4d‘s “try to make faces regular” function. This is probably not the easiest method to use, but I don’t know of any others. The quadrilaterals are all kites.
If you’d like to give Stella a try, for free, the website to visit is http://www.software.com/Stella.php/
The first polyhedron shown here is also a symmetrohedron. 42 of its faces are regular decagons, and twenty are equilateral triangles. There are also sixty each of two types of isosceles trapezoids. That’s 182 faces in all.
The second polyhedron shown, below, also has 42 regular decagons as faces, along with twenty equiangular hexagons, sixty isosceles triangles, sixty almost-square isosceles trapezoids, and 120 of another kind of isosceles trapezoid. That’s a total of 302 faces.
The third one I found, shown below, has 42 decagons, sixty convex hexagons, twenty equilateral triangles, sixty rectangles, and sixty isosceles trapezoids, for a total of 242 faces.
I made all three of these polyhedra using Stella 4d: Polyhedron Navigator, which you can try for yourself, free, at http://www.software3d.com/Stella.php.
This symmetrohedron has four faces which are regular hexagons, 24 which are regular pentagons, and four which are equilateral triangles. It also has twelve faces which are acute isosceles triangles, as well as twelve more which are obtuse isosceles triangles. I made it using Stella 4d, which you can try for free here.
To make this symmetrohedron, I augmented the faces of a rhombic dodecahedron with prisms, then formed the convex hull of the result. All faces except for the red rhombi are regular. This was made using Stella 4d, which you can try for free here.
I made this polyhedron by creating a zonohedron based on the edges and faces of the truncated tetrahedron. Only the blue hexagons are irregular. Stella 4d was used in its creation, and you may try this program for free at http://www.software3d.com/Stella.php.
This symmetrohedron includes, as faces, eight regular hexagons, six squares, and 24 isosceles triangles. I made it using Stella 4d, which you can try for free at this website.
I made this by zonohedrificaton of an octahedron, using its faces, edges, and vertices, and software called Stella 4d, which you can try for free right here.
This polyhedron has six regular octagons (red), a dozen octagons which are merely equilateral (yellow), eight regular hexagons, and 24 squares.
This symmetrohedron’s regular faces are twelve pentagons, thirty octagons, and twenty triangles. Its irregular faces include sixty yellow isosceles trapezoids, as well as sixty blue isosceles trapezoids. That makes 182 faces in all.
I used Stella 4d: Polyhedron Navigator to make this, and you can try this program yourself, as a free trial download, at http://www.software3d.com/Stella.php.
I made this using Stella 4d, which you can try for yourself at this website.
Symmetrohedra are symmetric polyhedra with many faces regular, but not necessarily all of them. The symmetrohedron shown above is the dual of the convex hull of the compound of the great rhombicosidodecahedron and its dual, the disdyakis triacontahedron. All of its faces are regular, except for the triangles, which are scalene.
The second symmetrohedron shown here is the dual of the convex hull of the great rhombcuboctahedron and its dual, the disdyakis dodecahedron. Like its “big brother” above, all of this symmetrohedron’s faces are regular, except for the scalene triangles.
These polyhedra were made using Stella 4d, a program you can try for free at this website.
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