The black polygons in this radial tessellation are regular decagons. The red figures are hourglass-shaped equilateral hexagons, which remind me of the distinctive red markings found on many black widow spiders.
In addition to the 42 regular decagons, the faces of this polyhedron include twenty equilateral triangles, sixty yellow trapezoids, and sixty blue trapezoids. That’s 182 faces in all.
The next picture shows what happens if all of the decagons have the same color, the triangles have another, and the trapezoids are hidden from view.
Both images were created using Stella 4d, software you can try for yourself at this website.
Ten rhombic triacontahedra fit perfectly into a decagonal ring. It’s not a “near-miss” — the fit is exact.
I made this with Stella 4d, software you can try for free, or purchase, at http://www.software3d.com/Stella.php.
The twelve purple faces of this faceted dodecahedron show up on Stella 4d‘s control interface as {10/4} star decagons, which would make them each have five pairs of two coincident vertices. I’m informally naming this special decagon-that-looks-like-a-pentagram (or “star pentagon,” if you prefer) the “antipentagram,” for reasons which I hope are clear.
Stella 4d, the program I use to make most of my polyhedral images, may be tried for free at http://www.software3d.com/Stella.php.
This tiling-pattern could be continued indefinitely, while still maintaining its five-fold radial symmetry, giving it the overall appearance of a pentagon.
The golden triangles, in yellow, are acute isosceles triangles with a leg:base ratio which is the golden ratio. Golden gnomons, shown in orange, are related, for they are obtuse isosceles triangles where the golden ratio shows up as the base:leg ratio, which is the reciprocal of the manifestation of the golden ratio which appears in the yellow triangles.
