It’s hard to get regular pentagons, regular star pentagons, regular decagons, and related polygons to tessellate the plane while maintaining radial symmetry. This is my latest attempt.
I just had the strange experience of encountering a polyhedral discovery of mine on the Internet from about eight years ago — one that I had completely forgotten, but had shared with others, who posted it online, and were kind enough to give me credit for the discovery. It’s the fifth polyhedron shown on this page: http://www.interocitors.com/polyhedra/Triamonds/ — and is shown with a .stel file, so I was able to use polyhedral-manipulation-and-imaging software, Stella 4d (available at www.software3d.com/Stella.php) to make a rotating image of it:
It’s a member of a class of polyhedra which have, as faces, only regular polygons with unit size, as well as “triamonds.” Triamonds are 1:1:1:2 trapezoids composed of three coplanar, equilateral triangles.
Back in 2006 or earlier, my guess is that I simply made a physical model out of card-stock paper and tape, and then took a photograph of it — something I haven’t done in a very long time, now that making moving pictures of virtual models has become so easy. Another possibility is that I used Zome, an excellent ball-and-stick modelling system available at www.zometool.com. Zome, like Stella, I still use — and I will be using Zome often with students, in class, when school starts next month. Fortunately, I have a lot of Zome!
Geometer’s Sketchpad and MS-Paint were both used to make the images on the faces of this polyhedron, and then Stella 4d was used to put it all together and create this rotating image. Stella may be bought, and/or tried for free, at www.software3d.com/Stella.php.
I suspect that this could be continued outward indefinitely, as a radial and aperiodic tessellation, using only the four polygons you see, repeatedly, here. However, I have no proof of this.