This decagonal mandala is split into fifty golden triangles (shown in yellow), and forty golden gnomons (shown in orange).
To make this polyhedron using Stella 4d (available here), one starts with the icosahedron, drops the symmetry of the model down from icosahedral to tetrahedral, and then stellates it once. The result is a chiral solid featuring four triangular faces and twelve kites:
The dual of this polyhedron, which is also chiral, has four triangular faces, and twelve faces which are isosceles trapezoids. It is a type of faceted dodecahedron — a partial faceting, meaning it is made without using all of the dodecahedron’s vertices.
If the isosceles triangles in this polyhedron were close enough to being equilateral that close inspection would be required to tell the difference, this would be a near-miss to the Johnson Solids. However, in my opinion, this doesn’t meet that test — so I’m calling this a “near-near-miss,” instead.
Software credit: visit this website if you would like to try a free trial download of Stella 4d, the program I used to create this image.
The two types of trapezoid are shown in blue and green. There are twenty-four blue ones (in eights set of three, surrounding each triangle) and twenty-four green ones (in twelve sets of two, with each set in “bowtie” formation).
This symmetrohedron follows logically from one that was already known, and pictured at http://www.cgl.uwaterloo.ca/~csk/projects/symmetrohedra/, with the name “bowtie cube.” Here’s a rotating version of it.
(Images created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)
The equiangular hexagons are very nearly regular, with only tiny deviations — probably not visible here — “from equilateralness.”
Little blurbs about posts on this blog get auto-tweeted on my Twitter, @RobertLovesPi. There’s also an A.I. on Twitter, @Hexagonbot, who retweeted my last two tweets about blog-posts here, but will not be retweeting the tweet about this one.
Why is this? Simple: @Hexagonbot is programmed to retweet any tweet which contains the word “hexagon,” which was in the titles of the last two posts here (also tessellations). This tessellation has no hexagons, though, and so the @Hexagonbot will not find it worthy of attention.
I cannot explain why hexagons get their own bot on Twitter, but other polygons do not have such bots. It’s simply one of the mysteries of the Internet.