A Cube-Based “Bowtie” Symmetrohedron Featuring Six Regular Hexadecagons, Eight Equilateral Triangles, and Two Dozen Each of Two Types of Icosceles Trapezoid

Image

A Cube-Based Symmetrohedron Featuring Six Regular Hexadecagons, Eight Equilateral Triangles, and Two Types of Icosceles Trapezoids

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.

dodecagons and hexagons

(Images created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)

Tessellation Using Regular Triacontakaihexagons, Equiangular Hexagons, Isosceles Triangles, and Isosceles Trapezoids

Image

Tessellation Using Regular Hexacontakaihexagons, Equiangulr Hexagons, Isosceles Triangles, and Isosceles Trapezoids
The equiangular hexagons are very nearly regular, with only tiny deviations — probably not visible here — “from equilateralness.”

Tessellation Using Regular Triacontagons, Isosceles Triangles, Equiangular Triangles, and Isosceles Trapezoids

Image

Tessellation Using Regular Triacontagons, Isosceles Triangles, Equiangular Triangles, and Trapezoids

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.