# A Concave Polyhedron Featuring Eight Regular Dodecagons

This 110-faced polyhedron has, in addition to the eight regular dodecagons, six rectangles, and 96 triangles. I made it using Stella 4d, a program you can try for free, as a demo version, at http://www.software3d.com/Stella.php. I wish I could remember how I made it!

Fortunately, I have many friends who are more knowledgeable than I, when it comes to mathematics. Perhaps one of them will be able to solve this mystery.

# A Non-Convex “Cousin” of the Cuboctahedron

My guess is that this is a faceting of the cuboctahedron, but I didn’t use faceting when I made it with Stella 4d (a program you can try here), so I am not sure about this. Based on its appearance, however, it is clearly related, in some manner, to the cuboctahedron, for the cuboctahedron is its convex hull.

# A Non-Convex Polyhedron with Sixty Non-Convex Pentagonal Faces

Created using Stella 4d:  Polyhedron Navigator, available for purchase (with a free trial download available, first) at www.software3d.com/Stella.php.

## Tessellation Using Regular Hexacontakaihexagons, Regular Dodecagons, and Two Different (and Unusual) Concave, Equilateral Polygons

### Image

Hexacontakaihexagons have 36 sides, and dodecagons, of course, have twelve. When a regular hexacontakaihexagon is surrounded by twelve regular dodecagons, in the manner shown here, adjacent dodecagons almost, but not quite, meet at vertices. The gaps between these near-concurrent vertices are so small that they cannot be seen in this diagram — a zoom-in would be required, with thinner line segments used for the sides of the regular polygons.

As a result, the yellow and purple concave polygons aren’t what they appear, at first, to be. They look like triconcave hexagons, but this is an illusion. The yellow ones, in sets of two regions that aren’t quite separate, are actually tetraconcave, equilateral dodecagons with a very narrow “waist” separating the two large halves of each of them. As for the purple ones, they appear to occur in groups of four — but each set of four is actually one polygon, with three such narrow “waists” separating four regions of near-equal area. These purple polygons are, therefore, equilateral and hexaconcave icosikaitetragons — that is, what most people would call 24-gons.

## On Classification of Concave Polygons By Number of Concavities

### Image

Concave triangles do not exist, so concavity does not appear in the examination of polygons by ascending side length until the quadrilateral. A quadrilateral may only have one concavity, as shown in the red figure. Any polygon with exactly one concavity is called a uniconcave polygon.

Beginning with pentagons, the potential for two concavities appears. A polygon with two concavities, such as the yellow pentagon shown here, is a biconcave polygon.

Triconcave polygons, such as the blue hexagon here, have exactly three concavities. It is not possible for a triconcave polygon to have fewer than six sides.

For a tetraconcave polygon, with four concavities, at least eight sides are needed. The example shown here is the green octagon.

For higher number of concavities, simply double the number of sides to find the minimum number of sides for such a polygon. This pattern begins on the bottom row in the diagram here, but does not apply to the polygons shown in the top row.