# A Symmetrohedron Featuring Twenty Regular Enneagons, Twelve Regular Pentagons, and Sixty Isosceles Triangles

I made this using Stella 4d: Polyhedron Navigator. You can get a free trial download of this program at http://www.software3d.com/Stella.php.

# A Tessellation Featuring Regular Enneagons and Triangles, as Well as Equilateral Three-Pointed Stars

Here’s a second version, with different colors:

## Tessellation Featuring Rhombi and Concave, Equilateral Dodecagons

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This particular tessellation is full of angles measuring 20 degrees, 40 degrees, and other angles which are not constructable using the traditional rules of Euclidean constructions. This is because this tessellation is based on a matrix which includes regular enneagons.

# Two Symmetrohedra Which Feature Enneagons

These two symmetrohedra were created using Stella 4d, software you can try for free right here.

# The Regular Enneagon, and Three Regular Enneagrams

The red figure above is a regular enneagon, or nine-sided polygon, and it has three regular enneagrams (or “star enneagons”) inside it. The light blue figure is called a {9,2} enneagram. The green figure can be viewed two ways: as a {9,3} enneagram, or as a compound of three equilateral triangles. Finally, the yellow figure is a {9,4} enneagram.

To see what these numbers in braces mean, just take a look at one of the yellow enneagram’s vertices, then follow one of the yellow segments to the next vertex it touches. Count the vertices which are skipped, and you’ll notice each yellow segment connects every fourth vertex, giving us the “4” in {9,4}. The “9” in {9,4} comes from the total number of vertices in this enneagram, as well as the total number of segments it has. The blue and green enneagrams are analogous to the yellow one. These pairs of numbers in braces are known as Schläfli symbols.

I should mention that some people call these figures “nonagons” and “nonagrams.” Both “ennea- and “nona-” refer to the number nine, but the latter prefix is derived from Latin, while the former is based on Greek. I prefer to use the Greek, since that is consistent with such Greek-derived words as “pentagon” and “hexagon.”

Finally, there is also an “enneagram of personality,” in popular culture, which some use for analyzing  people. Aside from this mention of it, that figure is not addressed here — nor is the nine-pointed star used as a symbol for the Bahá’í faith. However, it’s easy to find information on those things with Google-searches, for those who are interested.

# Tessellation Featuring Regular Enneagons and Hexaconcave Dodecagons

Some people call nine-sided polygons “nonagons,” rather than “enneagons,” but I prefer Greek prefixes to those based on Latin.