# A Regular Icosagon, With All Sides and Diagonals Shown

An icosagon is a twenty-sided polygon. Each segment-length was given its own color.

# A Euclidean Construction of a Regular Triacontagon

Steps of this construction:

1. Use the green circles and blue lines to construct the yellow pentagon, along with its green inscribed pentagram.
2. Construct the equilateral triangle shown in gray. This is needed to obtain a twelve degree angle. The triangle is needed for its sixty degree angle, because 72 – 60 = 12. (The 72 degree angle is found inside the pentagon.)
3. Identify the twelve degree angle shown in bold. A twelve degree angle is needed because 360 / 30 = 12.
4. Use the red circles to complete the thirty sides of the regular triacontagon, which is shown with bold black segments, inscribed inside a large, bold, red circle.

# 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.

# Zonohedra, Zonish Polyhedra, and Another Puzzle

In a recent post, I showed many images of zonohedra, then challenged readers to figure out, from the images, what zonohedra are: polyhedra with only zonogons as faces. Zonogons, I then explained, are polygons with (A) even numbers of edges, and with opposite edges always (B) congruent and (C) parallel. Here is another collection of zonohedra. (Individual images may be enlarged with a click.)

The next set of polyhedra shown, below, are not true zonohedra (as all the ones above are), but merely “zonish polyhedra.” From examination of the pictures above and below, can you figure out the difference between zonohedra and zonish polyhedra?

When you are ready to see the solution to the puzzle, simply scroll down.

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While zonohedra have only zonogons as faces, this restriction is “loosened” for zonish polyhedra. Such solids are formed by zonohedrifying non-zonohedral polyhedra, and letting at least some of the faces of the resulting polyhedra remain non-zonogonal. Zonish polyhedra  are called “zonish” because many (usually most) of their faces are zonogons, but not all of them — in each case, some non-zonogonal polygons (such as triangles and/or pentagons, with their odd numbers of edges) do appear. Non-zonogonal polygons are not required to have odd numbers of edges, of course: simply having opposite edges be parallel, but of different lengths, is enough to prevent a polygon (such as a hexagon, octagon, or decagon) from being a zonogon.

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Software credit: I used Stella 4d to make these images. This program may be tried for free at this website.

# Octagons Can Tile a Plane II

In April 2014, I found a tessellation of the plane which uses two kinds of octagons — both types equilateral, but only one type regular.

Now, I have found two more ways to tessellate a plane with octagons, and these octagons are also equilateral. However, in these new tessellations, only one type of octagon is used. One of them appears below, twice (the second time is with reversed colors), and the other one appears, once, in the next post.