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Constructing the Henkaipentacontagon: A Regular Polygon with 51 Sides

After completing the heptadecagon construction shown in the last post on this blog, I wondered if I could pull off a similar trick to the one mentioned there of combining the pentagon and triangle constructions to construct a regular pentadecagon — but using the heptadecagon and triangle, instead, to construct a regular polygon with (17)(3) = 51 sides, known as the henkaipentacontagon.

The answer: yes, I was able to, but certainly not in the simplest way possible, for I ended up having to go to 204, first, to get to 51. First, I extended two adjacent radii of the heptadecagon in the upper left as rays, just to give me room to work. Next, I placed point B₁on the lower of those two rays, to be used as the center of the large circle in which to construct my 51-sided regular polygon. I then constructed a line through B₁ which was parallel to the upper of these two rays, thus duplicating the ~21.17647º central angle of the heptadecagon, but in the center of my new, larger circle. Next, I constructed the yellow equilateral triangle with this heptadecagon-central-angle inside it, in such a way that the lower half of the yellow triangle would be a 30-60-90 triangle, ΔA₁B₁C_1, with the ~21.17647° angle inside, and sharing a ray with, this triangle’s 30° angle. By subtraction, that made a ~8.8235° angle, with its vertex at the center of the largest circle shown.

Next, I divided ~8.8235 into 360 . . . and, to my dismay, didn’t get a whole number as an answer, but 40.8, instead. I then noticed that one can multiple 40.8 by five, and obtain 204 as the product. Armed with this knowledge, I used my ~8.8235° angle, and 204 circles of equal radius, to locate 204 points, evenly-spaced, around the large circle.

51 is one-fourth of 204, so I connected every fourth point around the large circle with heavy blue segments, and made those 51 points (one fourth of the total) blue, as well. These blue points and segments are the sides and vertices of the regular henkaipentacontagon, shown inscribed, above, inside the largest circle in the diagram.

Since this polygon looks a lot like a circle, I then rendered a lot of things from the diagram above invisible, in order to produce this second image: the henkaipentacontagon alone, with different colors for its vertices and sides, all radii added, and three alternating colors for the 51 triangles each formed by two adjacent radii and a side.

Constructing the Heptadecagon

I have just completed my first construction of the regular heptadecagon — a construction that even the ancient Greeks were never able to figure out. They did figure out how to construct a regular pentadecagon (by combining the constructions for the regular pentagon and triangle), and I once replicated that discovery, meaning that I figured it out independently.

The regular heptadecagon construction, however, I did not figure out independently. I used instructions found here (http://www.mathpages.com/home/kmath487.htm), which built on the work of Carl Friedrich Gauss, who, in 1796, at the age of 19, became the first person in history to determine that such a construction is possible with the traditional Euclidean tools.

A word of warning, if you attempt to replicate this construction yourself: points M and G are merely close together, but are not in the same place. Point M is the center of the circle which passes through points D and V17, while point G is one of the two points of intersection of (1) the line passing through points O and V17, and (2) the circle centered at C, and passing through E.

Gauss (and other mathematicians, building on his work) also showed, later, that constructions are possible for regular polygons with 257 sides, as well as 65,537 sides. I might, someday, replicate the construction of the regular polygon with 257 sides.

A man named Johann Gustav Hermes once spent ten years completing a 200-page manuscript showing how to construct the regular polygon with 65,537 sides, and I believe he actually performed the construction, as well. I will not be constructing this polygon — ever. I will, however, figure out a proper name for it. Let’s see . . . it’s the heptakaitriacontakaipentacosioikaipentachilikaihexamyriagon. Try saying that five times in a row, quickly!