Tessellation Featuring Rhombi and Concave, Equilateral Dodecagons


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.

A Proposed New Unit for Angle Measure: The Euclid


A Proposed New Unit for Angle Measure:  the Euclid

Some angles are constructible, in the Euclidean sense that they may be constructed with the traditional geometricians’ construction-tools: a compass, and an unmarked straightedge. Examples include every angle shown above, such as the 108° interior angles of the purple regular pentagon, or the 60° angles of the yellow triangle. Angle LEN is constructible as well, and measures 48° — but to construct it, one must use compass-and-straightedge subtraction (the 108° pentagon angle HEK, minus the 60° triangle angle KEL). After constructing this 48° angle, I bisected it repeatedly, to show that angles measuring 24, 12, 6, and 3, and 1.5 degrees may be constructed as well. The 1.5° angle NET is shown with a blue interior.

Many other angles are non-constructible. For example, the angle between two adjacent radii of a regular enneagon (also called a nonagon) measures 40°, and so, because it has been proven that the regular enneagon cannot be constructed with the traditional Euclidean tools, it follows that 40° angles are non-constructible. If they were constructible, however, the subtraction-trick I used earlier to construct a 48° angle could be used, again, to construct an 8° angle (48° – 40°) — so 8° angles, therefore, are also non-constructible. Since repeatedly bisecting an 8° angle would yield angles measuring 4, 2, 1, 0.5, o.25, etc. degrees, all of these angle-measures are for non-constructible angles.

With the one degree angle on the non-constructible list, that throws into question the practice of using degrees to measure angles. As for other established units of angle measure, they have the same problem. It is not possible to construct an angle measuring one radian — nor one gradian, either. (Gradians are little-known angle-measuring units; a right angle measures 100 gradians.)

If an angle-measurement system is to be based on units which correspond to the measure of constructible angles, the blue angle above, measuring 1.5°, is ideal . . . and I am, therefore, using this angle as the definition for a new unit of angle measure:  the euclid. If an angle measures a whole number of euclids, it is constructible, and this cannot be said for the degree, radian, nor gradian. (By the way, leaving “euclid” uncapitalized, in this context, is deliberate, for I am using it as a unit. This follows the convention set by other units named after people. For example, “Newton” refers to Sir Isaac Newton, but “newton” refers to a unit of force.)

One full rotation would be a rotation of 240 euclids. A right angle is one-fourth of that, or sixty euclids. The interior angles of equilateral triangles measure forty euclids, and the interior regular-pentagon-angle of 108° becomes 72 euclids, in this new, proposed system.

360 has been used as the basis of the degree for reasons both historical and mathematical. Sixty, and its multiple 360, appear as important numbers in several ancient cultures, and 360 also has many whole-number divisors, having a prime-number factorization of (2)(2)(2)(3)(3)(5).

However, 240 has similar properties. As I have shown, it is based on the Euclidean construction-rules from ancient Greece. The number 240 also has many whole-number divisors, since its prime-number factorization is (2)(2)(2)(2)(3)(5).

Just in case this catches on, I have created a symbol for the euclid, to be used in superscript form, as the degree symbol is used:

euclid symbol

A simple “e,” by itself, would not do, for that would cause confusion with the important number e — the base of natural logarithms, among other things. That is why I included a circle, surrounding the letter “e,” for this symbol. In superscript form, this symbol for the euclid would resemble the well-known copyright symbol — but, fortunately, the copyright symbol is not, itself, copyrighted.

On the Constructible Angles


The Constructible Angles

The Ancient Greeks figured out how to combine the Euclidean constructions of the regular pentagon and triangle to obtain constructions for the regular pentadecagon, which has central angles (between adjacent radii) of 360/15 = 24 degrees. Here’s an example, showing how this can be performed:


Also, it’s easy to construct an equilateral triangle, and then bisect an angle of it, to obtain a 30 degree angle.

The existence of angle difference identities in trigonometry is tied to the fact that you can subtract angles, on paper, with Euclidean constructions. Therefore, an angle of 24 degrees may be subtracted from a 30 degree angle to obtain a 6 degree angle. This can be bisected to get a 3 degree angle, and then bisected again to obtain a 1.5 degree angle, then a 0.75 degree angle, and so on.

However, a one degree angle is impossible to construct. Were this not the case, a 24 degree angle’s constructibility would imply that of the 23 degree angle, by subtraction of a one degree angle. After that, subtract three degrees more, and you have a 20 degree angle . . . and with that, you can construct a regular enneagon, also known an a nonagon. But we know — it has been proven — that regular enneagons have no valid Euclidean constructions. Therefore, one degree angles are also non-constructible, by reductio ad absurdam.

Carl Friedrich Gauss’s much more recent proof (1796; he was 19 years old) that a regular polygon of 17 sides can also be constructed — the first significant advance in this field since the time of the ancient Greeks — adds more constructible angles. Building on his work, other mathematicians have also shown that regular polygons with 257 and 65,537 sides can also be constructed, adding yet more constructible angles, but they are all for angles measuring fractional numbers of degrees, since none of these numbers are factors of 360, which equals (2³)(3²)(5). It’s also possible to combine these possible constructions to construct more regular polygons, as was shown above for the pentadecagon. For example, one can construct a regular pentagon with 51 sides, since 51 = (17)(3) — but, again, combinations of this type only lead to possible constructions of angles with measures which are fractional numbers of degrees. For angles with degree measures which are integers, it’s multiples of three — and that’s it.

[Note regarding images: the photograph of a compass at the top of this page was not taken by me, but simply found with a Google image-search. The pentadecagon-construction image, though, I did make, using both Geometer’s Sketchpad and MS-Paint.]