Playing some kids’ geometric puzzle pieces (and then doing some pencil & paper checks), I realized something.
It started like this: I can line up a sequence of pentagons and equilateral triangles, end-to-end, and get a cycle (a segmented circle). There are 30 shapes in this cycle (15 pentagon-triangle pairs), and so the perimeter of the cycle is divided then into 30 equal straight segments.
[Here is a figure to show what i’m talking about]
You can do something similar with squares and triangles and you get a smaller cycle: 6 square-pentagon pairs, dividing the perimeter into 12 segments.
And then you can just build it with triangles – basically you just get a hexagon with six sides.
For regular polygons beyond the pentagon, it changes. Hexagons and triangles gets you a straight line (actually, you can get a cycle out of these, but it isn’t of segments like all the others). Then, you get cycles bending in the opposite direction with 8-, 9-, 12-, 15-, and 24-gons. For those, respectively, the perimeter (now the ‘inner’ boundary of the pattern – see the figure above for an example) is divided into 24, 18, 12, 10, and 8 segments.
You can also make cycles with some polygons on their own: triangles, squares, hexagons (three hexagons in sequence make a cycle), and you can do it a couple of ways with octagons (with four or eight). You can also make cycles with some other combinations (e.g. 10(edited from 5) pentagon-square pairs).
Here’s what I realized: The least common multiple of those numbers (the number of segments to the perimeter of the triangle-polygon circle) is 360! (at least, I’m pretty sure of it.. maybe here I have made a mistake).
This means that if you lay all those cycles on a common circle, and if you want to subdivide the circle in such a way as to catch the edges of every segment, you need 360 subdivisions.
Am I just doing some kind of circular-reasoning numerology here or is this maybe a part of the long-lost rationale for the division of the circle into 360 degrees? The wikipedia article claims it’s not known for certain but seems weighted for a “it’s close to the # of days in the year” explanation, and also nods to the fact that 360 is such a convenient number (can be divided lots and lots of ways – which seems related to what I noticed). Surely I am not the first discoverer of this pattern.. in fact this seems like something that would have been easy for an ancient Mesopotamian to discover..
edit for tldr * *
For those who don’t understand the explanation above (i sympathize): to be clear, this method gets you exactly 360 subdivisions of a circle but it has nothing to do with choice of units. It’s a coincidence, not a tautology, as some people are suggesting.. I thought it was an interesting coincidence because the method relies on constructing circles (or cycles) out of elementary geometrical objects (regular polyhedra).
The most common response below is basically what wikipedia says (i.e. common knowledge); 360 is a highly composite number, divisible by the Babylonian 60, and is close to the number of days in the year, so that probably is why the number was originally chosen. But I already recognized these points in my original post.. what I want to know is whether or not this coincidence has been noted before or proposed as a possible method for how the B’s came up with “360”, even if it’s probably not true.