Why the 360-gon Is Special (Hint: It's About Divisors)

That question doesn't make any sense. You can let an equilateral figure have as many sides as you want. There's no limit to it. It does look less and less like a polygon as you give it more sides, but at no point do you reach a boundary where it's like, "Oh, I can give it a thousand sides, but not a thousand and one. " So, let's talk about a question that would make sense instead. Every equilateral, equiangular polygon has a particular interior angle measure. A triangle, for example, has three angles that sum to 180°, so if we're looking at an equiangular, equilateral triangle, that means that one of those angles must be 60°. As we increase the number of sides for our particular polygon, something interesting happens. For the triangle, the square, the regular pentagon, and the regular hexagon, that individual interior angle measure is always an integer. But, when we get to the heptagon, the seven-sided figure, that angle measure isn't an integer anymore. What is the largest number of sides we can have for some regular polygon such that interior angle measure, the individual one, when measured in degrees, is an integer? Here's how I would approach it. The first thing that we need is what I call IAMS, the interior angle measure sum for a given polygon. There is a formula for this. We take the 180° basically of a triangle, and we multiply it by the number of triangles that we can fit into a given figure from the edges. That turns out to always be two less than the number of sides. So, for our total interior angle measure sum, we're always looking at 180 * n - 2. Now, if we're looking at the measure of just one of those in a regular polygon, we're going to divide that by n, and that's going to give us the individual angle measure for some regular n-gon. This value was an integer for something like n = 6, the hexagon, because when we replace this number with six, it would divide into our 180 evenly. On the other hand, for the heptagon, for our seven-sided figure, if we let n equal seven, it does not divide into 180 evenly, and it definitely doesn't divide into 7 - 2 = 5 evenly. So, we can tell that this question really simplifies to what's the largest n that is going to divide either the 180 or the n - 2 or both evenly. What immediately occurs to me is, why don't we just let n equal 180? If we're looking at the 180-gon, then we would divide whatever 180 * 178 is / 180, that's going to give me a 178° angle. But, this turns out not to be the largest integer angle that's possible, because there's another integer between 178 and 180. Surely, there's some way that we can get the interior angle measure of some regular polygon to actually equal 179.