@@rouslanrouslan2677 Well, Plato doesn't even mention the retrogradre motion of the planets. Aristotle does; but in his "De Caelo" doesn't know how to explain the phenomenon; and only later incorporates Eudoxus's results into the Metaphysics. We know about Eudoxus just because of Aristotle (not Plato).
> All the possible polygons... There are three basic sets of constructible polygons based on number of sides: Any Fermat prime, the product of two or more Fermat primes, and any of the numbers in the first two groups multiplied by some power of 2. The last group are relatively trivial - if you can construct a polygon with n sides, then bisecting any side will allow you to double the number of sides, and this can be done without limit. Therefore, it would've been more useful to focus on the polygons whose number of sides are a Fermat prime, or the product of two or more of them; they are all odd numbers, and in total there are 31 of them. The five Fermat primes are the most crucial, as by constructing two polygons with number of sides f and g, for example, it's possible to determine the smallest distance between a vertex on the f-gon and one on the g-gon, and construct an fg-gon.
I described the construction of 257 and 65537 sides regular polygons on the italian version of wikipedia. Here is the complete construction of the 257-gon (explanation and animated gif are mine): it.wikipedia.org/wiki/257-gono#Costruzione and here is the first step of the 65537-gon: it.wikipedia.org/wiki/65537-gono#Idea_della_costruzione
You should say all polygons possible up to 51 sides. You can double any of the examples (32x2=64, 34x2=68 etc.) There are also 2 more prime size polygons you can make: 257 and 65537. (The allowable primes are called the Fermat primes 2^(2^n)+1, and 65537 is the largest known example.
You are starting to encounter diminishing returns at 3:29 onwards because the shapes are starting to look more and more like circles instead of polygons.
