Couldn’t a sphere simply be thought of a a polygon with an infinite number of infinitesimally small polygons?
Yes, but how many of those infinitesimally small polygons meet at each vertex? You can’t arrange them so that the same number meet at every vertex, and also that each polygon is perfectly regular and has the same number of sides. If you could, then you’d have a sixth platonic solid, in my opinion.
If only Plato had known of Calculus. Alas, he was cataloging his solids a thousand-plus years too soon.
There’s a rather neat proof that there could only be at maximum 5 platonic solids with the geometry of our universe.
First, consider which regular polygons you’re going to use. A polygon needs to have at least 3 sides. For 6 sides, once you have 3 of them meeting at a vertex the angles add up to 360 degrees, so you can’t create a convex solid with those. And it’s more than 360 degrees for polygons with more than 6 sides. So we can only use triangles, squares, and pentagons.
For squares and pentagons, you can only have 3 polygons meeting at a vertex without exceeding 360 degrees. So those each define a possible platonic solid–you have to construct it to see if it’s actually workable and how many faces/edges/vertexes it has, but you can’t create two different platonic solids with the same shape of faces and the same number of faces meeting at each vertex.
Similarly, with triangles, you could have 3, 4, or 5 triangles meeting at each vertex. That gives you 3 more candidate platonic solids.
In a way, you can think of a circle as a (regular) polygon with an infinite number of sides, but that works because the circle is the limit of the polygons as the number of sizes increases to infinity. There’s no corresponding limiting process with Platonic Solids; there are no polyhedra that are arbitrarily close to a sphere while counting as a genuine Platonic solid.
In fact, this is the last proposition of the last book of Euclid’s Elements. Though, you have to be careful in your definition of “Platonic solid”. Euclid wasn’t careful enough, and his definition actually included at least two more. Possibly more than that, even, depending on your definitions for “polygon” and “solid”.
Just for some examples: Is this a Platonic solid? It meets Euclid’s definition, even though he didn’t realize it. How about this, or this, or this? All of these are considered, in some mathematical context or another, to be Platonic figures.
Remembering that unreferenced claims in Wikipedia are NOT proper citations … I’ve found:
From the article “Greek mathematics” … and several other places contained similar unreferenced claims … so it’s an open question whether Plato himself knew of the concept of the “infinity small” and whether he knew to apply such to his regular solids …
To a large degree, these five polyhedra are defined as Platonic Solids … and aren’t the only regular polyhedra to be found … for example there are four concave regular polyhedra known as the Kepler–Poinsot polyhedra …
My understanding is that derivatives and integrals had been around awhile … what Newton did was connect the two … such that ∫ f’(x) = f(x) + C …
Obviously, an infinite number meet at each vertex. They’re all infinitely small circles, which are polygons with an infinite number of vertices. It’s infinity all the way down. (Makes your head asplode if you think about it too much.)
No, because no number of circles can meet at a vertex.
Head definitely not asploded.
Huh? You can’t have more than 2 circles meet at a vertex, and then you have gaps around them.
Archimedes certainly did something similar to integrals, but he was also a couple of centuries after Plato. Might you be confusing him with Aristotle, Plato’s student?
But they’re infinitely small circles. In other words, they’re indistinguishable from points.
What does it mean for one point to meet another point?
It means my head asploded.
Doesn’t make any difference. If the sides of the polygon/circle don’t touch each other, and you shrink it down, they still won’t touch each other. No matter how small the circles are, you can’t tile a surface with them. There will ALWAYS be gaps. In fact, the area of the gaps (as a fraction of total area) will remain constant as you shrink the circles.
It means they are the same point.
It all depends on how you take the limits. Imagine, for instance, taking three equal circles tangent to each other. Now, take the gap in the middle that’s outside all three circles, and inscribe in that gap the largest circle that you can. This will create more gaps; again, inscribe in each gap the largest circle that will fit in that gap. Repeat this process infinitely.
Now, look at any point of contact between any two circles. For any given nonzero delta, you can find a circular neighborhood of radius epsilon such that there are an infinite number of circles within that neighborhood, all of which have radii which differ by less than delta. In a limiting sort of sense, then, it seems like it might be fair to say that each of those points is a vertex at which an infinite number of circles meet, all of the same size.
If you like, a flat disc, thought of as having two circular faces (top and bottom) is an infinitary Platonic solid whose faces have infinitely many “edges” (they are circles), with two meeting at each of infinitely many “vertices” (at each point along the edge, the top and bottom face meet).
[And for its dual, if you like, an infinitesimally thin but sizably long cylindrical rod is an infinitary Platonic solid with infinitely many “faces” each with two “edges”; its two vertices are its top and bottom, its infinitely many “faces” are infinitesimal strips running from top to bottom facing out in each longitudinal direction, with infinitely many “edges” also corresponding to each longitudinal direction.]
But if you want an infinitary Platonic solid whose faces are infinity-gons but more than two of them meet at each vertex, then this naturally takes you outside of Euclidean geometry and into hyperbolic geometry, landing at this figure: Order-3 apeirogonal tiling - Wikipedia. Similarly, to have four infinity-gons meet at each vertex yields Order-4 apeirogonal tiling - Wikipedia. And so on. If you were curious about infinitely many infinity-gons meeting at each vertex, you get Infinite-order apeirogonal tiling - Wikipedia. None of these close up into spheres; their topology is different.
Perhaps, but if you pick a point of contact between any two circles, any other particular circle you consider will not touch that point of contact. It seems odd, then, to consider that point as touching more than those two circles, since there’s not a single other circle one can point to which it touches. I’d describe this tiling as having infinitely many infinity-gons, each touching infinitely many other infinity-gons, but with just two infinity-gons meeting at each vertex.
(I’d also note that in this case, interestingly, the infinity-gons are not like the apeirogons I was discussing before, in that the adjacency structure of their vertices are different; instead of each vertex on an infinity-gon having a clockwise and counterclockwise successor (as with the apeirogons from before), we now have that between any two vertices on the same infinity-gon are infinitely many other vertices, with an inbetweenness structure isomorphic to the dyadic fractions modulo 1 rather than the integers.)