Or any other Catalan solid. But strictly speaking, a d10 isn’t two pyramids joined base-to-base: That would be a dual prism (bipyramid). But the d10 is actually a dual antiprism (trapezohedron).
As for the definitions of various solids, it’s interesting to note that Euclid’s definition of the Platonic solids wasn’t quite rigorous enough. His definition would also have included the pentagonal and triangular biprisms, since he forgot to specify that all of the vertices be the same.
Could it have to do with the highly efficient sphere packings that can be done in 24 or 48 dimensions?
http://www.itsoc.org/resources/review-articles/the-sphere-packing-problem/
Might be-- I’m pretty sure I’ve encountered that problem before. A Gardner Mathematical Games column, maybe?
That’s exactly where I first saw it, but probably before you were born. If I remember correctly, it was in one of his (ca. 1970) articles about the game of life, invented by John Conway. Conway is a mathematical genius and was a child prodigy. He and Neil Sloane have worked on the sphere packing problem, including breakthroughs in understanding 24 dimensions. It seems like a quintessentially abstract mathematical problem, but it has direct application to error correcting codes for communication.
I just discovered that that Gardner’s column is available online. The first paragraph mentions Conway and his discoveries about 24 dimensional sphere packing, including the solution to the “kissing” problem. Note below the number of spheres that can touch each other:
As others have pointed out, there are indeed only five Platonic Solids, which each use only one type of polygon as all faces. There are quite a few other polyhedra which use two or more types of regular polygons as sides.
But, in a sense, your statement is correct. Each and every regular polygon does have its own polyhedron – only it’s not regular. But its sides consist exclusively of regular polygons.
Actually, there are at least two different tyopes. There are quite a few of them, depending upon how imaginative you are.
In order to make the simplest type, imagine that you have a regular N-gon, a polygon with N sides, each of the same length, and with the same angle between adjacent faces. For the sake of illustration, let’s assume that you have a 100-sided figure. Now imagine a second one lying atop this first one, overlapping it with their verices above each other. Now slowly pull the upper one upwards until it is above the lower one by the length of one side. Connect the corresponsing vertices of the 100-gons, and you have a polyhedrom consisting of two parallel 100-gons and 100 squares.
You can construct a similar figure for any regular polyhedron, with any number of sides. Each such polyhedron will have sides that are only regular polygons, and there is no limit to the number of differrent ones you can create.
A variation of this is to lift the upper one slightly, rotate it so that the vertices of the upper one lie sort of above the midpoints of the sides of the lower one, and continue lifting unytil the upper one is at such a height that the triangles created by drawing lines from the vertex of the upper one to the two nearest vetrtices of the lower polygon form an equilateral triangle. At this point you can join both upper and lower polygons with 2N equilateral triangles (200 of them, in the case of two 100-gons). Again, you can create such a figure for each and every regular N-gon, no matter how many edges it has.
Even the definition of polyhedron is not self-evident. Imre Lakatos wrote a whole book on the history of Euler’s theorem (vertices - edges + faces = 2) which is true for many, but not all definitions, of polyhedron. I think it was called Proofs and Refutations. But even with the most restrictive (simply connected with flat faces, homeomorphic to a sphere) there are still infinitely many but only five regular.
Psst… see post #15.
Most of Gardner’s columns were before I was born, and all of them were before I was remotely old enough to appreciate them. But I spent many an afternoon in college with the department’s library of back issues.
I grew up reading Martin Gardner’s superb columns in my Dad’s Scientific American. It was a sad day when he stopped writing them.