The Corner

At Sixes and Sevens

Yeah, yeah, *five* Platonic solids.  They’re right there in the link I provided.  I think the four-ness of the tetrahedron had taken control of my brain; rather as eleven-ness took control in the classic Martin Gardner essay about the first Moon landing.  Gardner claimed that the famous garbling of Neil Armstrong’s words “That’s one small step for a man, one giant leap for mankind” resulted from there being twelve of them.  That first “a” had to be dropped in transmission to preserve the eleven-ness of the whole enterprise (officially known, of course, as “Apollo 11″).   The number five is critical in an oddly large number of math situations.  Not only are there five Platonic solids, five is also the largest number of sides the face of a Platonic solid can have.  If you go up a dimension, there are six Platonic “solids” in ordinary 4-dimensional space; but in five and every higher dimension, there are only three (the analogs of the tetrahedron, cube, and octahedron).    Also, if you compute the 2-dimensionsal “volume” of a circle (i.e., its area), the 3-dimensional volume of a sphere, the 4-dimensional “volume” of a 4-dimensional sphere-analog, and keep going up through the dimensions, the volume peaks at dimension 5 and then declines for ever.   Five is also the 5th Fibonacci number (1-1-2-3-5-…), the hypoteneuse of the smallest Pythagorean triangle (3-4-5), the number of Fermat primes known to exist, etc.  Algebraic equations of degree five and up have no general algerbraic solution, and so on.

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