...so being a father, and hoping that my children will not struggle through things that made me struggle with mathematical heuristics when I was a kid - although they may struggle with much worse than I did - I suggested my son read it this weekend, and he did. The paragraph, which is found on page 3 of a cute book called Sphere Packings Lattices and Groups (J.H. Conway and N.J.A. Sloan, Springer Verlag 1988 page 3) reads as follows:
We pause to assure the reader that there is nothing mysterious about n-dimensional space. A point in n-dimensional space Rn is simply a string of real numbers
x = (x1, x2, x3,...,xn).
A sphere in Rn with center (u1, u2, u3,...,un and radius ρ consists all the points satisfying
(x1-u1)2 + (x2-u2)2 + (x3-u3)2 +...+(xn-un)2 = ρ2
We can describe a sphere packing in Rn just by specifying the centers u and the radius. Everything is done with co-ordinates and there is no reason to draw pictures...
...and what follows below after this quasi-technical bit is the part I really, really, really wanted my son to read:
There has been a great deal of nonsense written in science fiction...
...about the mysterious fourth dimension. One should certainly not think that the fourth dimension represents time. In mathematics 4-dimensional space just consists of points with four coordinates instead of three (and similarly for any other dimension.
The bold is mine.
I remember as a child, and even as a teenager being confused about the relationship of the possibly inappropriately named fantasy stuff called "science fiction" and science/mathematics.
To be clear, I do find myself evoking spacetime when trying to educate my children about vectors, but as the text makes clear, there is nothing magical about spacetime as a 4 dimensional object. In a physical sense, one could imagine a sphere in which the "dimensions" are for instance, pressure, temperature, density and distance from the star Betelgeuse. (One might argue that I have made a poor choice since often in physics these quantities are often functions of one another, but let's um, not get technical.) In fact a vector need not represent anything at all.
This book is really well written by the way, and pays attention to little things like this. I picked up just for the hell of it.
It has fun passages like this, speaking about the face centered cubic lattice:
In this packing the spheres occupy π/√18 = 0.7405... The classical question of sphere packing now asks: is this the greatest density that can be obtained... As Rogers [Rog2] remarks, "many mathematicians believe, and all physicists know" that the correct answer is 0.7405...
Cute.
Anyway, that's what my son had to read to indulge his father this weekend, and if he so suffered through this point of mathematical heuristics, I thought I'd share it with you, since in general, I'm not a very nice guy.