# playing with pebbles

we’ll never know if ancient men did manipulate and actually think of numbers through pebbles. but for some reason i found myself thinking that if a primitive mind ever did so, perhaps it would have made sense to it to arrange the pebbles in different shapes and structures, like, for example, rectangles.

of course many numbers can be arranged as a rectangle of pebbles. but some cannot. those are the prime numbers, numbers than you can count by putting one pebble after the other in a line, or a one dimensional arrangement, but not in a rectangle. so, perhaps, instead of primes we could call them 1 dimensional numbers. from all of the non primer numbers, from those which can be arranged as rectangles, some would actually accept 3 dimensional arrangements into boxes as well… but not all would! so, there should be some numbers which we could call 2 dimensional numbers, which would look like prime numbers to beings of a three dimensional world which were unable to discern below the two dimensions of the plane. of course, in current terms, these are numbers which are the product of two primes. the numbers which cannot be arranged in 3 dimensions, but can in 2 (and therefore in 1), are:

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35…

the idea can be extended to 3-dimensional numbers, etc etc. in summary, we have:

 0d numbers 1 1d numbers (primes) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, … 2d numbers 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, … 3d numbers 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, … 4d numbers 16, 24, 36, 40, …