Mathematics and Planewalking

An essay on why mathematical study of the planes is relevant, by Rigora Inflexion, a prime sector of the Mathematicians   In my forays across the planes, I have discovered a number of interesting things:   The Spire on the Outlands: This spire is definitely defined by a mathematical equation. Trouble is, no-one can agree on what equation. (Of course, the Bleak Cabal claim that no such good sense exists, but I know what I've seen) Because space curves near the base of the spire, all attempts to measure its curvature have so far failed. Visual estimates from far off don't square with what's seen close to.   Infinity: Not even true-born planars know everything about infinity, even though it touches their lives so fundamentally. A prime called Georg Cantor seems to have had the dark of it quite well. He said that there was not just one kind of infinity, but at least two. He talked about 'countable' and 'uncountable' infinities.   Hold on, I hear you cry, how can you count infinity? Well, berk, it assumes you've got infinite time. But if you stumble on something uncountable, you couldn't count it even with infinite time. The classical example is of the whole numbers, or integers as they are called by the Mathematicians and Guvners. The integers are the definitive countable set. You can go 'One, two, three...' forever, and you'll never miss one if you've got long enough. But if you try to include all real numbers, including the so-called irrationals, like pi (bane of modrons!), you'll find that there's no way to account for all of them. You'll always miss infinitely many for every and any one you count.   To see it, just consider how many numbers differ from an infinitely long number like pi by just a single digit. Get the idea? Now, the question for the infinitude of planes is, which things in the planes are countable, and which uncountable? It is reasonably transparent that people are countable, and arbitrary points uncountable. But when it comes to layers of the Abyss, no-one's sure. If they are uncountable, then some must be empty. What's in an empty layer?   Space dimensions: Here's a topic that invariably confuses those not used to operating in strange environments. Many primes maintain that space is three-dimensional, and that time is a fourth dimension. It emerges that this is a pitifully inadequate explanation when you travel the planes. For example, all planes with a discernible form and more than one layer (except Bytopia, but I'll return to that) have at least four space dimensions: An up-down dimension, a forward/backward dimension, a left/right dimension and an up layer/down layer dimension.   There are exceptions. Mechanus has only one layer and therefore three space dimensions can describe it adequately. Bytopia has two layers, but they meet in a plane. This is like separating numbers into positive and negative. You still only have one line of numbers, but some are positive and others negative. Likewise, half of Bytopia is Shurrock and the other half is Dothion. In this way, Bytopia has two layers but only three space dimensions. Gehenna has four layers, but is a three-dimensional space and all the layers are finite.   The inner planes are always five-dimensional; for example, the plane of Fire has the dimensions up/down, left/right, forward/backward, magma/smoke and radiance/ash. The Astral Plane, just to be awkward, has no layers and no time, but is in fact a bizarre non-space. This leads to many bizarre results.   Limbo and the Abyss are so badly defined that their dimensionality is not a comprehensible concept. It has been suggested that there exist planes we can never visit that are home to two-dimensional beings who would treat a visit to the planes we experience much as a clueless prime experiences the Astral for the first time, especially if they were bodily adrift in the Void.   Speaking of the prime, it might be useful to point out that the prime may have a layered structure too. The prime most people think of is one of perhaps three main primes (one in which magic works but there is no phlogiston, one we know about already, and one from which I am writing this which has no inherent magic at all), and in addition to these, there are a number of 'pocket primes', realms resembling the primes we have mentioned, but actually more like demi-planes in size and inhabited by over-powers. At least, that's the theory.