Picture a four-dimensional cat.
That's what my brain got up to last night. It does this kind of thing all the time, leaving me rather bemused. Think of something odd. Pull together all kinds of information. Solve a problem. Draw a picture.
I lie in bed petting Tiny (because Tiny finds me too intimidating when I'm not on his level), and I should be sleeping; but instead I look at him and I think... "How many legs would a four-dimensional cat have?"
When you read this, remember that it's transcribed into words. I didn't think of it in words; words are too slow and clumsy. I think in concepts, relationships, and ideas, connecting to each other to form greater ideas. Words come in as key words, here and there: Cat. Legs. Dimensions. Balance. They pin down the corners of the ideas, but they don't make up their substance.
So I listen to Tiny purring, and I think...
Well, what are legs for? They're to balance you against gravity while you shift your weight to move. Cats have four. Humans have two, but we bounce from leg to leg while we walk, alternately falling and catching ourselves, and balancing as much on the two points at the heel and ball of each foot as we do from foot to foot.
A walking cat walks on four points of contact, one point per leg, instead of two points per foot. Three dimensions; four legs. Each pair of legs stabilizes the cat along one dimension: two dimensions against that third up and down pull of gravity. In three dimensions, balance is best achieved with three points--a three-legged stool at rest on a flat surface will always have all three legs touching the surface. Stability; three legs.
A cat has those three legs, plus a fourth for movement. When two legs are on the ground, there are always two more legs to offset a shift to one side or the other. When walking, three legs are on the ground and the cat is balanced even if it freezes in midstep. Stability and motion: four legs.
I picture a two-dimensional silhouette cat. It has two legs. A two-dimensional cat cannot have the stability that a three-dimensional cat does. Like a human, it must hop from leg to leg, using more than one point per foot to stay balanced. But unlike a human in three dimensions, a cat in two dimensions stays effortlessly balanced when standing still. Two legs are adequate for two dimensions.
A four dimensional cat would look like... what? To balance a three-legged stool, we need a two-dimensional plane and three points. To balance a four-legged, four-dimensional stool, to define it in four dimensions, we have to add another point. Four points: Stability.
But four legs is not ideal for a four-dimensional cat any more than three legs is ideal for a three-dimensional cat. When a three-dimensional, three-legged cat moves, no matter how slowly it moves, it must always lose stability during the step, because it must lift one of its three legs to walk, leaving only two, an unstable configuration. And when it balances on those two legs, the third leg can only be on one side or the other of those two legs, and so the cat can tip over on that side. (Realistically, of course, three-legged cats learn not to shift their weight in that direction, so as not to tip over. But this is a theoretical cat. A spherical cat, if you will.) Two legs gives the cat one degree of freedom, and adding another two legs balances the cat through that axis.
So a four-dimensional cat, like our four-legged stool, must have four legs for balance; but on each side of those four legs it must have one leg, placed along the fourth dimension. A four-dimensional cat with three legs is like a three-dimensional cat with two legs: It can balance along all dimensions but one. Add a fourth leg to the four-dimensional cat, and you have the equivalent of the three-dimensional cat with three legs: Stability. But to have movement, do you add one more leg, or two more?
When a cat picks up its leg to walk, it shifts the weight onto another leg. Which leg it shifts to depends on whether it's walking or pacing (same side or opposite side), but the point is, the legs are paired. When it lifts one, there's another leg to partner it, keeping its weight evenly distributed for easier and more efficient movement. So we need an even number.
So my conclusion is: A four-dimensional cat, living in a world with gravity, most likely has six legs.
It took me just over a half-hour to articulate this in words, but last night, when I thought about it for the first time, I went through all those steps in two or three minutes, jumping from the concept of stability, to friction, to images of the cat and the three-legged stool, to ideas about walking, to coordinate systems and four-dimensional vectors. Thinking about four-dimensional cats is a bit silly, but it's a typical enough daydream.
I don't think all autistic people think this way. Our experiences are as individual as our minds. I do know that I do, and I know that words are not as fundamental to me as they are to most people. There are people who believe that without a word to articulate a concept, you cannot conceive of the concept at all (I was first introduced to this idea when I read 1984), but I don't think those people have talked to a lot of autistics. Words make thinking more concrete, but wordless thinking exists and it is the source of many new ideas. When I think of a four-dimensional cat, I don't go through the process of generating all of that language. I don't need to. I don't even really need the "pin words"--cat, motion, stability, dimension, etc.--because those words are just simplifications of concepts much bigger than themselves. Just like the word "cat" cannot contain the idea of catness, the words I produce cannot contain my thoughts.
Do most people have thoughts bigger than their words? I think they must. We have whole universes inside our heads. Words are just the tiny trickle of proxy symbols meant to trigger bits of other people's universes when we say them and they hear them. When we don't have a word and we need to communicate, we make a word; but the idea was there before the word, and was no less real.