I know, I know, you're a three-dimensional person. (Or are you? On the Internet, nobody knows you're a pancake. Or Kansas.) But we're told that doesn't matter:

Up until the late 1990s, it was commonly thought that 3rd Dimensia was only a disorder for patients dealing with 2-to-3-dimensional crossover. But today, scientists and doctors know better. Be warned: 3rd Dimensia does not discriminate. It can strike anyone at anytime.One of the diagnostic questions is "Do you settle for just jumping over objects and projectiles?" -- I suspect that part of the reason video game characters can jump so high is because in two dimensions there is only one

*horizontal*dimension, so the more realistic option of swerving

*around*an oncoming enemy simply isn't available to them. If this is so, then I'd suspect characters in 3-D games can't jump as high (relative to their body height) as those in 2-D games; at some point the designers should have realized that real people don't jump that high, and designers probably feel more constrained by real physics in 3-D games than in 2-D games.

Try as I might, though, I couldn't find a reference on that site to my favorite fact about two-dimensional life -- namely that their digestion must work differently from ours, because topologically they are unable to have a digestive tract. Presumably they'd absorb nutrients through the skin, like unicellular life forms in our world do. I suspect larger 2-D life forms would have fractal surfaces, to get a large surface-area-to-volume ratio, similarly to how we have branching networks of blood vessels so that oxygen can get to all our tissues. This is one thing that Edwin Abbott Abbott got wrong.

## 6 comments:

I recently brought my wife with me to Kansas to visit my family, and she was shocked by how hilly the area was. Of course, Kansas is a few hundred miles wide, and other than the eastern 50 miles or so, it is quite flat.

Whenever the flatness of Kansas is brought up, I'm always reminded of how the earth is smoother than a billiards ball.

I don't see any obstacle to having a two-dimensional digestive tract. The point is that topological disconnectedness is less relevant than geometric disconnectedness. So, if your digestive tract is twisty enough, you won't fall apart even though you're really in two pieces. I guess you'd need two hearts, one on each side, though.

d. eppstein,

that's an interesting point you make about geometric disconnectedness. I hadn't thought about that.

Still, some sort of twistiness seems necessary -- either in the digestive tract or in the fractal surfaces I was envisioning.

I think whole organism can be assembled in jigsaw puzzle fashion, from hypothetical 2D chemicals, cells, and so on, with nutrients supplied through inter-tile environment. It can also have active moving parts and so on.

Last week I got flamed for similar discussions. At church I was trying to explain to an opthamologist that there were many dimensions. I used various measurements that he might make to illustrate the idea. I don't think he got it.

On the other hand, an experienced drummer can move her four limbs independently. Thus she can experience a space of at least 4 dimensions --- one for each limb. Configuration spaces give all sorts of higher dimensional examples.

Here is one especially for you Isabel: Consider the intersection of the cube [0,1]x[0,1]x[0,1]x[0,1] with the hyperplanes x+y+z+w=k, for k=0,1,2,3, or 4. The intersections will be a vertex, a tetrahedron, an octahedron, a tetrahedron, and a vertex respectively. If you want to understand this in many higher dimensions, then you can do so inductively. The intersection of the n-cube with the hyperplane \sum x_i =k is the topological join of the intersection of the (n-1)-cube with the plane \sum x_i = k-1 and the (n-1)-cube with the plane \sum x_i = k. In the last sentence the sums ranged from

i=1 to n-1. Just work with the vertices of the n-cube as binary sequences. This is just the geometric realization of Pascal's recursion.

As an alternative to having a fractal boundary, a two-dimensional organism could encapsulate food in vacuoles and let the vacuoles circulate around inside the body, releasing nutrients as they went. It would take energy to keep the circulation going, but there are probably energy costs associated with building and maintaining a complicated fractal outer surface too...

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