In fact, it's just about right!įirst off, a home-experiment: Close one eye. While 3D is a bit boring at time, it's also a nice comfortable kind of boring: not too hot, and not too cold. So if you ever find yourself wishing you could be some kind of magical four dimensional creature who could flit in and out of our ordinary 3D world at will, be careful what you wish for, since getting your wish might prove a lot more dangerous than you might think. A sun in such a world would be a tiny (from your $n=20$ perspective) object that would provide essentially no heat at all, since all the heat would be streaming off into all 20 of those dimensions. and that is a number that gets very, very small in a hurry. The size of an object, as measured by its total "area" in whatever form you see it (a dot for string land, a line for flatland, and an area or flat image for our world), will always be $1/s^$. Put that all together and you get a more general rule. So, like a piece of cloth cut in half both vertically and then horizontally, you end up with a total image size of only $1/(2*2)$, where each $2$ is the distance $s$ again. But since we see distant images as two dimensional, like images on movie screen, the total image size for something twice as far away as $s=1$ is also cut in half for its height, as well as its width. The circle trick works exactly as before, so an object twice as far away will again look half as wide. Now what about us? We live in a three-dimensional land, or $n=3$ if you use n to give the number of dimensions. Mathematically that comes out to $1/s$, where $s$ is the distance to the object on a more distant circle if the first circle is at $s=1$. That means that an object on that circle must look have the size as the same object on the inner circle, because it will take up only half as much room on the doubled circle length. If you use standard geometry or just measure it, you will find that the second circle is twice as long as the first one. To figure that one out, draw two circles, one an inch from a center, and the second one two inches away. How about two dimensions, which was called Flatland in a famous 1884 book? You had better be on good terms with all of your neighbors in string land, no matter how far away they are! That is, it doesn't matter how far away the object is, it's size (and impact) will still be multiplied by exactly one. Mathematically, that comes out to a factor of 1 for any length $s$. The explosion "looks the same" at the mouth of the tunnel as it does a mile in. An explosion in a tunnel is channeled entirely into just two directions, and only very slowly loses force through friction as it moves. It's also why they don't let explosives trucks into long tunnels. This is why optical fibers are so great for communications, incidentally, since they are one-dimensional worlds where the light just keeps on doing the exact same thing no matter how far it travels, and looks just the same when it arrives. It may take longer to arrive, but that's about it. If you think about it a bit, the answer is "none" - they both look like dots, since light in string land can only travel one way and never changes angles or intensity. Here's an example: For a one-dimensional or "string land" creature, what would be the apparent difference in size between a dot nearby and a dot many miles away? Yes, it's very much physics related: The perceived smallness of distant objects is a direct function of how many space dimensions we live in.
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