# From the third dimension to the fourth dimension

## Last revised 2003-01-19

In realmspace, there is a character named Bob. He is basically your average guy - two legs, two eyes, a brain, all the usual body parts. His view of his world is a 2 dimensional plane, one dimension more than Fred has. Here is Bob in his native habitat, and his view of that habitat:

Since Bob has one dimension more than Fred has, his writing system can be more elaborate. He can write on a 2 dimensional surface. Instead of only one line of text on a page, there can be multiple lines. This allows a lot more text to me stored in a realmic book. Here is what a page from one of his books looks like:

Bob fully understands the concept of a cube. A cube is a sequence of squares laying top to bottom flat on top of each other. For any one square in this cube, every point on one surface of the square touches every other point on the surface of the square above it, and on the other side, touches every point on surface of the square below it. Here is an illustration:

Bob also understands the concept of a sphere. A sphere is a sequence of concentric circles laying top to bottom on each other, as with a square. Like a square, adjacent slices of the circle touch each other at every point. Here is an illustration:

Bob has troubles imagining a tetracube, however. Here is the best he can do to imagine a tetracube:

A tetracube is actually a sequence of cubes, where every part of a cube in this tetracube (all the insides of it) touch all the insides of the cube to its right and to its left (or its up and down, or its front and back). In actuality, the direction of the adjacent cubes are not imaginable in terms of 3d directions. I will call one of the directions into tetraspace Upsilon (for up) and the other Delta (for down).

If Bob was viewing Fred's planar world, he would see that there are two sides to it. Bob could either be on top of Flatland or below it, but in order to get to the other side, Bob must go through Flatland. There is no other way to get to the other side. Planespace separates realmspace into two separate parts. In the same way, a tetral character (Let's call her Emily) would see that she is on one side of realmspace, and she'd have to pass through Bob's world to get to the other side of it. Realmspace separates tetraspace into two separate parts.

In the first illustration below, an arrow shows the path from one side of Flatland to the other. If the arrow was an object, Fred would only see a small point. In the second illustration, a tetral arrow that passes from one side of realmspace to the other would intersect realmspace with a small point. Bob doesn't see any other part of the arrow but this:

One day, Bob is innocently sitting in his living room (as Fred was that one day), and all of a sudden, in the middle of the air, a sphere appears and grows rather rapidly. Its size levels off at a point, then starts shrinking slowly. Its shrinking quickens until it completely disappears. Bob just experienced the passage of a tetrasphere (four dimensional sphere) through realmspace.