Two other great reads for helping to understand *n*-dimensional spaces are *Flatland* by Edwin A. Abbott and its companion, *Sphereland* by Dionys Berger.

I’ve done a lot of reading on hyperspace, but I think it was in *Flatland* that I read the following analogy:

A point has zero dimensions. Slide it in any direction to create a line.

A line has one dimension and two terminal points. Slide it perpendicular to that dimension to create a square.

A square has two dimensions, four lines, and four terminal points. Move it perpendicular to those two dimensions to create a cube.

A cube has three dimensions, six squares, twelve lines, and eight terminal points. Move it perpendicular to those three dimensions to create a tesseract (hypercube, 4-cube).

A hypercube has four dimensions, eight cubes, 24 squares (I think; here the math fails me and I don’t have time to look it up) . . . and 16 terminal points. Move it perpendicular to those four dimensions to creat a 5-cube.

. . . and so on. The number of each element proceeds in a logical progression as you add dimensions.

Another illustration goes something like this:

The cross-section of a line is a point.

The cross-section of a cube (assuming perpendicularity) is a square.

The cross-section of a hypercube (4-cube) is a cube.

The cross-section of a 5-cube is a 4-cube.

etc.

One of these books describes “trapping” a higher-dimension creature in the next-lower-dimension world and how the lower-dimension inhabitants would see it. For example, in Flatland (2-D), a 3-D being could step into Flatland from “above” (which Flatlanders cannot understand, because they have only two dimensions), and Flatlanders would see it as a suddenly appearing point that would “grow” into an irregularly shaped 2-D blob (really a cross-section of its body). They might surround it with a 2-D string and think that they have trapped it, but the 3-D creature could simply lift itself free out of Flatland and “disappear.”

Similarly, a 4-D creature could “drop” into our world, appearing as a slowly growing 3-D blob (a cross-section of its 4-D body). We could put the blob in a bag and tie it shut, but the 4-D creature could simply pull itself “out” of our space and “disappear” to our eyes.

But the 4-D creature would be similarly stymied by a 5-dimensional creature visiting its 4-D world.

Freaky stuff!