8 cubes

[blazes816]

Hey, I just had an idea. Mind you i'm new to this, How do we know that the tesseract is made up of 8 cubes? Is it because a square is 4 lines, and cube is 6 squares, so logically a tesseract is 8 cubes? And i'm asking about the number, not shape.

-B8

[PWrong]

One way is to draw the tetracube and count the cubes.

Or you could simply list them: front, back, left, right, top, bottom, ana, kata.

Another way is to look at the equations for a tetracube:
0<x<1, 0<y<1, 0<z<1, 0<w<1

The eight cubes are as follows:
x=0, 0<y<1, 0<z<1, 0<w<1
x=1, 0<y<1, 0<z<1, 0<w<1
0<x<1, y=0, 0<z<1, 0<w<1
0<x<1, y=1, 0<z<1, 0<w<1
0<x<1, 0<y<1, z=0, 0<w<1
0<x<1, 0<y<1, z=1, 0<w<1
0<x<1, 0<y<1, 0<z<1, w=0
0<x<1, 0<y<1, 0<z<1, w=1

[pat]

I think a slightly simpler notation is that the corners of the tetracube are < ±1, ±1, ±1, ±1 >. Now, consider holding one coordinate constant.... there are eight corners with that coordinate. For example, if we keep the second coordinate at -1, we have the eight corners < ±1, -1, ±1, ±1 >. Those form a cube. There are eight such cubes (each of four coordinates at each of two values { +1, -1 }).