2x2x2x2, part 4

So we’ve seen that you can represent a 2x2x2 cube on the page by placing its third dimension next to its first two dimensions. You can also represent a 2x2x2x2 hypercube on the page using a similar trick.

It takes sixteen little cubes to make a 2x2x2x2 hypercube. We can visually represent those sixteen little cubes as follows:

0 1
2 3

4 5
6 7

8 9
a b

c d
e f

I’ve represented each hypercube as a tiny square, and all the hypercubes are labeled. The labels are in base sixteen. In base sixteen, after you run out of numbers, you continue on with letters, using a, b, c, d, e, f to represent 10, 11, 12, 13, 14, 15.

There are four dimensions: x, y, z and w. Each tiny 2×2 square represents x and y (columns for x, rows for y). Each large collection of squares represents a value of z and w (columns for z, rows for w).

So you can see that every one of the 16 locations represents a unique combination of x,y,z and w. For example, the top left corner represents 0,0,0,0. The top right corner represents 1,0,1,0 (in other words, x and z both have value 1, while y and w both have value 0).

It’s important to keep in mind what we are doing here: We are discussig a 4D object, but we sre visually flattening it out, so that we can show it on the page.

More tomorrow.

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