Hypercubes
In Geometry we can have different dimensions.
The general idea of a cube in any dimension is called a hypercube, or n-cube.
A 0-cube is a point, a 1-cube is a line,
a 2-cube is a square, a 3-cube is a cube, etc
Points, Lines, Surfaces, ...
The magic binomial x+2 can tell us how many points, lines, surfaces etc for each dimension:
(x+2)0 = 1
For zero dimensions we have 1 point
(x+2)1 = x + 2
We have x (a line of length "x") and 2 points
(x+2)2 = (x+2)(x+2) = x2 + 4x + 4
We have x2 (representing the area), 4 lines each of length x, and 4 points
Now (x+2)3 = (x+2)(x2 + 4x + 4) = x3 + 6x2 + 12x + 8
We have an x3 (the volume), 6 surfaces, 12 lines and 8 points.
Verify it for yourself ... how many faces are there on a cube? How many lines, how many points?
But we can go further ... into higher dimensions!
A Tesseract is the 4D version of a cube: a 4-cube.
(x+2)4 = (x+2)(x3 + 6x2 + 12x + 8) = x4 + 8x3 + 24x2 + 32x + 16
So a 4-cube has:
- 1 4D space
- 8 cubes
- 24 surfaces
- 32 lines
- 16 points
We may have trouble imagining what it looks like, but we can know its facts!
How on Earth does this Work?
It is pure magic ... and the fact that x+2 describes a line with two points.
Maybe if we get more general it would help?
More General
Let us step away from pure cubes and allow different sizes:
(x+2)1 = x + 2
We have a line of length "x" and 2 points
(x+2)(y+2) = xy + 2x + 2y+ 4
We have a rectangle of area xy, with 2 lines of x, 2 lines of y, and 4 points
(x+2)(y+2)(z+2) = xyz + 2xy + 2yz + 2xz + 4x + 4y + 4z + 8
We have a cuboid with volume of xyz, 2 surfaces each of xy, yz and xz, 4 lines each of x, y and z, and 8 points.