# 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**

**x**(a line of length "x") and

**2**points

(x+2)^{2} = (x+2)(x+2) = **x ^{2} + 4x + 4**

We have **x ^{2}** (representing the area),

**4**lines each of length

**x**, and

**4**points

Now (x+2)^{3} = (x+2)(x^{2} + 4x + 4) = **x ^{3} + 6x^{2} + 12x + 8**

We have an **x ^{3}** (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)(x^{3} + 6x^{2} + 12x + 8) = **x ^{4} + 8x^{3} + 24x^{2} + 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**

(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.