So, I've been thinking about hyperdimensions lately. Hyperdimensions, in its simplest form, is the extension of dimensions from the normal one-hyperdimensional line (0D, 1D, 2D, 3D...) into a two-hyperdimensional plane. This can take many forms; complex numbers, for example, are one way of expressing this. I'm going to attempt to come up with one hyperdimensional way of thought.

So, first up, you can express vectors (for the purposes of this, the same thing as coordinates) as follows:

\begin{bmatrix} x \\ y \end{bmatrix}

The above represents a point on the plane with the position x across and y up. Pretty clear. A point in 6-dimensional space would be expressed something like this:

\begin{bmatrix} x \\ y \\ z \\ w \\ v  \\ u  \end{bmatrix}

A point on a line would simply be \begin{bmatrix} x \end{bmatrix}

Now, there are some neat things that you can do with this, using things called matrices. A matrix is basically a grid of numbers, and it can represent a few things depending on what you're using it for:

  • A coordinate
  • A set of coordinates
  • A transformation

However, if we treat all dimensions as simply being points in a hyperdimensional space, we can do something interesting: we can say that \begin{bmatrix} x \\ y \end{bmatrix} describes a vector with a dimensionality of \begin{bmatrix} 2 \end{bmatrix}. That six-dimensional vector you saw above would have a dimensionality of \begin{bmatrix} 6 \end{bmatrix}.

You might have noticed something: the dimensionality of a vector is just another one-dimensional vector! And, if you can write dimensionality as a one-dimensional vector, why not a two-dimensional one? Let's try it.

So, firstly, notice that the vector \begin{bmatrix} 2 \end{bmatrix} can also be written as \begin{bmatrix} 2 \\ 0 \end{bmatrix}. They both represent the same thing: moving two dimensions across in the dimension line, then moving 0 dimensions up.

There's no good reason why we can only move 0 dimensions up, though. Let's say we move one dimension up. The dimensionality of our vector is now \begin{bmatrix} 2 \\ 1 \end{bmatrix}. What does this look like?

I think it looks like this: \begin{bmatrix} { x }_{ 1 } & { x }_{ 2 } \\ { y }_{ 1 } & { y }_{ 2 } \end{bmatrix}.

In the next blog post on this, I'll show you some more things about hyperdimensions. Remember to leave feedback and questions in the comments - I love explaining things!

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