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So, remember when I said that a matrix can be used to show a set of points? Well, let's talk about that.

A line segment has two points. One of these is at \begin{bmatrix} 0 \end{bmatrix}. The other is at \begin{bmatrix} 1 \end{bmatrix}. The pointframe of a line segment can be represented by the single matrix \begin{bmatrix} 0 & 1 \end{bmatrix}.

A square's pointframe can be represented by the single matrix \begin{bmatrix} 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \end{bmatrix}. A cube's pointframe can be shown with the single matrix \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \end{bmatrix}. So on and so forth. What do you notice about these matrices? They represent a single vector with a dimensionality of \begin{bmatrix} d \\ { 2 }^{ d }-1 \end{bmatrix}.

Which brings me to a very interesting observation: every single unique pointframe, with any number of dimensions, is represented uniquely by a single point with a dimensionality expressible in the form \begin{bmatrix} { x }_{ D } \\ { y }_{ D } \end{bmatrix}.

Let me emphasise that: every single shape is out there in this unexplored area. And I haven't even scratched the surface yet.

So, time to explain what hyperdimensionality is.

A square's pointframe has a vector location of \begin{bmatrix} 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \end{bmatrix}. It has a dimensionality of \begin{bmatrix} 2 \\ 3 \end{bmatrix}. It has a hyperdimensionality of \begin{bmatrix} 2  \end{bmatrix}.

The hyperdimensionality is the dimensionality of the dimensionality of the vector. Notice that, by only taking the single top entry of the dimensionality vector, we find the traditional dimensionality of the shape - a square, for example, is traditionally 2D.

A traditional point has a hyperdimensionality of \begin{bmatrix} 1  \end{bmatrix}. The pointframe of any shape has a hyperdimensionality of \begin{bmatrix} 2  \end{bmatrix}.

So, tell me, ladies and gentlemen: what does a hyperdimensionality of \begin{bmatrix} 3 \end{bmatrix} represent?

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