GEOMETRY:

HIGHER DIMENSIONS

The hypercube exists in four dimensions of space

What does this mean?

When it comes to visualising and using the hypercube, none of this is essential

but it adds depth and is just really fun

There's also scope for using these deeper geometric insights musically

but thats for another day

Ok let's understand this chart:

Colour Wheel

If we can understand the jump from 2D to 3D

- How a square becomes a cube

We can understand the jump from 3D to 4D

- How a cube becomes a hypercube

If we drew a cube on a piece of 2d paper

We could draw it like this, a little square inside a big square:

Colour Wheel

Its a small 2D square inside a bigger 2D square

A 2D person would think of it as simply one flat square inside another flat square in 2D space

But we know

It's acctually

A 2D square behind a 2D square in 3D space

The hypercube is exactly the same only one dimension higher

Colour Wheel

We've got a little 3D cube inside a bigger 3D cube

We can see the 3D/4D analogy really clearly:

Colour Wheel

For us 3D people, thats the only way we can see the hypercube: little cube inside a big cube

but a 4D person would see the hypercube as having four dimensional depth with the bigger cube 'in front' of the smaller cube

We'll never truly see or grasp this

in the same way a 2D person will never see or grasp the 3D depth of a cube

But the 2D person can understand that the cube is composed of 6 squares

Even if they have to think of some squares being 'inside' others (which they aren't)

(We can do the same with 4D)

Here are the 6 faces of the cube according to 2D people:

Colour Wheel

It's important to note that the last four squares appear shrunken and slanted to the 2D person

but to us 3D people, we know this is only a matter of perspective,

these squares are in fact not shruken or slanted

In 3D, all six squares are equal is size and shape

All this applies up a step in 4D

Here's the 8 cubes of the hypercube from our 3D perspective:

Colour Wheel

Just like with the 3D version before,

the front, back, left, right, top and bottom cubes appear shruken and slanted

But just like with the 3D cube, this is only because of our limited perspective

In 4D, all 8 cubes have equal size and perspective

Just like a 3D cube allows us to connect together multiple 2D squares,

the 4D hypercube allows us to connect together multiple 3D cubes.

By this same logic, the metaprism exists in 5D, because it allows us to connect together multiple 4D hypercubes

Colour Wheel

I do sometimes think with a bit of a 4D mind when dealing with hypercubes,

But other than its construction, I've not explored the metaprism much with a 5D perspective

Story moment:

When i showed my mathematician friend joe my chord wheel prototype (back in 2016)

The wheel was able to spin and had other spinning wheels upon the big wheel

He said,

"That's not a series of flat spinning wheels, its a shadow of a rotating 5D structure,

Every spin of a wheel is acctually a rotation in space"

He was right.

Anyway

There's loads of great resources on the internet/youtube about 4D and hypercubes if you want to learn more about this,

But 99% of the time im just thinking of all these things as simple intuitive 3D objects as we see them :)

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