Home > Uncategorized > Visualising a #Hypercube, ps: the earth is flat.

Visualising a #Hypercube, ps: the earth is flat.

S_Arch

The arch in la defense, Paris is designed to match the shadow cast by a shape called a hypercube. Unfortunately for mere mortals, a hypercube is a shape that is very hard to visualise, because it stretches into an additional dimension.

So, what is a hypercube, and why did I say the earth was flat?

Well, a cube is to a square what a hypercube is to a cube.

A Square has 2 dimensions, a cube has 3 dimensions, and a hypercube has 4 dimensions. Unfortunately, we have no perception of a 4th dimension, so it’s really hard to imagine a hypercube, but it can be described easily.

Dimension Descriptor Points Edges
0 Singularity 1 0
1 Line 2 1
2 Square 4 4
3 Cube 8 12
4 Hypercube 16 32

Now, lets take things back, and think of a simpler shape, a bit closer to home, – actually, earth.

A circle has 2 dimensions, a sphere has 3 dimensions, and a hypersphere has 4 dimensions.

For most of us, – and I’m excluding any astronauts among us, then the earth is effectively a 2 dimensional plane. If you travel north from Canada, you end up in Russia. If you travel south from Argentina, you end up in Australia, so, to take a ptolemaic view of the world, then the map of the earth is as follows:

flat earth

And that map extends outwards in all directions, to infinity.

But what’s wrong with this map?, If you travel north from Canada, you end up in Russia. If you travel south from Argentina, you end up in Australia?, that’s exactly what happens. If you wanted to travel by land, then this map is perfectly accurate.

The issue is perspective, in the same way that Ptolemy’s map of the solar system was observationally correct, but from an incorrect perspective. (Thanks to the church)
AT02FG06

So, to take this analogy further, to the hypersphere, an astronaut travelling in any direction, up, down, left, right, for long enough, would reach his starting point again.

Like if you left London by boat, travelled to America, Asia, Middle East, Europe, then back to London again. In the same way, a space ship on a hypersphere could travel past our galaxy, through another few million galaxies, then arrive home again, without changing direction.

The effect of light travelling from one side of the universe and arriving at the other has been observed, which has led astronomers to believe that is a suggested shape of our universe is a hypersphere (*actually a hypertorus, like a giant 4D doughnut).

Since the sphere of the earth is so big in our perception, it appears two dimensional, so the 3D curvature of the earth is not perceivable, to the average person. In the same way, the universe is so large, that the 4D curvature of the universe is not perceivable to humans. Even if the universe were smaller, the curvature would still be beyond our comprehension, but the effects of which would be visible. If the universe were the size of our solar system, then we’d look out, and see our own sun on a starry night (many times over)

So, taking it back to the Hypercube.

If you were an ant, walking around a dice, and wanted to map the surface of your new world, then you’d get a map like this:

ant-on-a-dice

Notice, that the pattern repeats for ever, and that this little ant, can never escape his home. every corner he turns, either forward, back, left or right, there is just another face with a number on it.

It doesn’t bode well for the actors in the movie “Hypercube” who also, would not be able to escape their 4D prison, when restricted to three dimensions of movement.

inside a hypercube

And it’s important to mention that this pattern repeats for ever in all directions, up, down, left, right, forwards, and backwards. The same map would apply for waking on the inside or outside of a hypercube.

Although this is effectively a ptolemaic view of a hypercube, it’s the best our 3D minds can visualise at the moment.

 

 

 

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