Hypercube construction

27 February 2008 at 11:22 am (Mechatronics, Un-Usual Post, Videos)

I’ve decided to write this post in order to try to give an answer to Eggshell Robotic question “Tesseract / Hypercube – Mechanical Possible?“… The main reason to make a post on my blog rather then making a comment on Eggshell Robotic’s post is mainly the fact that I wrote a lot… and I use images and links… so here it is :

A hypercube is basically 8 cubes organized using the same logic as we use to build a square (2 dimensions) with strokes (1 dimension), and to build a cube (3 dimensions) with squares (2 dimensions)… We take the first cube, join its 6 faces with one face of 6 other cubes… then we go to the 4th dimension by joining all faces that are next to each other… The 8th cube is used to close the hypervolume…

It might be easier to explain it comparing to the cube construction… We have 6 squares (2D) we take one to use as center. We join 4 other squares around the one in the center… (still in 2D)… Now we rotate the 4 new squares to the upper dimension in order to link theme (now we are in 3D) but the cube is not closed, therefore we have a 6th square to close the missing face of the cube… Here is a site showing part of the process.

Now… Imagine a creature that has a 2D perception of the world…. He can only move in 2 dimensions. Now let’s imagine that one of those 2D perceptive creatures live on one of those 4 faces that will rotate to the third dimension of our future cube… When we add his world (the square) to the rest of the cube path, our dear creature will see a huge augmentation in his world… Imagine, in an imperceptible laps of time he sees his world size multiplied by 5 (scientists would get crazy!)… But when we build the 3rd dimension by rotating the cube faces, the creature will see his world size become 5 times smaller and it will not be able to have a perception of the rest of the cube (scientists get crazy again and then they will start research and they will create some hyperspace theory advancing the 3rd dimension)… In fact this creature can’t build a cube… Because he lives in a 2D world (and a cube is, by definition in 3D)… so even if he gets to build a cube, he will only see a bi-dimensional section of it… and he will observe that the rest of the material used to build the cube just “disappears”…

With us it works the same… We have a 3d perception (for spacial dimensions, the 4th being time – or the ability to percept the changes of our space)… If we want to build a hypercube we need 4 spacial dimensions (plus time)… So we could build an hypercube but we won’t be able to see it in its totality… We only see a 3D projection of the hypercube…

What we see in this animation is a hypercube that remains still in our 3 dimensions and that is manipulated in the 4th dimension… To have that resultant movement, we have to be able to manipulate the 4th dimension of the hypercube, if we suppose that we reach to build a hypercube, we would be able to manipulate the 4th dimension… To do so, we need hyper dimensional engines… Really complicated (in my eyes…)

But what forbids us to inspire ourselves with the hypercube transformations to find new ways to move? Nearly nothing (except the physical constraints and costs)… To be honest I like the idea presented in Eggshell Robotic… And who knows, maybe sometime in the future we don’t play hyper-rubicube?

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