Undergrad Can Complex Shapes Be Rotated in 4D?

[LightningInAJar]

TL;DR

higher dimensional shapes. Simple and complex.

I have seen videos of a 4D rotation of a cube or tesseract. Was wondering if complex shapes can be processed into 4D rotation versions of themselves?

 

[Ibix]

What do you mean by a "complex shape" in this context? Just arbitrary combinations of geometric primitives?

The point about a cube is that it's generalisable to $n$ dimensions because there's a simple rule for generating one. You just write down every possible combination of $n$ zeroes and ones and you have your corner coordinates. Connect each corner to every other corner whose coordinates are the same except for one number. Done. You can then rotate and project that into as few or as many dimensions as you like.

Similar rules generate other geometric solids (although probably not all of the standard 3d ones work in arbitrary dimensional space). But if there's no generalisable rule (which there won't be for most shapes, possibly including whatever you mean by "complex shapes"), what is the 4d equivalent?

 

[LightningInAJar]

By complex I mean not a Euclidean Solid which I assume have convenient characteristics. Can a .STL 3D model file be turned into a 4D rotation? I am curious what more complicated objects look like expanded.

 

[Ibix]

A tesseract isn't a "4d rotation". It's a 4d generalisation of a cube. The problem with a general shape is that there is no generalisation to 4d because there's no generalisable rule for generating it.

Think about a square. The rule for generating it is to write down all possible combinations of two zeros and ones - (0,0), (0,1), (1,0), and (1,1) - then connect points whose coordinates differ in only one place. That exact same rule generates a cube if you change "two" to "three". Now think of an arbitrary polygon drawn on a sheet of paper. How do you generalise the rule for drawing that particular arbitrary polygon to 3d?

The same is true generalising a 3d shape to a 4d one. If you can write a general rule for creating the shape that works in an arbitrary number of dimensions then you can generalise it to a 4d equivalent. But this is not possible for arbitrary polygons.

That said, you can always embed a 3d object in 4d space, just as you can imagine a 2d object in 3d space. You simply take your (x,y,z) coordinate triples and make them (x,y,z,0). You could then rotate this and view its 3d projection. The result would be to scale the object along one of its directions, and possibly to distort it slightly if perspective effects are simulated.

 

[benorin]

There's a small amount of n-dimensional geometry in my insight article A Path to Fractional Integral Representations of Some Special Functions.

 

[LightningInAJar]

So a "3D projection" of a 4D shape within 3D space is possible?

 

[Ibix]

Of course - I used to have a 3d projection of a 4d cube hanging in my room. And the animations you've seen are 2d projections of a 4d object.

 

[LightningInAJar]

Are there free softwares to create such an animation? I want to take some .stl model files and expand them out into 4D.

 

[Ibix]

I don't know, but I doubt it. You could try searching. If you know any OpenGL or enough python to drive Blender or something like that it would be easy enough to write.