>3 dimensional visualisation

I’ve always been fascinated with the problem of thinking about more than 3 dimensions. Humans only needed to survive in a 3 dimensional world and are therefore best suited to think about 3 dimensional shapes. Although we don’t really think in 3 dimensions. We don’t have a 3 dimensional volume in our head that we can slice and rotate or manipulate in any way we see fit. It’s more like a association structure. We store things like: this edge is connected to this one with this angle. And due to our need to work with objects: If I push my finger here, this is what it feels like and this is how the object will respond. To illustrate why we do not think in 3 dimensions, consider the tetrahedron. Most people think of the tetrahedron as a triangle connected to a single point above it:

Tetrahedron formed by starting with a triangle base.

But the exact same shape can be formed by taking two equal length line segments, setting them up perpendicular and on a small distance from each other:

Tetrahedron formed by starting with two perpendicular lines.

When I asked some people about it, they always responded first that they didn’t think it is the exact same shape. After a bit of visualization, they came to the realization that it is. This means that one 3d shape can be stored in very distinct ways.  This is also the basis for a lot of optical illusions. Basically, we don’t store exactly what we see, but we store what we need to work with the object.

Now is this possible for four dimensions? You can kind of get an intuitive idea of four dimensional objects by thinking of the fourth dimension as time. A hypersphere would then look like a regular sphere that quickly pops into existence, slows down it’s growth until maximum size, and then speeds up shrinking until it pops away again. A unrotated(with it’s edges parallel to the coordinate system’s axes) cube is also easy. But when you rotate it and then take the 3 dimensional slice, things get more complicated:

Slice through a 4 dimensional hypercube.

Another approach I tried is to start with rotation on one axis and then add more rotations in more and more dimensions:

A transparent 6 dimensional cube projected in 3d, projected again in 2d.

The red,green and blue values correspond to the distance in the 4th,5th and 6th dimension towards the camera. Rotations are shown on the left, where each rotation is designated by their corresponding plane. Multidimensional rotations do not have axes but have planes. A rotation of a coordinate around a plane will result in that coordinate having the same distance to that plane after rotation.

Playing with the two applications will get you a step further in thinking about 4 dimensional objects. But if a person is actually capable of doing complicated operations on them, that remains to be seen.

I’ll post the 6d rotation example on openprocessing if I can get it to work with openGL in applet form.

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