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Tuesday, August 17, 2010

Programming 4D rotations, SO(4).

Just had a ( minor ) cognition. I was trying to figure out a formula for rotating around a 4-dimensional line. As will become clear soon I did not make much progress. But let's start from the beginning.

If you follow my blog you might have seen videos of a Mathematica program I wrote. You can select a cube or tetrahedron, then from a list of all the rotationaxes of that polyhedron and rotate the object with a rotation slider control. The rotation is visible as well as a projection of it in 2D. - I wanted to extend this to 4-D. So take a 4D polyhedron, i.e. a tesseract. Show the rotation-axes (...) rotate it and show projections in 3D and 2D. - In order to do so I would need 4D rotationformulas, I supposed that they would depend on an axis and an angle just like in 3-D.

The 4D space is simply created by adding an ( imaginary ) axis. Points in 4D have four coordinates. Concepts like distance and angle are similar to the defintions for 2D and 3D. In 2D we rotate around a centre of rotation, a point. But extend 2D to 3D and you see that a 2D rotation is actually a 3D rotation around the z-axis. Thinking along these lines there is no such thing as rotating around a line in 4D. In 4D and above we rotate around planes and the six standard rotational planes in 4D are the XY, XZ, YZ, UX, UY and UZ planes. - This makes perfect sense because if you rotate a cube in 3D along one of the standard rotation axes X, Y and Z the equivalent 4D rotations are those in the YZ, XZ and XY planes. The remaining three rotations seem to distort the cube or make it disappear all together, while in 4D-reality the cube would remain fixed of course.

It turns out that the formula for rotation are simple. Setting up a hypercube is not too difficult either. Haven't added it to the program though.

What might prove difficult though is finding ( computable ) data of the symmetry groups of the 4D-polyhedra.

This website 4D Euclidean Space was very helpful in learning and compiling this data.

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