We can understand a bit more about the nature of 4D rotations by considering their axes and their poles.

In 2D any rotation leaves a single point (which is zero dimensional) unchanged.


In 3D a rotation leaves a single line(1D) unchanged.

When this axis of rotation passes through a sphere it intersects it at a pair of (0-dimensional)points – sometimes called the poles of the rotation. This can be seen in 2D as the fixed points of a Mobius transformation.

Now in 4D the axis of rotation isnt a line, but a plane(2D). Just as the 1D axis of a 3D rotation intersects a 2-Sphere at the 2 poles, the 2D axis of a 4D rotation must intersect the 3-Sphere at its poles. But what are the poles of the 3-sphere like ? Well instead of being 0D points they are 1-Dimensional lines. In fact they are linked circles. In these animations you can see that the objects are swirling through and around one of these circles much like the way smoke flows in a smoke ring. The opposite pole is actually a straight line through the centre of this circle. Rotation about this pole is visible in the way the objects turn as they swirl.

It is also possible choose the axes of rotation so that the poles both project to circles.

By changing some of the constants in the code it is possible to make objects rotate about one pole by a different amount than it rotates about the other pole. For example, to make the non-orientable surfaces shown earlier, I set one rotation to half that of the other, so that when the line goes 360° about one pole, it has only gone 180° about the other, hence a path on the surface swaps sides as it goes around.
Now if the rotation about each pole is by an equal amount you have something called an isoclinic rotation, and this is where things start to get truly mind-bending…

Continue to Page 4 of 4-Dimensional Rotations

5 Responses to “4-Dimensional Rotations – Page3”

  1. That elephant looks weird lol. Like the animation and the rotations of 4-d images of animals in a 2-D depiction of space. A.K.A the computer monitor.

  2. Avanti Shrikumar Says:

    “Just as the 1D axis of a 3D rotation intersects a 2-Sphere at the 2 poles, the 2D axis of a 4D rotation must intersect the 3-Sphere at its poles.”

    I found the reference to “the 2D axis” a little misleading, because you are talking about _double_ rotations where there are actually _two_ 2D planes involved. In a _simple_ 4D rotation (where there is just one 2D plane), the plane would intersect a 3-sphere in just one circle(=”pole”) (a single plane can’t intersect the 3-sphere in two completely disconnected circles, can it?). Of course, I’m no theoretical math major…but I think I’m right about this one; the wiki link you gave explicitly calls isoclinic rotations a kind of double rotation.

    Good animations, though.

  3. Daniel Says:

    Hi Avanti,

    I see you are quite right. Thank you for pointing this out.
    I’ll have to think about this a bit to try and write a more correct description.

  4. Lineo Says:

    just watch the classical hypercube angle were one of it’s cubes(sides) looks like a small cube inside a big cube.we all know it well.and when you watch it rotating in the fourth direction it looks like everting the cube.yeah i guess everting is the closest thing you could call it in 3 dimenions.it is also what you can see in the third picture.don’t think it really is everting it just looks like everting in our three dimenions, thats all

  5. poll Says:

    how do you animate the horse?

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