Message #551

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Something interesting and strange about permutations
Date: Wed, 13 Aug 2008 01:21:04 -0500

In reference to how to think about rotations, I thought I’d also share again
the link to the wikipedia article on the 4-dimensional rotation group
SO(4)<http://en.wikipedia.org/wiki/SO(4)>.
The reason being that all of this discussion of how to think about rotations
applies only to "simple rotations". In the 4D case, there are additionally
"double rotations", which leave *only the origin fixed* during the motion
(in contrast with all 3D rotations), and "isoclinic rotations", which are a
special case of double rotations with different properties (double rotations
generally leave only 2 planes "invariant as a whole" or invariant in the
sense of being rotated in themselves, while isoclinic and simple rotations
leave an infinite number of planes invariant as a whole). And one can even
further distinguish between two types of isoclinic rotations.

All of the more complex rotations are built up from two simple rotations, so
you might say that the extra classification is unnecessary. However, during
my investigations while coding Magic120Cell, I was presented with thinking
about how to make any arbitrary 4D rotation in a single step, and found
these additional motions and their various properties worthwhile to think
about. In 5D, I imagine the different possible combinations of simple
rotations produce yet more unique behaviors.

As for the simple rotations, I do like that there is the possibility of
looking at them in dual ways (to allow any extra insight one might gain from
the various perspectives). For myself, my mental model in 3D is still
biased towards thinking of rotations as acting about an axis, probably
because of how standard schooling presents this. My mental model in 4D is
heavily biased towards thinking about rotations as the motion through a 2D
plane instead of about a fixed subspace. This is likely due to my
understanding being built up from my coding efforts and this being the more
natural way to perform the calculations.

Take Care,
Roice

On Mon, Aug 11, 2008 at 5:16 PM, David Vanderschel <DvdS@austin.rr.com>wrote:

>
> >Regarding rotations, I really don’t think that it is
> >helpful to try to think of N-D rotations as involving
> >rotation axes. The fact that 3D rotations are easily
> >visualized as happening *about* an axis is really
> >just a quirk of three dimensions. A better way to
> >think of rotations is that they always occur *within*
> >a 2D plane.
>
>
> I don’t think that there is an important distinction
> to be made here. The (n-2)-dimensional subspace
> orthogonal to the plane of rotation is also called the
> "fixed space" for the rotation. In 3D, it is just a
> line. In 4D, the fixed space for a rotation is a 2D
> subspace. Defining a rotation in terms of its fixed
> space or its plane of rotation are essentially
> equivalent, since the two subspaces are always related
> by orthogonality. When Nelson wrote, "The axis of
> rotation of a figure in N-space will always be
> composed of a segment of N - 2 dimensions.", his "axis
> of rotation" would more appropriately be referred to
> as the "fixed space for the rotation" and his
> "segment" would more appropriately be referred to as
> "subspace" or "hyperplane".
>
> >In other words, while an object moves under the
> >influence of any single rotation in any number of
> >dimensions, any point of that object will move in a
> >circular arc within a single 2D plane. In 3
> >dimensions there will be a single rotation axis that
> >cuts through the centers of rotation of all those
> >parallel planes but in 4 dimensions there can be more
> >than one axis that does that, so try to forget about
> >axes and just look for the planes of rotation.
>
>
> I think this is poor advice. It is often very useful
> to be able think about a rotation in terms of its
> fixed space. Indeed, the reasoning for what to use
> for a rotation often involves thinking about the
> aspects of state that you do NOT want to change.
> I.e., it is a constraint on the fixed space that
> may motivate the choice of rotation plane.
>
> .
>
>
>