h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Ach, one piece of botanical threefold symmetry we missed out whilst in PR is the banana! You can always cut it lengthwise in 3.

And, possibly, the tomato as well - cut it alongside its equator and, lo and behold! there are 3 similar sections.

You've got me thinking about symmetries in 3D.

Each fruit has a 2D plane describing a 3-symmetric pattern.

Can a 3-symmetric pattern exist in 3D?

Tomatos usually have 2, 3 or 5 sections. Fibbonacci.

TRiG.

The 2D-plane in which you have a 3-symmetry was what I meant, really.

As for 3-symmetric patterns in 3D.... Well, I would say that the above qualifies, really (axial rotation of period 3). Now if you require your transformations not to have any fixed subspace, that wouldn't work in 3D, as long as you want the transformations to preserve length.

Fixed subspace? What does that imply?

'Fixed subspace' is a line (or a plane) in which your transformation doen't do anything. With a rotation around an axis, that would be the axis: points on it do not move around -- they are 'fixed' under the rotation.

And in a reflection, the reflecting surface is a 'fixed plane'.

Now, any length-preserving transformation in 3D would be, either a reflection (if it swaps left and right), or a rotation around an axis (if it doesn't swap left and right).

If it's a rotation, then you can cut the object along a 2D-plane perpendicular to this fixed axis, and you get a rotational symmetry of the resulting section.

(As to why the explanation required 2 postings, this is beyond my comprehension )

So transformations of this sort are defined by 2D operations?

Yes, that's a way to put it

Excellent, but would transformations of 4D objects be defined by 3D operations?

Actually, they would be defined by 2D transformations. It is explained in http://tinyurl.com/2nnvkr but I will summarise.

Take some n-dimensional space. Any length-preserving transformation, which we assume here to preserve orientation (not a very restrictive assumption) can be decomposed as follows.

You can divide your higher-dimensional space in little 2D planes and some straight lines, all of which are pairwise orthogonal* and well-behaved with respect to the transformation. Well-behaved means that, on each of the 2D plane, it acts like a rotation (allowing for half-turns), and it fixes each straight line.

If you like, you can replace the choice of these straight lines by a single (higher-dimensional) subspace, on which the transformation you started with fixes all points.

Let's describe an amusing corollary of this decomposition theorem. If there exists a fixed line for such a transformation, then the decomposition above would be made only of 2D subspaces. Therefore, the total dimension of the space you start with would be twice the amount of 2D subspaces, hence an even number. In particular, this cannot happen in 3D space - in other words, there must be a fixed line in a isometry of 3D space.

* this is an analogue of having an axis and a perpendicular plane. Unfortunately you cannot picture two 'mutually orthogonal planes' in 3D.



Was this more or less clear?

Oh, and to specialise this to 4D space: by what I explained above, you can cut up the 4D space in two totally* perpendicular planes in such a way that on both planes, the transformation acts like a rotation or doesn't move anything.

* totally perpendicular meaning: any line from the one plane is perpendicular to any line from the other.

For example: two intersecting walls in a room are *not* totally perpendicular.

A fascinating reply, thanks!



When I can answer 'yes' to that question, my work here will be done.

Icy