Symmetry Groups in Euclidean and Hyperbolic Spaces: A Comparison

[mnb96]

Hello,
how do symmetry groups in the Euclidean space differ from the symmetry groups in the hyperbolic space (in the Poincaré disk) ?
I've been told that in the hyperbolic case one has at disposal a richer "vocabulary" to describe symmetries, but I don't see how, and maybe I misunderstood.

Can anyone give me a hint and/or some source?
Thanks

 

[arkajad]

You may like to start with http://www.plunk.org/~hatch/HyperbolicTesselations/"

 

[mnb96]

Now I am wondering if in the hyperbolic case we have the same families of symmetries like Reflections, Rotations and so on...or if we can actually define something new.

I'm just trying to figure out where is this "extra richness" in the hyperbolic case that we cannot find in the Euclidean plane.

 

[arkajad]

Euclidean plane, as a Riemannian manifold, is flat. In the hyperbolic case you have different kinds of rotations and reflections.

 

[mnb96]

Euclidean plane, as a Riemannian manifold, is flat.

Does this directly imply that we can define new families of symmetries that do not exist in the Euclidean plane? (e.g. they are not reflections/rotations etc..).
If yes, could you mention one, or some?

Thanks!

 

[arkajad]

Think of a scalar product of signature (p,q). You will have subspaces with all kind of signatures. Each kind gives you a different submanifold of the corresponding Grassmann manifold. A whole ZOO.

 

[arkajad]

You will have non-compact subgroups and the interplay between compact and non-compact.
The richness comes, for instance in the natural compactification. In the Euclidean case $$R^n$$ has a natural one-point compactification. In the hyperbolic case the "conformal infinity" has some interesting structure.

 

[apeiron]

Does this directly imply that we can define new families of symmetries that do not exist in the Euclidean plane? (e.g. they are not reflections/rotations etc..).
If yes, could you mention one, or some?

Thanks!

Isn't there an extra symmetry across scale - so a fractal or conformal symmetry, as arkajad mentioned?

In the euclidean plane, the tesselations are all the same size. In the hyperbolic plane, they are of every size - symmetric around a powerlaw axis of scale.

http://www.mcescher.com/Gallery/recogn-bmp/LW436.jpg

 

[mnb96]

I find it difficult to spot this extra symmetry from the pictures, especially from Escher's pictures. It seems to me that all these illustrations exhibit an ordinary rotational symmetry. In the case of the picture you mentioned 3-fold symmetry.

I guess I am wrong, but I don't know where.

 

[arkajad]

You also have scaling symmetry. Look at that:
x^3y+y3^z+z^3x = 0
Can there be an equation more simple than that? It describes „The Riemann Surface of Klein with 168 Automorphisms”. And yet this simple equation, when analyzed, gives rise to beautiful 168 triangles representing "the fundamental domain".
The method of drawing is extremely simple. ;)
It is described in a paper "The Riemann Surface of Klein with 168 Automorphisms"
by
HARRY E. RAUCH1 AND J. LEWITTES
Research partially sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. AF-69-1641
...
….We call attention to an incorrect answer to (iii) by Hurwitz ([7], p. 159, criticized in [1]) and an abortive attempt on (i), (ii), and (iii) by Poincare in [16], p. 130, all noticed after the completion of our work.2

2. Klein's surface
Klein originally obtained his surface S in the form of the upper half-plane identified under the principal congruence subgroup of level seven, Gamma(7), of the modular group Gamma. In this form it is necessary to compactify the fundamental domain at its cusps. Klein's group then appears as Gamma/Gamma(7), which is simple and of order 168.

We need, however, another representation given by Klein, one which we recognize today as the unit circle uniformization of S. In the unit circle draw the vertical diameter L1 and another diameter L3 making an angle of Pi/7 with L1 and going down to the right. In the lower semicircle draw the arc L2 of the circle which is orthogonal to the unit circle and to L1 and which meets L3 at the angle Pi/3. Let t be the non-Euclidean triangle enclosed by L1,L2,L3 and let Ru R2i R3 be the non-Euclidean reflections in L1,L2,L3respectively. R1,R2,R3 generate a non-Euclidean crystallographic group, which we denote by (2, 3, 7)', with t as a fundamental domain. The images of t under (2, 3, 7)' are a set of non-Euclidean triangles each of which is congruent or symmetric to t according as the group element which maps t on it has an even or odd number of letters as a word in R1 R2i R3. These triangles form a non-Euclidean plastering or tesselation of the interior of the unit circle. The union of t and its image under R2 is a fundamental domain (with suitable conventions about edges) for the triangle group (2, 3, 7), which is the group generated by ….. A convenient fundamental domain for N is the circular arc (non-Euclidean) 14-gon Delta shown in Fig. 1. It will be noticed that t appears as the unshaded triangle immediately below and to the right of P0. There are 168 unshaded triangles, which are the images of t under Gamma168 in Delta and 168 shaded triangles, which are the images of t under anticonformal elements of (2, 3, 7)'. ….

And here, attached, is my own rendering:

 

[mnb96]

Thanks a lot arkajad and apeiron!
Now I start to see better what's going on, although I admit the topic seems to be much deeper than I initially thought, and I need to spend some time to analyze it better.

At least, it is now clear that in the hyperbolic case (e.g. in the Poincaré disk) we can see a sort of "scale-symmetry" because the tessellations in the hyperbolic plane are more "exotic" than in the Euclidean plane.

@arkajad: I seem to have difficulties finding an electronic version of that paper.

Thanks a lot!

 

[arkajad]

@arkajad: I seem to have difficulties finding an electronic version of that paper.

Thanks a lot!

I don't think there is an electronic version. It's 10 pages, but the geometry part is first 4 (the rest is homology an such things). If you are really interested I can scan them for you.

 

[mnb96]

I don't think there is an electronic version. It's 10 pages, but the geometry part is first 4 (the rest is homology an such things). If you are really interested I can scan them for you.

Thanks a lot, but I don't want to bother you unless I feel it is really necessary.
I still have to study more basic stuff, then if I really need to access that paper I might ask you again.
Thanks. See you.

 

[arkajad]

@mnb96
Alan F. Beardon, "Algebra and geometry" has a nice chapter 15 - "Hyperbolic geometry". I think you may like it.

14 Group actions 284
14.1 Groups of permutations 284
14.2 Symmetries of a regular polyhedron 290
14.3 Finite rotation groups in space 295
14.4 Groups of isometries of the plane 297
14.5 Group actions 303
15 Hyperbolic geometry 307
15.1 The hyperbolic plane 307
15.2 The hyperbolic distance 310
15.3 Hyperbolic circles 313
15.4 Hyperbolic trigonometry 315
15.5 Hyperbolic three-dimensional space 317
15.6 Finite Mobius groups 319

 

[mnb96]

I got the book you suggested from the library.
It seems well written and will be useful.
Thanks!