User talk:Martin von Gagern

Welcome

Hello dear visitor!

Feel free to drop me a line here. -- Martin von Gagern 22:21, 7 Mar 2005 (UTC)

Wallpaper diagram

Your diagrams that are used in the wallpaper group article are very helpful. I did notice one small problem: Wallpaper group diagram p4g.svg is missing a set of glide axes. There should be dotted lines going diagonally between the green squares. Just thought I'd let you know. --Lasunncty (talk) 20:52, 15 August 2008 (UTC)[reply]

Thanks for the information. I just updated the corresponding SVG images. I don't feel like maintaining the PNGs forever, so I left those alone. --Martin von Gagern (talk) 14:35, 4 November 2008 (UTC)[reply]

Hyperbolic symmetry diagrams?

Hi Martin, In 2006 you made some really great Euclidean plane wallpaper group diagrams. Your diagrams replaced some crude markups I colorized from a book like File:Wallpaper group-cell-p4m.png. I'm in the process of testing out some similar diagrams for *642 hyperbolic symmetry: File:Tetrahexagonal_tiling_subgroups.png, just hand-made in MSPaint, labeling with Coxeter notation for subgroups of [6,4] and Orbifold notation for the resulting group. I'm still looking into the locations of the glide reflections, so those are not all correct. Anyway, I'm curious if you could make some SVGs like this. I drew a partial hyperbolic tiling of *642 as a straight-line Beltrami–Klein model, but might be more clear inside a hexagon with circle arc edges of Poincaré disk model. What do you think? Is this something you're interested in? I see some incomplete hyperbolic work from [1]. Tom Ruen (talk) 23:26, 30 March 2013 (UTC)[reply]

I'm very much interested in hyperbolic geometry. The problem is that there are infinitely many symmetry groups in hyperbolic geometry. So the completeness of the Euclidean enumeration of wallpaper groups cannot possibly be achieved for the hyperbolic case. A reasonable subset might be possible, though. I guess I'd prefer the Poincaré disk model, since orders of rotation are more apparent in that model due to its conformality. I also find the translative cells less important in the hyperbolic case. In the Euclidean case you know that one translative cells together with two translation vectors will generate the whole ornament, but in the hyperbolic plane that's not the case. So I'm not sure how useful these images actually are. I did at some point have an application where I could draw hyperbolic ornaments and also show all the symmetry features. In most groups I ended up with glide reflections all over the place. I could try to restore that version of morenaments hyp, and check whether this is really the case or just a bug in my implementation, but I think I checked back then. Another interesting project might be hyperbolic versions of Commons:Symmetry Blendings. I'd like to do work on both of these, but don't have the resources just now. In the long run, yes, and until then I'll be glad to offer assistance if anyone else wants to have a go at these. Martin von Gagern (talk) 20:51, 31 March 2013 (UTC)[reply]

I understand. Completeness isn't possible, but some representative samples can be useful, like my chart above for subsets of *642, picked because even orders allows many permutations of mirror removals jsut like [4,4]. There are not perhaps a great set of artwork to analyse with overlays, but I have one of Escher's at Order-6_square_tiling#Example_artwork, and so something like that would be fun with your graphics as blending or overlays. You might be interested to see User:Tamfang's uniform tilings at [2], all reflectional symmetry for (p q r), <=8, with black&white checkerboard images showing fundamental domains. So anyway, I'll keep working with my crude graphics, but glad for your help sometime in some high quality SVG ones perhaps! :) Tom Ruen (talk) 21:41, 31 March 2013 (UTC)[reply]

p.s. Maybe easy for you, it would be good to have 7 pure Frieze diagrams like this File:SymBlend fmm.svg minus blend. I have my cheap ones at List_of_planar_symmetry_groups#Frieze_groups. Tom Ruen (talk) 05:09, 1 April 2013 (UTC)[reply]

p.s. Using User:Tamfang's hyperbolic tilings [3], I made some composites (via edge-detection, color changing, and overlays) to represent subgroups of [6,4] (*642) symmetry at User:Tomruen/temp11. I use "checkerboard" patterns to show out rotational symmetry, and isolate each of 3 mirror types by 3 colors. I'm not sure what notation you use, but I'm actually doing this in part to demonstrate Coxeter's bracket notation subgroups. All even-order elements allow for a full set of permutations of subgroups. These bitmap operations could be automated with some care for any *pq2 or *pqr family. [6+,4+] (Orbifold 32×) is the only one where glides are important. So far I just showed the glides as thin lines, although perhaps a dithering filter could simulate somewhat irregular dashed lines. Tom Ruen (talk) 23:55, 11 April 2013 (UTC)[reply]

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