Adam P. Goucher wrote:
Firstly, what's the common name for A_2? I usually say `hexagonal lattice' (using the Voronoi partition), whereas Dan appears to use `triangular lattice' (taking the dual partition). My copy of SPLAG is 110 miles away, and that tends to be the authoritative resource on this topic.
Sloane?
There might be a transponder difference; I think triangular is a bit more popular among math'ns here, but hexagonal is widely understood to mean the same thing.
Dan Asimov wrote:
Actually, the chain mail thing should overlook every other (triple) intersection point, contrary to what I wrote below.
Here's a bare-bones, line-point symmetric way to see it, perhaps:
Consider a triangular lattice L in the plane, and 3-color it, say by numbering the nodes 0, 1, 2 so that every smallest triangle has all three numbers. Briefly imagine Voronoifying this lattice with hexagonal cells numbered 0, 1, 2.
Now forget about the points labeled 2. Do not think of the number 2 !!!
The points labeled k, for k = 0 or for k = 1, each form a triangular lattice themselves. Call these L_0 and L_1.
Now 7-color, separately, each of L_0 and L_1 (using 14 colors in all).
Redraw the previous hexagonal Voronoi cells of the points of L_0 and of L_1. (Just in case we need them later, pick a third heptad of colors to 7-color the hexagons of L_2.)
Your description so far is precisely what my program does, apart from the trivial difference between hexagons and discs. There is a palette of 21 colours, as implied by the /21 in the argument to the Hue command. The third group is invisible since those discs have radius 0 in the program.
(I cheated in several places to make the code as compact as possible -- 100 characters is not many at all.)
The only real difference is that all hexagons are the same size. I'd look for 14 colors that seem visually well-separated. Maybe take the vertices and face centers of the RGB cube.
Finally, mod out the plane by the appropriate sublattice of the original lattice L, so that (the images of) exactly 7 hexagons of L_0 and 7 hexagons of L_1 appear in the quotient torus T.
The (images of the) 7 hexagons of L_0 may be thought of as the 7 points of the Fano plane, and the (images of the) 7 hexagons of L_1 as the 7 lines. This higlghts the duality between points and lines.
Now we can bring back the (images of the) 7 hexagons of L_2 to get a *third* group of 7 in the torus T, with 3-way symmetry among the 3 groups.
Yes, indeed: S_3 symmetry, in fact. Bizarrely, I think that this reduces the overall amount of symmetry of the graph from PGL(2,7) (order 336) to order-252.
I may be confused about this, but I don't think you can turn the thing over (the 7-color tiling of the hexagonal torus by 7 hexagons is chiral). If so I see a total symmetry group of size 21*6 = 126. (Unless you allow permuting the within-group colors, in which case multiply by (7!^3)/42 = 3048192000.) --Dan
On Oct 2, 2014, at 1:46 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Adam's picture is very nice, but I think having the "points" and "lines" colored with the same or similar colors is a bit misleading.
Maybe 14 distinct colors, 7 for the points and 7 for the lines might be better.
The universal cover of the picture I originally intended would be what you get if you start with a triangular lattice of circle centers on the plane, and then increase the circles' common radius until they first have triple intersections. It would resemble a picture of chain mail. Then the circles are the lines, and their intersections are the points. (Ignore the centers.) Now 7-color the "lines", and separately 7-color the "points", in a repeating way.
--Dan
On Oct 2, 2014, at 4:19 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Can anyone supply a picture of Dan's construction, transplanted from T to a polygonal domain in the plane with identifications along the boundary?
Here's a picture generated from a 100-character Mathematica program:
https://twitter.com/wolframtap/status/517631202837405696
Identify discs of equal colour. Large discs represent lines; small discs represent points. Adjacency represents incidence.
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