Fano Plane and Hexagonal Tiling - a deep connection or a shallow one?

August 11, 2015 1:24 AM

I've been chewing on the Fano plane a lot these last few weeks, and have just noticed something that seems really interesting to me. Notably, that the seven lines of the plane map very nicely to the 7-color hexagonal tiling.

Is there some trivial mathematical reason why this will naturally be the case, or is there a more profound connection between these two things? Have I somehow stumbled across something new? Googling has offered me no obvious answers (and in fact, has also not shown any images of Fano planes using my coloring scheme, mapping the 3-bit numbering of points to RGB coloration). If I'm treading familiar ground, is there somewhere I can learn more about this connection? If I'm in undiscovered territory... where do I go from here?

posted by NMcCoy to Science & Nature (10 answers total)

this is probably obvious, sorry... if you move the "triangle corners" of the fano plane onto the circle then you get something that's basically a dot in the middle of a ring of 6 dots. then that has the same symmetry as a hexagon in the middle of six hexagons (although the colouring in your hexagons doesn't match - CMY need to be rotated).

but since the fano plane does have the "corners" then the two are not identical, because the moving some points off the ring to the corners reduces the symmetry (from six-fold to three-fold).
posted by andrewcooke at 3:10 AM on August 11, 2015

I think it's just because a regular hexagon can be considered to be made out of 6 equilateral triangles (see the pic on the Wikipedia page). So if you tile triangles with lines down the center, you can also see it as a series of hexagons. Likewise, you could turn it into a rhombuses and with clever coloring make a Necker Cube. There are lots of quilt patterns that play with these geometries.
posted by tchemgrrl at 7:38 AM on August 11, 2015

Response by poster: I may not have clearly specified the intent of my question.

On the Fano plane, there are seven points and seven lines. Each point lies on exactly three lines; each line includes exactly three points. Any two points share exactly one line, any two lines share exactly one point.

On the hex tiling, there are seven colors and seven tiles. Each color appears on exactly three tiles; each tile includes exactly three colors. Any two colors share exactly one tile; any two tiles share exactly one color.*

There is a direct mapping between these relationships. Is there some underlying mathematical reason why this is so?

*Note that this relationship would hold with any 7-color tiling overlaid with an offset hex grid in that manner; I've matched the colors up with my Fano coloring for clarity. It's not my choice of colors that causes this relationship to happen.
posted by NMcCoy at 12:08 PM on August 11, 2015

oh sorry, i misunderstood the hexagonal diagram.
posted by andrewcooke at 12:27 PM on August 11, 2015

Response by poster: (There's also another interesting relationship/property going on with the hex grid, but I wanted to keep things simple to start off with. To be specific, with the particular 7-color tiling I chose, half of the vertices sum to 0 (due to how the Fano plane numbering is constructed); these are the ones I centered my offset grid on. The other vertices do not sum to 0; in fact, they sum to colors in a pattern identical to the arrangement of the tiling itself. I suspect this is related to the fact that the Fano plane is the dual of itself, but haven't delved into a thorough analysis of it yet.)
posted by NMcCoy at 12:48 PM on August 11, 2015

Response by poster: Hm, a couple of points of clarification: I said "this relationship would hold with any 7-color tiling" but that's not in fact correct - it specifically refers to the 7-color 3-uniform coloring. There are other ways to tile a hex grid with 7 colors that don't result in Fano-icity.

Also, on more analysis, that secondary offset thing doesn't seem to be connected to the self-dual property but it does highlight some interesting relationships. (This is constructed from "if the black dots on the hex grid were colored with the same spatial relationships as the colored dots, what color would they be?")
posted by NMcCoy at 3:32 PM on August 11, 2015

Best answer: This might be of interest, though it is not exactly the same.
posted by eruonna at 4:34 PM on August 11, 2015 [2 favorites]

Actually, with some more thought, it looks like that construction maps nicely to your (or vice versa).
posted by eruonna at 4:38 PM on August 11, 2015

Response by poster: Oh, wow, eruonna, that is incredibly relevant. My colored "off-grid" points on the "another interesting relationship" hex grid are precisely the faces of the Fano plane! And the sum-color of the dot on that intersection is the single point that lies on none of the plane's lines (which, of course, is isomorphic to "the one color that shows up on none of those three hexes"). The interesting thing here is that the hex-structure on that page isn't the same as mine, though it turns out to embed the same information in a different way. I think there's more to be discovered here, but that is definitely a valuable find, thank you!
posted by NMcCoy at 6:15 PM on August 11, 2015

I guess I'd say the answer is "yes and no." On the one hand, the points of the Fano plane are naturally associated with the points of the projective plane over F_2. But the colors in your tiling are naturally associated with the elements of the finite field F_7. (Just because your tiling is labeling the elements of the lattice Z[w] by what they are mod 1+2w, where w is a cube root of 1.) Both sets have 7 elements, but there's not really any natural correspondence between the two.

On the other hand, you can set up another version of the Fano plane on F_7 by letting the lines be the sets of the form

(x+1,x+2,x+4).

This works, more or less, because 1,2,4 are the quadratic residues in F_7^*. See this paper of Zagier for a description of the general picture of which this is a part.
posted by escabeche at 9:42 PM on August 15, 2015