Topology question - 7 points and 6 lines with 3 points each

What's the name of the graph conventionally drawn as an equilateral triangle, with the midpoints marked (thus 3 lines of 3 points are formed), the centroid marked and connected to all other points (thus 3 more straight lines of 3 points are formed), and then finally the circle joining the 3 midpoints is drawn (creating a 7th line through 3 points). When you've completed that, you can see all points are equivalent in that they they are all members of 3 lines.

I'm pretty sure it's been used in some gaming context historically.

Fano plane?

http://mathworld.wolfram.com/FanoPlane.html

I thank you! I should have asked earlier, I'd wasted 10 minutes on google, and it only took 2 minutes on LG! Cheers!

Fano plane has 7 lines with 3 points each. Not calling the circle connecting the 3 middles a line (even straight line!) is a pure and open fail to see the Fano plane as it is!

Ooops. At least my description manages to count them all.

But why are you sure I was forgetting that one? ;-)

For a description of the relationship with octonions, see Baez

Article at Wikipedia about Projective Geometry

and a discussion (shameless advertisement) about a game that uses another, larger projective plane: What are the mathematical/computational principles behind this game?

Great links - thanks!

Fire and Ice is a very good game that uses 7 Fano planes arranged in a Fano plane array.

I was asking myself if the next one the 13 points finite PP based on PG(2,3) had a nice graph, and what about the more complex 21 points PG(2,4) one?

I found "Embeddings of Graphs in Finite Projective Planes" from Oscar Vega which answers also some question like is it possible to embed any graph in a large enough finite PP? For example the non-planar famous K(5) can be embedded in the 133 points finite PP or order 11... hence it is in fact a "planar" graph.

@Kerry - yeah, I'm pretty sure it was the fire and ice link in the super-tic-tac-toe thread that made me think of the Fano plane in the first place. (To be honest, I couldn't really make out the picture on the photo, I wasn't sure if it was really a Fano plane.) Thanks for closing the loop.

Aficionados of the Fano plane might be interested in the so-called game Configurations The game mats have a pattern on each side, so half are missing in the photo.

What in the Fano plane is the equivalent of rotating 180 degrees on a game board?

I'm going to implement a variant of NineStein on a Fano plane, and William has suggested a symmetry feature, which is very easy to implement on a ttt-board - rotate by 180 degrees and flip colours. (Centre board being a special case which must be symmetrical in itself)

But the Fano plane has a richer and wilder set of symmetries, and it's not obvious an equivalent exists. For a start, there's no way of pairing the boards as they are odd in number, and it lacks the concept of a "central" board, as all points are equivalent. I don't think it's right to just call the point drawn in the centre special, that seems mathematically artificial.

Thoughts?

I would say: First, define a point A as the center of the rotation. If you rotate a point B it will land on the third point of the line AB.
Technically, a rotation by 180 degrees is the same as mirroring at a point. So it only seems naturally that A, B and the image of B should lie on one line.

The only question is which point should be the center. But as every point is technically the same it really doesn't matter.

I like Zandor's suggestion. And I'd keep the center point fixed, which moves each of the other ones to the diametrically opposing point. (Just so you get visual as well as mathematical symmetry.)

Altertatively, if you choose a corner point as "A" you slide all close boards away and all distant board closer. That's kind of interesting looking.

But, if you choose an edge point, you make a mirror image along the line through "A" and the center and exchange the two points on that line. That doesn't feels as interesting.

How-many of the 168 automorphisms are self-inverses? All of those are equally valid candidates. Perhaps one should be selected randomly, and then it's up to the human to work out which boards are flipped versions of which other ones. That's the only mathematically pure way.