On Friday, October 10, 2014, our Math 4600 class worked together to determine the coordinates for the projective plane of order 7 represented by the Spot It® game. Here lines are represented by the 57 symbols, and points by the 57 cards. (If you have the original edition of Spot It® which has only 55 cards, here are the missing two points. Blue Orange now makes several editions of the Spot It® game based on different symbol sets.) Students randomly selected a quadrangle as the initial quadrangle with coordinates (0,0,1), (0,1,1), (1,0,1) and (1,1,1); after making this arbitrary choice of starting quadrangle, all coordinates of the remaining points and lines are then uniquely determined.
We use the following convention for points and lines: Let F={0,1,2,3,4,5,6} be the field of order 7 (the integers mod 7). Points are nonzero row vectors P=(x,y,z) over F; lines are nonzero column vectors L=(a,b,c)T over F. Nonzero scalar multiples of a point (or line) represent the same point (or line). The point P=(x,y,z) lies on the line L=(a,b,c)T iff the usual matrix product PL=0, i.e. iff xa+yb+zc=0. The points with nonzero z-coordinate were scaled into the form (x,y,1); these may be represented as the usual points (x,y) of the affine plane. Click on any point of the plane for a color-coded view of all 8 lines through the selected point.
I am grateful to Professors Leanne Holder and Allen Holder, Rose-Hulman Institute of Technology, for introducing to me the Spot It® game and its connection to the projective plane of order 7. Thanks also to Noam Kedem for spotting some incorrect symbols in my pages dynamically displaying points and lines.
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revised October, 2014