# The Smallest Projective Space and other lines of 3

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 You might be familiar with the Fano plane. One way to obtain it is to select triples from 1 to 7 that have a binary bit sum of 0. It's an example of a projective plane. Projective plane rules Every point pair defines a line. Every line pair defines a point. There are 4 points not on a line. Code: Select[Subsets[Range, {3}], BitXor @@ # == 0 &] These can be arranged so that each triplet (123 145 167 246 257 347 356) is on an arc or a line. So how to draw a bunch of nice arcs and lines? Here's some code I whipped up. ArcDraw2D[{threepoints_}] := Module[{colinearitytest, circle, center, radius, angles, ends}, colinearitytest = Chop[N[Det[Append[#, 1] & /@ threepoints]]]; If[colinearitytest != 0, circle = Circumsphere[threepoints]; center = N[circle[]]; radius = N[circle[]]; angles = Mod[N[Arg[(# - center).{1, I}]/Pi], 2] & /@ threepoints; ends = {angles[], angles[]}; If[angles == Sort[angles] || angles == Reverse[Sort[angles]], Circle[center, radius, Pi ends], Circle[center, radius, Pi {Max[ends], 2 + Min[ends]}]], Line[threepoints]]]; Now, let's extend that. Select[Subsets[Range, {3}], BitXor @@ # == 0 &] gives the next step up: triples from 1-15 with a BitXor sum of 0. I'll rearrange the output a bit. pg23 = {{1, 4, 5}, {8, 9, 1}, {2, 10, 8}, {14, 12, 2}, {5, 11, 14}, {1, 3, 2}, {2, 7, 5}, {5, 13, 8}, {8, 6, 14}, {14, 15, 1}, {2, 6, 4}, {14, 7, 9}, {5, 15, 10}, {1, 13, 12}, {8, 3, 11}, {6, 11, 13}, {7, 4, 3}, {15, 9, 6}, {13, 10, 7}, {3, 12, 15}, {15, 8, 7}, {13, 2, 15}, {3, 14, 13}, {6, 5, 3}, {7, 1, 6}, {9, 3, 10}, {10, 6, 12}, {12, 7, 11}, {11, 15, 4}, {4, 13, 9}, {9, 2, 11}, {10, 14, 4}, {12, 5, 9}, {11, 1, 10}, {4, 8, 12}}; Now a bit of set-up for the graphic. numbers = {8, 2, 14, 5, 1, 3, 6, 7, 15, 13, 11, 4, 9, 10, 12}; locations = Flatten[MapIndexed[ RootReduce[{Sin[2 (#2[] - 2) Pi/5], Cos[2 (#2[] - 2) Pi/5]} {2, .5, -1.378}[[#2[]]]] &, Partition[numbers, 5], {2}], 1]; newloc = Last /@ Sort[Transpose[{numbers, locations}]]; colors = {Black, Gray, Cyan, Green, Red, Yellow, Blue}; Then we can go right to the graphic. Graphics[{AbsoluteThickness[1.6], MapIndexed[{colors[[Ceiling[#2[]/5]]], ArcDraw2D[newloc[[#1]]]} &, pg23], MapIndexed[{{Black, Disk[#1, .12]}, {White, Disk[#1, .11]}, Style[Text[#2[], #1], 20]} &, newloc]}, ImageSize -> {500, 500}] If you look close, you can see the Fano plane from the opener in here. There are 15 Fano planes of 3 different types. To see these in 3D, take a look at 15 Point Projective Space.An alternative geometry allows lines that don't intersect, otherwise known as parallel lines. Affine plane rules Every point pair defines a line. Every point-line pair defines a line. There are 4 points that define 6 lines. One affine plane example is the Game of Set, where each of 81 card has three possibilities for the attributes number, color, shape, and shading. In a line of three Set cards, the values for a given attribute will be either all the same or all different. There are 1080 ways to get a Set triplet. Select[Subsets[Range[1, 81], {3}], Total[Mod[Total[IntegerDigits[#, 3, 4]], 3]] == 0 &] Is there another nice affine plane where all lines have 3 points? Yes, within elliptic curves. Take triplets from the integers that add to zero. Now place them on the plane so that all triplets are on a straight line. You might get an image like the following: If you pick any triplet with a sum of 0, you'll find it is on a straight line. To add new numbers, pick two pairs with the opposing sum, such as {1-,-15} and {-2,-14} for 16. The new number will go to the intersection point of the two lines. Answer - another post of yours has been selected for the Staff Picks group, congratulations !We are happy to see you at the tops of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming! Answer