# Examples This directory contains some examples (starting from... 1 example, which might remain alone for a long time or forever!). ## `pg2` - projective geometries of dimension 2 This example generates (finite) [projective geometries][pg] of dimension 2 and order provided on the command line (defaulting to 2, i.e. the [Fano Plane][fp]). Dimension 2 projective geometries also go under the name of [projective planes][pp]. [pg]: https://en.wikipedia.org/wiki/Projective_geometry [fp]: https://en.wikipedia.org/wiki/Fano_plane [pp]: http://mathworld.wolfram.com/ProjectivePlane.html Example: \$ perl pg2 elements in field: 2 0. (1, 3, 5) 1. (0, 3, 4) 2. (2, 3, 6) 3. (0, 1, 2) 4. (1, 4, 6) 5. (0, 5, 6) 6. (2, 4, 5) errors in check: 0 The *elements in field* corresponds to the order of the projective geometry. As anticipated, the defualt value is `2`. Then, the list of *lines* is provided, as collections of *order*+1 *points* each. So, for example, *line* `0` is comprised of *points* `1`, `3` and `5`. For duality, you can also consider each of them as *points*, listing the *lines* it belongs to. As a matter of fact, the arrangement is such that this property always holds, i.e. if *point* `x` belongs to line `y`, then point `y` belongs to line `x`. The *errors in check* is a verification that the generated list of *points*/*lines* actually is a projective geometry, i.e. that all lines have the same *order*+1 *points* and that each *point* belongs exactly to *order*+1 lines. As a curiosity, the game [Dobble][dobble] (known in some countries as *Spot It*) is a game based on *PG(2, 7)*: \$ perl pg2 7 elements in field: 7 0. (1, 8, 15, 22, 29, 36, 43, 50) 1. (0, 8, 9, 10, 11, 12, 13, 14) 2. (7, 8, 21, 27, 33, 39, 45, 51) ... 54. (3, 13, 15, 24, 33, 42, 44, 53) 55. (4, 12, 15, 25, 35, 38, 48, 51) 56. (7, 9, 15, 28, 34, 40, 46, 52) errors in check: 0 where: - each *point* is associated to a picture - each *line* is associated to a card - only 55 cards out of the 57 possible ones are included in the game - each picture is included in at most 8 cards (because 2 cards were left out) - each card contains exactly 8 pictures - any two cards share exactly 1 picture (corresponding to the notion that two *lines* intersect in exactly one *point*) You can consider the dual of course... this is left as an exercise! [dobble]: https://boardgamegeek.com/boardgame/63268/spot-it