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 of dimension 2 and order provided on the command line (defaulting to 2, i.e. the Fano Plane). Dimension 2 projective geometries also go under the name of projective planes.

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 (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!