# 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!