# finite plane

A finite plane (synonym linear space (http://planetmath.org/LinearSpace2)) is the finite (discrete) analogue of planes in more familiar geometries^{}. It is an incidence structure where any two points are incident^{} with exactly one line (the line is said to “pass through” those points, the points “lie on” the line), and any two lines are incident with at most one point — just like in ordinary planes, lines can be parallel^{} i.e. not intersect in any point.

A finite plane without parallel lines is known as a projective plane^{}. Another kind of finite plane is an affine plane, which can be obtained from a projective plane by removing one line (and all the points on it).

## Example

An example of a projective plane, that of order $2$, known as the *Fano plane* (for projective planes, order $q$ means $q+1$ points on each line, $q+1$ lines through each point):

An edge here is represented by a straight line, and the inscribed circle is also an edge. In other words, for a vertex set $\{1,2,3,4,5,6,7\}$, the edges of the Fano plane are

$$\{1,2,4\},\{2,3,5\},\{3,4,6\},\{4,5,7\},\{5,6,1\},\{6,7,2\},\{7,1,3\}$$ |

Notice that the Fano plane is generated by the triple $\{1,2,4\}$ by repeatedly adding $1$ to each entry, modulo $7$. The generating triple has the property that the differences of any two elements, in either order, are all pairwise different modulo $7$. In general, if we can find a set of $q+1$ of the integers (mod ${q}^{2}+q+1$) with all pairwise differences distinct, then this gives a cyclic representation of the finite plane of order $q$.

Title | finite plane |
---|---|

Canonical name | FinitePlane |

Date of creation | 2013-03-22 13:05:34 |

Last modified on | 2013-03-22 13:05:34 |

Owner | marijke (8873) |

Last modified by | marijke (8873) |

Numerical id | 18 |

Author | marijke (8873) |

Entry type | Definition |

Classification | msc 51E20 |

Classification | msc 05C65 |

Classification | msc 51E15 |

Classification | msc 05B25 |

Related topic | LinearSpace2 |

Defines | Fano plane |