### Abstract

We discuss n4 configurations of n points and n planes in three-dimensional projective space. These have four points on each plane, and four planes through each point. When the last of the 4n incidences between points and planes happens as a consequence of the preceding 4n-1 the configuration is called a theorem. Using a graph-theoretic search algorithm we find that there are two 84 and one 94 theorems. One of these 84 theorems was already found by Mbius in 1828, while the 94 theorem is related to Desargues ten-point configuration. We prove these theorems by various methods, and connect them with other questions, such as forbidden minors in graph theory, and sets of electrons that are energy minimal.

Original language | English |
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Pages (from-to) | 75-92 |

Number of pages | 18 |

Journal | Journal of the Australian Mathematical Society |

Volume | 88 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2010 |

### Keywords

- Configuration
- Desargues
- Fano
- Forbidden minor
- Geometry
- Mbius
- Minimal energy electrons
- Pappus
- Projective space
- Theorem