Theorems of points and planes in three-dimensional projective space

David Glynn

    Research output: Contribution to journalArticle

    4 Citations (Scopus)

    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 languageEnglish
    Pages (from-to)75-92
    Number of pages18
    JournalJournal of the Australian Mathematical Society
    Volume88
    Issue number1
    DOIs
    Publication statusPublished - Feb 2010

    Keywords

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

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