User:Jeremydover/sandbox/Desarguesian plane of order 3
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![]() | This is not a Wikipedia article: It is an individual user's work-in-progress page, and may be incomplete and/or unreliable. For guidance on developing this draft, see Wikipedia:So you made a userspace draft. Find sources: Google (books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
Projective plane of order 3 | |
---|---|
Order | 3 |
Lenz-Barlotti Class | VII.2 |
Automorphisms | |
Point Orbit Lengths | 13 |
Line Orbit Lengths | 13 |
Properties | Desarguesian Self-dual |
Construction
[edit]This plane can be constructed using the standard vector space construction with a three-dimensional vector space over the finite field GF(3).
Ovals and Hyperovals
[edit]This plane has odd order, so by Qvist's theorem it does not have a hyperoval. Being a Desarguesian, this plane does have ovals, and by Segré's theorem, all ovals in this plane are conics, being the set of absolute points of an orthogonal polarity.
Subplanes
[edit]Since this plane is a Desarguesian plane of prime order, it has no proper subplanes.
Unitals
[edit]Since the order of this plane is not square, it does not have a unital.