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'finite plane'
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| Title of object: |
finite plane |
| Canonical Name: |
FinitePlane |
| Type: |
Definition |
| Created on: |
2002-10-07 05:29:24 |
| Modified on: |
2002-10-22 08:49:57 |
| Classification: |
msc:05C65, msc:05B25 |
| Keywords: |
finite geometry combinatorics |
| Defines: |
Fano plane |
| Synonyms: |
finite plane=finite projective plane |
Revision comment (for changes between this and next version):
| Changes for correction #3296 ('Display not working'). |
Preamble:
\documentclass{article}
\usepackage{amssymb, amsmath, amsthm, alltt, setspace}
\usepackage{graphicx}
\newtheorem*{thm}{Properties of finite planes}
\theoremstyle{definition}
\newtheorem*{defn}{Definition}
\theoremstyle{definition}
\newtheorem*{rem}{Remark}
\theoremstyle{definition}
\newtheorem*{nott}{Notation}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}
\newtheorem*{eg}{Example}
\newtheorem*{ex}{Exercise}
\newtheorem*{prop}{Proposition}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\leftbb}{[ \! [}
\newcommand{\rightbb}{] \! ]}
\newcommand{\bt}{\begin{thm}}
\newcommand{\et}{\end{thm}}
\newcommand{\Rel}{\mathbf{R}}
\newcommand{\er}{\thicksim}
\newcommand{\sqle}{\sqsubseteq}
\newcommand{\floor}[1]{\lfloor{#1}\rfloor}
\newcommand{\ceil}[1]{\lceil{#1}\rceil} |
Content:
Let $\mathcal{H} = (V, \mathcal{E})$ be a linear space. A \emph{finite plane} is an intersecting linear space. That is to say, a linear space in which any two edges in $\mathcal{E}$ have a nonempty intersection.
Finite planes are rather restrictive hypergraphs, and the following holds.
\begin{thm*}
Let $\mathcal{H} = (V, \mathcal{E})$ be a finite plane. Then for some positive integer $k$, $\mathcal{H}$ is $(k+1)-$regular, $(k+1)-$uniform, and $|\mathcal{E}| = |V| = k^2 + k + 1$.
\end{thm*}
The above $k$ is the \emph{\PMLinkEscape{order}} of the finite plane. It is not known in general if finite planes exist of \PMLinkEscape{order} other than $k$ a power of a prime. The terminology "finite plane" is suggestive, as we can think of the edges as a finite collection of lines in Euclidean space in general position so that they all intersect pairwise in exactly one point. The added restriction that all pairs of vertices determine an edge (i.e. "any two points determine a line"), however, makes it impossible to depict a finite plane in the Euclidean plane by means of straight lines, except for the trivial case $k = 1$. The finite plane of \PMLinkEscape{order} $2$ is known as the \emph{Fano plane}. The following is a diagrammatic representation:
\begin{center}
\includegraphics{fano.ps}
\end{center}
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
\begin{center}
\begin{array}{c}
\{1, 2, 4 \} \\
\{2, 3, 5 \} \\
\{3, 4, 6 \} \\
\{4, 5, 7 \} \\
\{5, 6, 1 \} \\
\{6, 7, 2 \} \\
\{7, 1, 3 \}
\end{array}
\end{center}
Notice that the Fano plane is generated by the ordered
lows because every pair of vertices is contained in precisely one edge. For a fixed edge $e$, ${e \choose 2}$ counts all the pairs of vertices in $e$, and this summed over all edges gives all the possible pairs of vertices, which is ${n \choose 2}$. The second one holds because given any vertex, this vertex forms exactly one edge with every other vertex, so $|e| - 1$ counts the number of vertices $v$ shares in vertex $e$, summed over all edges gives all the vertices except $v$. |
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