Near polygon

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In mathematics, a near polygon is an incidence structure introduced by E.Shult and A. Yanushka in 1980.^[1] Shult and Yanushka showed the connection between the so called tetrahedrally closed line-systems in Euclidean spaces and a class of point-line geometries which they called near polygons. These structures generalise the notion of generalized polygon as every generalized 2n-gon is a near 2n-gon of a particular kind. These structures were extensively studied and connection between them and dual polar space ^[2] was shown in 1980s and early 1990s. Some sporadic simple groups, for example the Hall-Janko group, act as automorphism groups of some near polygons.

Definition

A near 2d-gon is an incidence structure ( $P,L,I$ ), where $P$ is the set of points, $L$ is the set of lines and $I\subseteq P\times L$ is the incidence relation, such that:

The maximum distance between two points (the so called diameter) is d.
For every point $x$ and every line $L$ there exists a unique point on $L$ which is nearest to $x$ .

A near quadrangle is same as a generalized quadrangle and it can be shown that every generalized 2d-gon is a near 2d-gon with the following two additional conditions:

Every point is incident with at least two lines.
For every two points x, y at distance i < d, there exists a unique neighbour of y at distance i − 1 from x.

A near polygon is called dense if every line is incident with at least three points and if every two points at distance two have at least two common neighbours. It is said to have order (s, t) if every line is incident with precisely s + 1 points and every point is incident with precisely t + 1 lines.

Examples

All generalized polygons.
All dual polar spaces.
The Hall–Janko near octagon^[3] associated with the Hall–Janko group.

Regular near polygons

A finite near $2d$ -gon S is called regular if it has an order $(s,t)$ and if there exist constants $t_{i},i\in \{0,1,\ldots ,d\}$ , such that for every two points $x$ and $y$ at distance $i$ , there are precisely $t_{i}+1$ lines through $y$ containing a (necessarily unique) point at distance $i-1$ from $x$ . It turns out that regular near $2d$ -gons are precisely those near $2d$ -gons whose point graph is a distance-regular graph. A generalized $2d$ -gon is a regular near $2d$ -gon of order $(s,t;0,0,\ldots ,0)$ .

Notes

↑ Shult, Ernest; Yanushka, Arthur. "Near n-gons and line systems".
↑ Cameron, Peter J. "Dual polar spaces".
↑ http://cage.ugent.be/~hvm/artikels/187.pdf

References

Shult, Ernest; Yanushka, Arthur (1980), "Near n-gons and line systems", Geom. Dedicata 9: 1––72, doi:10.1007/BF00156473, MR 566437 .

Cameron, Peter J. (1982), "Dual polar spaces", Geom. Dedicata 12: 75–85, MR 645040 .

De Bruyn, Bart (2006), Near Polygons, Frontiers in Mathematics, Birkhäuser Verlag, doi:10.1007/978-3-7643-7553-9, ISBN 3-7643-7552-3, MR 2227553 .

Brouwer, A.E.; Cohen, A.M. (1989), Distance Regular Graphs, Berlin, New York: Springer-Verlag., ISBN 3-540-50619-5, MR 1002568 .

Cameron, Peter J. (1991), Projective and polar spaces, QMW Maths Notes 13, London: Queen Mary and Westfield College School of Mathematical Sciences, MR 1153019

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