Finite-Geometry Models

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Excerpts, with commentary by Steven H. Cullinane, from
http://bms.ulb.ac.be/Bulletin/98-2-3.pdf/polster.pdf

Burkard Polster

Pretty pictures of geometries

Bull. Belg. Math. Soc. 5 (1998), 417-425

This research was supported by the Australian Research Council.
Received by the editors August 1997.
Communicated by James Hirschfeld.

1991 Mathematics Subject Classification:
51-01 -- Geometry, instructional exposition
51Exx -- Finite geometry and special incidence structures.

Key words and phrases: finite geometries, pictures.

Abstract

We present four construction principles that allow us to produce many beautiful plane and spatial models of some of the most important small finite geometries.

Introduction
....
In this note we describe four of the most useful construction principles for constructing pictures of small incidence geometries which capture large parts of the abstract beauty of the geometries they depict. In order to illustrate these construction principles, we use them to construct models for some of the most important small geometries such as the Fano plane, the generalized quadrangle of order 2, the Desargues configuration and the projective space PG(3,2).

Why are good pictures important? Two of the main reasons that come to mind are the following:

To convey some of the abstract beauty of the objects we study to people outside our field. This seems to be especially important today as it becomes more and more important to "justify" and "sell" the kind of research we are fascinated by.

Many of us think in terms of pictures of various degrees of abstraction. The kind of pictures we want to concentrate on in this note are immediately accessible and can serve to lure students into studying incidence geometry and as a first step in teaching students pictorial thinking in geometry.

Construction principles for pictures of small geometries

Whenever we are trying to create an appealing model of an abstract geometry, we are trying to merge its abstract symmetries with spatial symmetries.... We found that most... pictures, including the traditional ones, can be made up using... simple rules.....

Construction principle [1]:
number right --> everything right

Given a small, highly symmetrical geometry with n points, look for the same number of points arranged into a highly symmetrical spatial object. Try to merge the two structures such that the symmetries of the spatial object translate into symmetries of the geometry.

Commentary by S. H. Cullinane:
See The Relativity Problem
in Finite Geometry

and Geometry of the 4x4 Square.

Construction principle [2]:
subset geometries

Given a subset geometry on... n points, try to translate automorphisms of symmetrical arrangements of n points in the plane or in space into "good" models of the geometry which exhibit as many of these automorphisms as possible.

Commentary by S. H. Cullinane:

See The Diamond 16 Puzzle.

Construction principle [3]:
subgeometry --> full geometry

Try to extend "good" models of subgeometries of a given geometry to a "good" model of the full geometry.

Commentary by S. H. Cullinane:
See Generating the Octad Generator

and Geometry of the 4x4 Square.

Construction principle [4]:
geometry --> subgeometry

Try to find models of subgeometries of a given geometry "right in the middle" of a good model of the geometry.

Commentary by S. H. Cullinane:
See The Diamond Theorem,
Diamond Theory,
and Solomon's Cube.

The proofs underlying properties described in these notes involve extending symmetries of centrally located 2x2 or 2x2x2 subgeometries to their larger 4x4 or 4x4x4 supergeometries.

References [by Polster]

[1] A. Beutelspacher. 21 - 6 = 15: a connection between two distinguished geometries. Amer. Math. Monthly 93:29-41, 1986.

[2] A. Beutelspacher. A defense of the honour of an unjustly neglected little geometry or a combinatorial approach to the projective plane of order five. J. Geom. 30:182-195, 1987.

[3] D.R. Hughes and F.C. Piper. Design Theory. Cambridge University Press, 1985.

[4] R.H. Jeurissen. Special sets of lines in PG(3,2). Linear Algebra Appl. 226-228:617-638, 1995.

[5] S.E. Payne. Finite generalized quadrangles: a survey. Proceedings of the International Conference on Projective Planes (Washington State Univ., Pullman, Wash., 1973), pp. 219-261. Washington State Univ. Press, Pullman, Wash., 1973.

[6] S.E. Payne and J.A. Thas. Finite generalized quadrangles, Research Notes in Math. 110. Pitman, Boston, 1984.

[7] B. Polster. A geometrical picture book. Universitext series, Springer-Verlag, to appear.

Burkard Polster
Department of Pure Mathematics
The University of Adelaide
South Australia 5005

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