Centre for the Mathematics of Symmetry and Computation

Finite Geometry

The Centre uses its breadth of knowledge of permutation groups, graph theory, computation and algebraic combinatorics to study finite geometries.

Geometry is the study of shapes, proportion and configuration. To the lay-person, Geometry is synonymous with the mathematics of the ancient Greek civilisation, featuring the pioneers of Euclid, Pythagoras and Plato.

In the enlightenment period, there was a great drive from mathematical and philosophical thinkers towards formalising mathematics, with a focus towards a reduction of all principles and truths to a finite collection of axioms, and an answer to the classical problem of the independence of Euclid’s Parallel Postulate. This led to the creation of new types of "geometry" besides the familiar Euclidean geometry; for example, hyperbolic geometry, Riemannian geometry and finite geometry.

Finite geometry is geometry with a finite number of objects (for example, points and lines), and is intimately linked with experimental design, information security, particle physics and coding theory.

Apart from being an exciting and interesting areas in combinatorics, finite geometry has many applications to algebraic geometry, group theory, codes, graphs and designs. In particular:

  • The theory of maximal distance separable (MDS) error-correcting codes is equivalent to the study of arcs of projective spaces.
  • Many of the finite simple groups can be described as symmetries of finite geometries, and it remains a hot topic in group theory to expand our knowledge of the Classification of Finite Simple Groups using finite geometry.
  • Finite geometry and algebraic combinatorics is playing an ever increasing role in evolutionary biology, or more specifically, combinatorial phylogenetics.
  • Many of the known constructions of extremal graphs, where the term extremal can be interpreted in many ways, emanate from constructions of finite geometries.

Finite projective planes

For an example of a finite geometry, consider the sets

{1,2,4},{2,3,5},{3,4,6},{4,5,7},{5,6,1},{6,7,2},{7,1,3}

of size 3 in the set {1,2,3,4,5,6,7}, you can make a finite projective plane from these objects by having the numbers 1 up to 7 begin the points, and these sets of size 3 being the lines. Here is a picture of it:

 

Fano plane

Note that in order to draw this example, it was necessary that one of the lines was drawn as a circle.
This geometry is the smallest nontrivial example of a non-Euclidean geometry. It is different in that every pair of lines meet in a point; there are no parallel lines.

The re-discovery of projective geometry in the medieval age brought about a revolution in art and drawing. Albrecht Dürer was one of the artists in this period who exploited the use of projection and perspectivity in his art.

 

Painting of Albrecht Durer

 

In short, projective geometry was enormously influential in scientific thought and invention in the past 500 years, and nowadays is a rich and vibrant field of research.

There are famous open problems which are long standing and which have helped sculpt the modern discipline. For example, it is still open whether or not every finite projective plane has order the power of a prime.

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Last updated:
Monday, 5 September, 2016 7:53 PM

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