Kummer configuration

In geometry, the Kummer configuration, named for Ernst Kummer, is a geometric configuration of 16 points and 16 planes such that each point lies on 6 of the planes and each plane contains 6 of the points. Further, every pair of points is incident with exactly two planes, and every two planes intersect in exactly two points. The configuration is therefore a biplane, specifically, a 2−(16,6,2) design. The 16 nodes and 16 tropes of a Kummer surface form a Kummer configuration.[1]

There are three different non-isomorphic ways to select 16 different 6-sets from 16 elements satisfying the above properties, that is, forming a biplane. The most symmetric of the three is the Kummer configuration, also called "the nicest biplane" on 16 points.[2]

Construction

Following the method of Assmus and Sardi (1981),[2] arrange the 16 points (say the numbers 1 to 16) in a 4x4 grid. For each element in turn, take the 3 other points in the same row and the 3 other points in the same column, and combine them into a 6-set. This creates one 6-set block for each point, and shows how every two blocks have exactly two points in common and every two points have exactly two blocks containing them.

Automorphism

There are exactly 11520 permutations of the 16 points that give the same blocks back.[3][4] Additionally, exchanging the block labels with the point labels yields another automorphism of size 2, resulting in 23040 automorphisms.

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See also

References
  1. Hudson, R. W. H. T. (1990), Kummer's quartic surface, Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-39790-2, MR 1097176
  2. Assmus, E.F.; Sardi, J.E. Novillo (1981), "Generalized Steiner systems of type 3-(v, {4,6},1)", Finite Geometries and Designs, Proceedings of a Conference at Chelwood Gate (1980), Cambridge University Press, pp. 16–21
  3. Carmichael, R.D. (1931), "Tactical Configurations of Rank Two", American Journal of Mathematics, 53: 217–240, doi:10.2307/2370885
  4. Carmichael, Robert D. (1956) [1937], Introduction to the theory of Groups of Finite Order, Dover, p. 42 (Ex. 30) and p. 437 (Ex. 17), ISBN 0-486-60300-8
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