# 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.

## See also

## References

- Hudson, R. W. H. T. (1990),
*Kummer's quartic surface*, Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-39790-2, MR 1097176 - 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 - Carmichael, R.D. (1931), "Tactical Configurations of Rank Two",
*American Journal of Mathematics*,**53**: 217–240, doi:10.2307/2370885 - 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