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# 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]

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## 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.