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Following the method of Jordan (1869), but see also Assmus and Sardi (1981), 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-element block for each point.
Consider two points on the same row or column. There are two other points in that row or column which show up in the blocks for both starting points, therefore those blocks intersect in two points. Now consider two points not in the same row or column. Their corresponding blocks intersect in two points which form a rectangle with the two starting points. Thus all blocks intersect in two points. By examining the blocks corresponding to those intersection points, one sees that any two starting points are present in two blocks.
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the Kummer configuration, named for Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of ...
, is a geometric configuration
In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the ...
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
Incident may refer to:
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* ''Incident'' (film), a 1948 film noir
* Incident (festival), a cultural festival of The National Institute of Technology in Surathkal, Karnataka, India
* Incident (Scientology), a ...
with exactly two planes, and every two planes intersect in exactly two points. The configuration is therefore a biplane
A biplane is a fixed-wing aircraft with two main wings stacked one above the other. The first powered, controlled aeroplane to fly, the Wright Flyer, used a biplane wing arrangement, as did many aircraft in the early years of aviation. While ...
, specifically, a 2−(16,6,2) design. The 16 nodes and 16 trope
Trope or tropes may refer to:
Arts, entertainment, and media
* Trope (cinema), a cinematic convention for conveying a concept
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s of a Kummer surface
In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb^3 with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variety o ...
form a Kummer configuration.
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 best biplane" on 16 points.
Construction
Following the method of Jordan (1869), but see also Assmus and Sardi (1981), 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-element block for each point.
Consider two points on the same row or column. There are two other points in that row or column which show up in the blocks for both starting points, therefore those blocks intersect in two points. Now consider two points not in the same row or column. Their corresponding blocks intersect in two points which form a rectangle with the two starting points. Thus all blocks intersect in two points. By examining the blocks corresponding to those intersection points, one sees that any two starting points are present in two blocks.
Automorphisms
There are exactly 11520 permutations of the 16 points that give the same blocks back. Additionally, exchanging the block labels with the point labels yields another automorphism of size 2, resulting in 23040 automorphisms.See also
*Klein configuration In geometry, the Klein configuration, studied by , is a geometric configuration related to Kummer surface
In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb^3 with the max ...
References
{{Incidence structures Configurations (geometry) Algebraic geometry