Kummer configuration

<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the Kummer configuration, named for <a href="/facts/Ernst_Kummer/dGnaTQyG">Ernst Kummer</a>,  is a <a href="/facts/Geometric_configuration/2EJN52Ud">geometric configuration</a> 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 <a href="/facts/Incidence_(geometry)/dlzk9bUl">incident</a> with exactly two planes, and every two planes intersect in exactly two points. The configuration is therefore a <a href="/facts/Block_design/R523MRYh">biplane</a>, specifically, a 2−(16,6,2) design. The 16 nodes and 16 <a href="/facts/Trope_(mathematics)/n4C5j95u">tropes</a> of a <a href="/facts/Kummer_surface/q0LnNkSx">Kummer surface</a> form a Kummer configuration.
</p><p>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.
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Kummer configuration open-in-new

Kummer configuration