In geometry, the Schläfli double six is a configuration of 30 points and 12 lines, introduced by . The lines of the configuration can be partitioned into two subsets of six lines: each line is disjoint from ( skew with) the lines in its own subset of six lines, and intersects all but one of the lines in the other subset of six lines. Each of the 12 lines of the configuration contains five intersection points, and each of these 30 intersection points belongs to exactly two lines, one from each subset, so in the notation of configurations the Schläfli double six is written 12₅30₂.

Construction

As Schläfli showed, the double six may be constructed from any five lines ''a''₁, ''a''₂, ''a''₃, ''a''₄, ''a''₅, that are all intersected by a common line ''b''₆, but are otherwise in

general position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are ...

(in particular, each two lines ''a''_''i'' and ''a''_''j'' should be skew, and no four of the lines ''a''_''i'' should lie on a common

ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, t ...

). For each of the five lines ''a''_''i'', the complementary set of four out of the five lines has two

quadrisecant In geometry, a quadrisecant or quadrisecant line of a space curve is a Line (geometry), line that passes through four points of the curve. This is the largest possible number of intersections that a Generic property, generic space curve can have ...

s: ''b''₆ and a second line ''b''_''i''. The five lines ''b''₁, ''b''₂, ''b''₃, ''b''₄, and ''b''₅ formed in this way are all in turn intersected by another line, ''a''₆. The twelve lines ''a_i'' and ''b_i'' form a double six: each line ''a''_''i'' has an intersection point with five of the other lines, the lines ''b''_''j'' for which ''i'' ≠ ''j'', and vice versa. An alternative construction, shown in the illustration, is to place twelve lines through the six face centers of a cube, each in the plane of its face and all making the same angles with respect to the cube's edges.

Related objects

A generic

cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather th ...

contains 27 lines, among which can be found 36 Schläfli double six configurations. The set of 15 lines complementary to a double six, together with the 15 tangent planes through triples of these lines, has the incidence pattern of another configuration, the

Cremona–Richmond configuration In mathematics, the Cremona–Richmond configuration is a configuration of 15 lines and 15 points, having 3 points on each line and 3 lines through each point, and containing no triangles. It was studied by and . It is a generalized quadrangle wit ...

. The

intersection graph In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types o ...

of the twelve lines of the double six configuration is a twelve-vertex

crown graph In graph theory, a branch of mathematics, a crown graph on vertices is an undirected graph with two sets of vertices and and with an edge from to whenever . The crown graph can be viewed as a complete bipartite graph from which the edges ...

, a bipartite graph in which each vertex is adjacent to five out of the six vertices of the opposite color. The

Levi graph In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particulap. 181 From a collection of points and lines in an incidence geometry or a projective configuration, we fo ...

of the double six may be obtained by replacing each edge of the crown graph by a two-edge path. The intersection graph of the entire set of 27 lines on a cubic surface is the complement of the

Schläfli graph In the mathematical field of graph theory, the Schläfli graph, named after Ludwig Schläfli, is a 16- regular undirected graph with 27 vertices and 216 edges. It is a strongly regular graph with parameters srg(27, 16, 10, 8). ...

References

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External links

* {{DEFAULTSORT:Schlafli double six Configurations (geometry)