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Schläfli double six
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 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). For each of the five lines ''a''''i'', the complementary set of four out of the five lines has two quadrisecants: ''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 ''ai'' and ''bi'' 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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