The Double Six

A generic cubic surface contains 27 straight lines.

Now, a double six configuration on a cubic surface is a set of 2*6 lines a_i, b_j, i, j = 1, ..., 6, out of the 27, where:

• a_i does not meet a_j for i <> j
• a_i does not meet b_i
• a_i meets b_j for i <> j.

E.g.: a₁ meets b_j for j = 2, 3, 4, 5, 6:

Here is a generic cubic surface with 2*6 lines upon it that form a double six:

On a generic cubic surface, there are 36 such configurations. With the double six notation introduced by Schläfli, another double six would be e.g.:

In the Cubic Surface Program, the set of 2*6 lines corresponding to the six points and the six conics (through five of the six points each) are called 1, 2, ..., 6, and 1', 2', ..., 6' and form such a double six:

The first image above shows such a double six on a generic cubic surface. In the second image, only the lines 1' and the lines 2, 3, 4, 5, 6 (which meet 1') are shown.

But the concept of a double six is independent of cubic surfaces. The following interactive geometry tool'-construction shows 2*6 lines that form a double six. On each of the six faces of the cube there is a green and a red line. The intersection points of lines are indicated by small circles.
You can drag the red and the pink points to change the construction in order to see and understand the incidence relation of a double six: each red line meets 5 of the green lines, but no red one; conversely, each green line meets 5 of the red lines, but no green one.
I, personally, think, that the best way to see it is to fix one line and the five points on it with the eyes and then drag one of the pink points. But note that this is no 3d-construction, so that you will not always have a correct view on the cube when you drag the points.

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• A. Henderson: The 27 lines upon the cubic surface, Cambridge, University Press, 1911.

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© Oliver Labs, The Cubic Surface Homepage, Algebraic Geometry Group, University of Mainz, Germany