The Double Six

Schläfli found that the 27 lines can be described in terms of only 12 lines.

The structure of these 12 lines is as follows:

There are 6 lines which are pairwise skew (non-intersecting), denoted as a₁,a₂,…,a₆ and drawn in red, say.

There are 6 further lines, also pairwise skew, denoted as b₁,b₂,…,b₆ and drawn in blue, say.

The property satisfied by these 12 lines is that a_i intersects b_j if and only if i ≠ j.

Such a configuration of 12 lines is called a Schäfli double six.

We often denote a double six as

a₁

a₂

a₃

a₄

a₅

a₆

b₁

b₂

b₃

b₄

b₅

b₆

The remaining 15 lines of the surface can be expressed in terms of the double six.

For each ordered pair i,j with 1 ≤ i,j ≤ 6 and i ≠ j, a new line

c_ij = a_ib_j ∩a_jb_i

can be found. Here, a_ib_j denotes the plane spanned by a_i and b_j and a_ib_j ∩a_jb_i is the line of intersection of the planes a_ib_j and a_jb_i.

Recall that two planes intersect in a line in projective three-space.

The important fact about double sixes is that they determine the surface.

Thus, we can think of them as the "back-bone" of the surface.

In addition, once a double six has been distinguished, the incidences between lines can be described completely in terms of the labeling of lines as a_i, b_j, and c_ij. We omit the details.

There are 36 ways in which a double six can be chosen among the 27 lines (disregarding the fact that a given double six can be permuted in 6! ×2 different ways - rearranging the columns and switching a_i with b_i for i=1,…,6).

File translated from T_EX by T_TH, version 4.08.
On 4 Jun 2017, 11:23.