The seven points

Which class is which depends on the marking. The outer automorphism exchanges each class $a$ with its class $b$, so the points and the lines of the Fano plane are assigned their classes only once the group of the plane is identified with the group of the projective line. The seam table and this journal’s labels use the identification of Seams, under which the points are class $a$. The volume’s own chart, its table of observers and pairs in Chapter X, uses the other: there the stabilizer of the clock 1 fixes the bisection $\{0,1,2,4\}\mid\{3,5,6,\infty\}$, which lies in class $b$. Read in that chart, the names exchange: the clocks are class $b$ and the lines class $a$, and so on for $A_4$ and $V_4$.

The points of the Fano plane, and its complete quadrilaterals; the bisection $\{0,1,2,5\}|\{3,4,6,\infty\}$ of $\Proj^1(\F_7)$ and its orbit, and the perfect matchings in the orbit of $\{01,24,36,5\infty\}$; the subgroups $V_4^a$, $A_4^a$ and $S_4^a$; the conics for $\alpha=\zeta+\zeta^2+\zeta^4$ and the self-polar triangles from $V_4^a$ in Klein’s plane; the $K_4$’s of Coxeter vertices whose antiflags have a common point; the unit lines of the octonions; and Coxeter’s seven octavian orders.

In Thurston’s congruence link complement it is the complementary pairs of tetrahedra of class $a$, the points of the Fano plane of the cells. Among the thirty octonion lattices, read as the points and planes of $\mathrm{PG}(3,2)$, the seven octavian orders are the planes through the point of Kirmse’s lattice. In the lattice $E_8$ of class $a$ over $\Z[\lambda]$, the seven lines through the residue of the sixteen vectors modulo $\lambda$ have stabilizers of class $S_4^a$, the module of the points, as do the points of Klein’s $\mathbf 3$ reduced at the prime of $\Q(\zeta)$ over $\lambda$.

At Klein’s lattice a point $p$ of the Fano plane carries a cube: the neighbour $M_p$ at $(\alpha)$ has an orthonormal basis, on which the stabilizer $S_4$ of $p$ acts as the rotation group of the cube with vertices $\pm u_1\pm u_2\pm u_3$. Carried along the two trees at 7, this object behaves differently: along the Bianchi group’s tree it is carried in exactly one way, since it is rigid and every mixed square is proper, and along Mumford’s in none, since an improper twist turns it into the object of the lines.