The object of size 14, class b

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 oriented quadrangles of the Fano plane; the four-subsets of $\Proj^1(\F_7)$ in the orbit of $\{0,1,2,4\}$; the groups $V_4^b$ with a cyclic order of their involutions; the oriented self-polar triangles from $V_4^b$; the oriented $K_4$’s of Coxeter vertices whose antiflags have a common line; the 14 lattices $\Z^8+\tfrac12C$ in the octonions whose Fano plane shares exactly one line with the octonion table; and the 14 ideal tetrahedra of Thurston’s congruence link complement whose cusp sets lie in the orbit of $\{0,1,2,4\}$.

The fourteen octonion lattices whose Fano plane shares one line with the table are the fourteen tetrahedra of class $b$ of Thurston’s manifold: for each, exactly four of the eight lattices sharing no line with the table share three lines with it, these four-sets are the planes of an affine 3-space on the eight lattices, and the seam from the eight lattices to the cusps carries them onto the tetrahedra of class $A_4^b$, with incidence preserved.