clock

Part I, The Sentence and the Kernel · defined in Chapter I, The Founding Sentence

One of the seven points of the Fano plane, chosen as the direction of time. Relative to it the other six points are letters and the four lines missing it are reports.

Plate W.5The clock 1, ringed: the four lines missing it, the incircle 246 and the sides 257, 347, 356, are its reports, the six other points its letters, and the three lines through it, in gold, its axes.

Each letter lies on exactly two of the report lines and any two report lines meet in exactly one letter, so with the clock deleted the Fano plane is the complete graph $K_4$ : reports as vertices, letters as edges, and the three lines through the clock as the axes. The stabilizer of the clock in $\GL(3,2)$ has order 24 and acts on the four reports as the report group $S_4$ , so each of the seven clocks carries its own copy of the kernel graph. A clock is not a device that ticks. It marks which direction is time, and elapsed time is a count, the age of a register.

In the octonion attachment the seven points are the imaginary units and the clock is the unit chosen as time. Left multiplication by it sends every letter to $\pm$ its antipode and squares to $-1$ on the letters: a complex structure whose complex lines are the axes, and with the unit 1 a fourth line, the lepton’s. Its stabilizer in $G_2$ is $\SU(3)$ , so marking the clock is also the act that later makes colour a stabilizer.

Proposition(One clock is the kernel graph) proved

(1) Fix a clock $p$ . Each letter lies on exactly two report lines, and any two report lines meet in exactly one letter. The incidence is the complete graph $K_4$ , reports as vertices and letters as edges; antipodal letters are opposite edges and the axes are the three perfect matchings. (2) The stabilizer of $p$ in the collineation group $\GL(3,2)$ , of order 168, has order 24 and acts faithfully on the four report lines: it is the report group $S_4$ . (3) There are seven clocks and twenty-eight anchored observers.

Proof

A point $a\ne p$ lies on three lines, one of which is $\{p,a,a+p\}$ ; the other two miss $p$ . Two distinct lines meet in one point, not $p$ when both miss it. $\GL(3,2)$ is transitive on the seven points, so the stabilizer has order $168/7=24$ ; faithfulness is a finite check. Each point is missed by four lines.

As mathematics

A point of $\mathrm{PG}(2,2)$ , a nonzero vector of $\F_2^3$ : the object of the seven points, with stabilizer $S_4^a$ , which is self-normalizing, so the object is rigid. The points and the lines share a permutation character, $1+\chi$ with $\chi$ irreducible of degree 6, but they are not one object for a fixed marking: the stabilizer of a point fixes one point and no line, and the polarity between them is a seam only over the outer automorphism $g\mapsto(g^{\mathsf T})^{-1}$ . A clock and a report are two kinds of direction, never one.

Of the thirty lattices $\Z^8+\tfrac12C$ in the octonions exactly seven are closed under multiplication, Coxeter’s octavian orders, and the stabilizer of each fixes exactly one unit $e_c$ : the bridge from an octavian order to a clock is built. At Klein’s lattice a point $p$ carries a cube, since 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 $\pm u_1\pm u_2\pm u_3$ .

Its name in another field	Bridge
a nonzero vector of $\F_2^3$	built
an imaginary octonion unit $e_c$ , with its complex structure $L_{e_c}$	built
an octavian order, one of Coxeter’s seven integral octonion lattices	built
a cube of Klein’s lattice, the neighbour $M_p$ at $(\alpha)$	built
a complementary pair of tetrahedra of class $a$ in Thurston’s link complement	built
the direction of time	a reading

A second sense

The volume uses the word in two other senses and warns against confusing them. The internal clock operators of the tower’s composites (Chapters II, VI and VII) are rotation generators of internal frames: clock directions, not durations. The vector clocks of Fidge and Mattern and the Poisson clock of Chapter VI keep time, and a register’s time is its age, the count of its committed occurrences, never its clock. Seams uses the word only in its physical readings, where a clock is a point $c$ of the Fano plane or the unit $e_c$ .