anchored observer

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

A clock together with an anchor; there are twenty-eight. On the sky it is a pair of points, and in the lift an edge.

{0, ∞}(1, 246)d₀⟨z ↦ 2z⟩

Projective line

a 2-subset of P¹(F₇)

Fano plane

an antiflag (p, L), p ∉ L

Graphs

a vertex of the Coxeter graph

The group

a Sylow 3-subgroup of PSL(2,7)

generator: z ↦ 2z
on the eight points: (1 2 4)(3 6 5)
fixes: {0, ∞}, and nothing else

Klein quartic

a bitangent of x³y + y³z + z³x = 0

the line: x + y + z = 0
touching: at (1 : ω : ω²) and (1 : ω² : ω), the two points fixed by ρ of the subgroup

Choose a vertex of the Coxeter graph, or step through all twenty-eight.

Plate W.2One anchored observer in five theories at once: the antiflag

(1,246)

, the pair

\{0,\infty\}

, the Sylow subgroup

\langle z\mapsto2z\rangle

, the bitangent

x+y+z=0

and a vertex of the Coxeter graph; stepping moves all five names together.

Chapter I counts them: the stabilizer of a clock $p$ in $\GL(3,2)$ is the report group $S_4$ , and each of the seven clocks is missed by four lines, so there are $7\times4=28$ anchored observers. Chapter X finds them on the sky. The stabilizer $H_x$ of one observer has order six and acts faithfully on its three rods, so $H_x\cong S_3$ , and the only thing an observer’s own symmetry can do is permute its rods. The rotation of the rods fixes exactly two points of $\Proj^1(\F_7)$ , and this pair is a $\PSL(2,7)$ -equivariant bijection from the observers onto the 28 pairs of sky points; for the base observer $(1,246)$ it is $\{0,\infty\}$ . A clock’s four observers have disjoint pairs, the body diagonals of a cube.

Chapter XIII finds them in the lift: its twenty-eight edges are the pairs of cusps, each with stabilizer $S_3$ , and each carries a canonical rest frame $T_{ab}$ . Two observers at different clocks meet when they are joined in the Coxeter graph.

Definition(Clock, anchored observer)

Let $\Pi$ be the Fano plane, whose points are the seven nonzero vectors of $\F_2^3$ and whose lines are the seven triples summing to zero. A clock is a point $p$ of $\Pi$ . Relative to $p$ the letters are the six points $a\ne p$ , the antipode of $a$ is $\bar a=a+p$ , the axes are the three lines through $p$ , and the reports are the four lines missing $p$ . An anchored observer is a pair $(p,L)$ with $L$ a line missing $p$ : a clock together with the report on which the observer stands, its anchor. The three letters on $L$ are the anchor’s rods; the other three are its odd letters.

As mathematics

An antiflag $(p,L)$ of $\mathrm{PG}(2,2)$ , $p\notin L$ , so that $\F_2^3=p\oplus L$ and the stabilizer is $\GL(p)\times\GL(L)\cong S_3$ , the normalizer of a Sylow 3-subgroup. The subgroups of order 6 of the group of order 168 form a single class of self-normalizing subgroups, so the twenty-eight are one rigid object: between any two of its incarnations there is exactly one seam, and the seams agree along every route between theories.

The Coxeter graph is intrinsic to it. Among the orbital graphs of the group on the twenty-eight exactly one is cubic and connected, and every seam carries its edges onto the edges of the others: on the pairs it joins two disjoint pairs with cross-ratio $-1$ , on the antiflags it joins $(p,L)$ and $(q,M)$ when $p\ne q$ , $L\ne M$ , $p\notin M$ and $q\notin L$ .

Its name in another field	Bridge
a 2-subset of $\Proj^1(\F_7)$	built
a Sylow 3-subgroup of $\PSL(2,7)$	built
a bitangent of the Klein quartic	built
an odd theta characteristic of the Klein quartic	built
a vertex of the Coxeter graph	built
an edge of Thurston’s congruence link complement	built
a vector of norm 3 in Klein’s lattice, up to sign	built
a class of records modulo $\Gamma(\mathfrak p)$ , a class of rest frames	built
an observer	a reading