meeting

Part III, Observers of Observers · defined in Chapter XII, The Coxeter Graph

Two anchored observers $(p,L)$ and $(q,M)$ at different clocks meet when they are joined in the Coxeter graph: when the triples of points outside $L\cup\{p\}$ and outside $M\cup\{q\}$ are disjoint, equivalently when their pairs of sky points are disjoint and harmonic, or when their report axes are orthogonal in the crystal read modulo seven. The relation is cubic, so each observer has three meetings, and there are forty-two in all, two over each of the twenty-one pairs of clocks.

Plate W.22The Coxeter graph on the twenty-eight observers: its forty-two edges are the meetings, two over each pair of clocks, and the base observer

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with its three meetings is lit.

Adjacent observers have $L\cap M=\{p+q\}$ . When their tops are $(p+q,p)$ or $(q,p+q)$ , each appends the antipode of its own top at its own clock, an exchange that is the same for every choice of transport between the charts; with an explicit completion it is trace preserving and covariant, and both appended letters are received ones. The Coxeter graph is the invariant connected relation between different clocks with the fewest edges, and permission for such observers to meet is an added clause: Chapter XII proves that a clean cross-clock meeting exists and where it lives, not that the world contains it.

With the octonion product kept, a meeting fixes its product-keeping comparison up to colour, and around any closed path such comparisons compose to pure colour. Flat comparisons exist and are unique up to colour gauge, because the graph with its twenty-four heptagons as faces is simply connected; the heptagons are the plaquettes of the gauge links.

Theorem(the complementarity graph) proved

The complementarity relation on the 28 anchored observers is the Coxeter graph. It is cubic, with 42 edges, girth seven and diameter four, and it is distance-regular with intersection array $\{3,2,2,1;1,1,1,2\}$ . From any observer the numbers of observers at distances 0,1,2,3,4 are 1,3,6,12,6. Its automorphism group is $\PGL(2,7)$ , which acts distance-transitively; the relativity group $\PSL(2,7)$ has seven orbits on ordered pairs, of sizes 1,3,3,3,6,6,6 from a given observer, and each lies inside one distance class.

As mathematics

An edge of the Coxeter graph, the unique connected cubic orbital graph of $\PSL(2,7)$ on the object of size 28: distance-regular with intersection array $\{3,2,2,1;1,1,1,2\}$ , girth 7 and automorphism group $\PGL(2,7)$ . Every seam between incarnations of the twenty-eight carries its edges onto the others’: two disjoint pairs of $\Proj^1(\F_7)$ with cross-ratio $-1$ ; two antiflags with disjoint triangles; two Sylow 3-subgroups $P$ , $Q$ such that $tu$ has order 4 for all elements $t\in P$ , $u\in Q$ of order 3. The forty-two edges are the object with stabilizer $C_4$ , also incarnated by the forty-two imaginary points of $\Proj^1(\F_{49})$ .

In the link complement, four cusps that are not the cusps of a tetrahedron contain exactly one pair of disjoint edges with harmonic ends, and the half-turn of a face about its base pair is the involution of a Coxeter edge. In Klein’s lattice neighbours $w,w'$ have $h(w,w')=\pm\sqrt{-7}$ : they are orthogonal modulo $\sqrt{-7}$ , not in the lattice.

Its name in another field	Bridge
an edge of the Coxeter graph	built
two disjoint pairs of $\Proj^1(\F_7)$ with cross-ratio $-1$	built
two antiflags with disjoint triangles	built
four cusps of the link complement that are not the cusps of a tetrahedron	built
the half-turn of a face of the link complement	built
two norm-3 vectors of Klein’s lattice orthogonal modulo $\sqrt{-7}$	built
an imaginary point of $\Proj^1(\F_{49})$	built

A second sense

In Chapter IV a meeting is an encounter of two registers in the arena, a coincidence of positions, which for two free registers in three dimensions recurs only finitely often; from Chapter XII on, and in the axioms, it is the Coxeter relation between observers. Seams keeps positions and observers apart: the observers’ network has girth seven, so the arena’s smallest loops are not the network’s, and no meeting can curve them.

Built from: anchored observer
In the dictionary: gaugethe object of size 42, cyclic
In the volume: IThe Founding SentenceXIIThe Coxeter GraphXIIIThe Level-Seven ShadowXVIIITwo Parents of the SkyXXIIOne SpeedXXIIILight, Vacuum and HandednessEp.Forcing, Not Sacred Geometry