sky

Part III, Observers of Observers · defined in Chapter X, The Finite Celestial Sphere

The finite celestial sphere: the eight points of the projective line over the field with seven elements, on which the observers’ group $\PSL(2,7)$ acts.

Plate W.19The sky: the eight points

0,1,\dots,6,\infty

of the projective line over

\F_7

, with its twenty-eight pairs, the anchored observers, drawn as chords.

The twenty-eight anchored observers are exactly the pairs of points of the sky. A clock pairs the eight points into four opposite pairs, as the diagonals of a cube pair its corners, and its symmetry group turns them as the rotations of that cube do. A point of the sky is a tour: the seven pairs containing it belong to seven observers, one at each clock, carried through all seven by a Singer cycle. The eight points are the null lines of the crystal’s metric read modulo seven, and each observer’s pair is the two ends of its report’s axis.

The finite geometry contains no rapidity and no boost, and $\PSL(2,7)$ is not a subgroup of the Lorentz group. The sky is the reduction, at a prime above seven, of the celestial sphere of $3+1$ dimensions, and in the lift its eight points are the eight cusps, the light directions. It has a second arithmetic parent, Mumford’s lattice, compact at the real place, and the two share the sky and nothing beyond it.

Theorem(observers are pairs) proved

For each anchored observer $x$ , the subgroup of order three in its stabilizer $H_x$ fixes exactly two points of $\Omega$ ; write $\operatorname{pair}(x)$ for them. The map $x\mapsto\operatorname{pair}(x)$ is a $\PSL(2,7)$ -equivariant bijection from the anchored observers onto the unordered pairs of points of $\Omega$ .

Proof

$H_x\cong S_3$ has a unique subgroup $C_x$ of order three, and it fixes exactly two points. Since $H_{gx}=gH_xg^{-1}$ , also $C_{gx}=gC_xg^{-1}$ , whose fixed points are $g\cdot\operatorname{pair}(x)$ , so the map is equivariant. Its image is a nonempty invariant set of pairs, hence all 28, and a surjection between two sets of 28 elements is a bijection.

As mathematics

$\Omega=\Proj^1(\F_7)=\{0,1,\dots,6,\infty\}$ with the Möbius action of $\PSL(2,7)$ , which is two-transitive; the stabilizer of a point is a Borel subgroup $7{:}3$ , self-normalizing, so the object of the eight points is rigid. Its other incarnations include the eight Sylow 7-subgroups, generated by Singer cycles of the Fano plane, the eight flex triangles of the Klein quartic, and the eight $E_8$ lattices $\Z^8+\tfrac12C$ in the octonions that share no line with the table. Over $\F_7$ the group is the derived orthogonal group of a three-dimensional quadratic form, and the eight points are its null lines.

The same eight points are seen at three places: as the link of a vertex of the tree of $\PGL(2,\Q_7)$ at the prime 7, as the cusps of the congruence link complement at the complex place, and, in Mumford’s arithmetic, as the eight lines of a null plane modulo $\sqrt{-7}$ . Both arithmetic parents reduce to the same line life of the group, and at 7 they share the sky but not its completion.

Its name in another field	Bridge
the cusps of Thurston’s congruence link complement	built
the Sylow 7-subgroups of $\PSL(2,7)$	built
the flex triangles of the Klein quartic	built
the eight $E_8$ lattices in the octonions sharing no line with the table	built
the link of a vertex of the tree of $\PGL(2,\Q_7)$	built
the finite celestial sphere	a reading

Builds: cusplift
In the dictionary: lifecompletioncontinuumthe sky
In the volume: VIIISpace as a TallyXThe Finite Celestial SphereXIRulial RelativityXIIThe Coxeter GraphXIIIThe Level-Seven ShadowXIVWhy OctonionsXVSpin from the Double CoverXVIIITwo Parents of the SkyXXIIOne SpeedXXIIILight, Vacuum and HandednessXXIVA Number Nature Could RefuteEp.Forcing, Not Sacred Geometry