axis

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

A letter together with its antipode; there are three.

Plate W.4The three axes of the kernel graph as its three pairs of opposite edges:

A=\{01,23\}

in blue,

B=\{02,13\}

in gold,

C=\{03,12\}

dashed. Each is the pair of fibres of one binary reading.

Equivalently an axis is a perfect matching of the four reports. Each axis consists of the two fibres of one of the minimal register’s three binary readings, the sign $s$ , the switch $\ell$ and their product $s\ell$ , so the three axes are the three ways to forget one of the register’s two bits. On the Fano plane the axes of a clock $p$ are the three lines through $p$ . In the report form the letter differences along a matching are orthogonal, so the three axes are mutually perpendicular spatial directions, for every member of the family of forms.

Chapter V separates places from displacements. The report group acts on the three axes through $S_4\to S_3$ , and the kernel is the Klein four-group of double transpositions: the displacements, the zero and one step along each axis, whose two cycles are the two letters of that axis. Relabelling can permute the three steps; it cannot turn no step into a step.

Proposition(The octahedron and the three forgettings) proved

(1) Under adjacency the six letters form the octahedron: each letter is adjacent to four letters and antipodal to exactly one. (2) Each axis consists of the two fibres of one of the register’s three binary readings $s$ , $\ell$ and $s\ell$ . A letter is one answer to one binary question about the reports, its antipode is the other answer, and the three axes are the three ways to forget one of the two bits.

Proof

A pair of reports shares a report with every other pair except its complement. The reports form $\Z_2\times\Z_2$ under componentwise multiplication; its three nontrivial characters are $s$ , $\ell$ and $s\ell$ , and the two fibres of each are a perfect matching. Distinct characters give distinct matchings, and four objects have exactly three perfect matchings.

As mathematics

With the clock $c$ fixed, an axis is a line through $c$ ; with $c$ it is a flag of the Fano plane, one of twenty-one, the object with stabilizer $D_8$ . On $K_4$ it is a pair of opposite edges. Fano’s axiom fails in characteristic 2, and in its dual form the three diagonal lines of the complete quadrilateral of reports, the three axes, pass through one point, the clock; over the reals they would form a triangle.

At the prime above three the axes are the three non-split lines of $\mathfrak{sl}_2(\F_3)$ , each pairing the four points of $\Proj^1(\F_3)$ off. In Klein’s lattice the twenty-one flags are the twenty-one pairs $\pm v$ of vectors of norm 2: $s_v$ is the half-turn about a four-fold axis of the clock’s cube, and modulo $(\alpha)$ it is the transvection with centre $p$ and axis $\ell$ .

Its name in another field	Bridge
a perfect matching of $K_4$	built
a nonzero displacement, a double transposition in $V_4\lhd S_4$	built
a non-split line of $\mathfrak{sl}_2(\F_3)$	built
a pair $\pm v$ of norm-2 vectors of Klein’s lattice, with the half-turn $s_v$	built
a spatial axis	a reading

Built from: letterantipode
In the dictionary: imprintthe object of size 21
In the volume: IThe Founding SentenceIIIThe KernelIVWorld, Kernel, ObserverVFour Reports, Six LettersVIIThe Branchial TreeVIIISpace as a TallyIXWhat Space ForgetsXThe Finite Celestial SphereXIRulial RelativityXIIThe Coxeter GraphXIVWhy OctonionsXVIThe QuartetXVIIITwo Parents of the SkyXXThe Commuting SquaresXXIVA Number Nature Could Refute