antipode

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

Of a letter: the letter made of the two other reports.

Plate W.3The letter 01 and its antipode 23 in gold: the two opposite edges of the kernel graph, which share no report; each of the other four letters shares a report with 01.

Two letters are adjacent when they share a report and antipodal when they are disjoint, and under adjacency the six letters form the octahedron: each is adjacent to four and antipodal to exactly one. The reports form the group $\Z_2\times\Z_2$ , a letter is one answer to one binary question about them, and its antipode is the other answer. On the Fano plane, relative to a clock $p$ , the antipode of the letter $a$ is $\bar a=a+p$ , the third point of the line through $p$ and $a$ .

The antipode is what only a partner can write. Two consecutive letters of a log are equal, adjacent or antipodal, and each case has one origin: persistence, the register’s own move, or a received exchange. A private writer never appends the antipode of its top letter, which shares no report with it. In the lift a letter is a rest frame, and the step to its antipode is a boost with $\gamma=2$ , against $\gamma=\tfrac32$ for an adjacent letter.

As mathematics

For a clock $c$ the antipode is the translation $a\mapsto a+c$ of the six points other than $c$ : an involution without fixed points whose orbits are the three lines through $c$ , with $c$ removed. On $K_4$ it sends each edge to the opposite edge, and on the octahedron $K_{2,2,2}$ , the line graph of $K_4$ , it is the antipodal map. With its clock a letter is an ordered pair $(c,a)$ of distinct points, one of the forty-two of the object with stabilizer $V_4^b$ , and its antipode is $(c,a+c)$ .

Every collineation is linear on $\F_2^3$ , so $g(a+p)=ga+gp$ : antipodes at one clock go to antipodes at the image clock, which is why every rule written with antipodes reads the same in every chart.

Its name in another field	Bridge
the opposite edge of $K_4$	built
the antipodal vertex of the octahedron $K_{2,2,2}$	classical
the other fibre of one of the characters $s$ , $\ell$ , $s\ell$ of $\Z_2\times\Z_2$	built
the unit rest frame at inner product 2, a boost with $\gamma=2$	built

Built from: letterreport
Builds: axis
In the dictionary: the object of size 42, class b
In the volume: IThe Founding SentenceIVWorld, Kernel, ObserverVFour Reports, Six LettersVIIThe Branchial TreeXThe Finite Celestial SphereXIRulial RelativityXIIThe Coxeter GraphXIIIThe Level-Seven ShadowXIXRulial InvariantsXXThe Commuting SquaresXXIIILight, Vacuum and Handedness