Plexocentric Mathematics

Plexocentric Mathematics is the formal discipline governing the construction, manipulation, and adjudication of coordinate systems in which all vectors originate at a single fixed point, the Centroid, and terminate upon the surface of a closed polyhedron called a Plexohedron. It is a branch of Apocryphal Mathematics, the wider inquiry into operations on objects that do not exist, and it is the working arithmetic of the Department of Cartographic Anomalies, which has maintained the discipline's standards since at least the Fourth Recension.

The practice is distinguished from ordinary positional geometry by a single prohibition, repeated in every recension: no vector may pass through the Centroid, and no vector may terminate short of a facet. A vector that does neither is called irresolute and is not admissible in any certified survey.

Origins and the Ninth Concordance

The discipline was not named until the Ninth Concordance, whose Annex VII formally described plexocentric operators and their permitted compositions. The Annex is eleven sentences long and is generally held to be the shortest founding document in the archive; the Third Concordance, by contrast, devotes some forty pages to the same operators without ever using the word plexocentric, a circumstance the Directorate of Cartographic Review has ruled "regrettable but not voiding" (Directorate Minute 4,112).

Annex VII was drafted by the clerk Sennet Orrow, who had been assigned the task after two senior drafters declined it on grounds of conscience. Orrow's working notes survive in the Scriptorium Annex and record his central difficulty plainly:

"The operators behave correctly when written and incorrectly when used. I have therefore confined the definition to the written case."

— Sennet Orrow, working notes, Scriptorium Annex, bundle 9

This distinction between the written and the used case remains the discipline's characteristic hazard.

Core Operations

Plexocentric mathematics recognizes three primitive operations. Projection carries a point outward from the Centroid until it meets a facet. Collapse returns a facet-point to the Centroid along its own line. Rotation moves a facet-point along the surface of the plexohedron without changing its radial distance, which in practice means it does not move at all, since radial distance is fixed by the surface. Rotation is retained in the canon because removing it would require an amendment, and amendments to the Ninth Concordance are heard only in the winter session.

Composition is permitted in any order except Collapse-before-Projection, which yields the null survey and has been the subject of forty-one disciplinary findings since the practice was standardized.

The Factional Dispute

A longstanding disagreement divides the discipline into two schools. The Plexocentric faction holds that the center of any informational sphere is constantly shifting based on observational context, and therefore that the Centroid is a convenience rather than a fact. Against them stand the Sphere-Adherents, sometimes called the Fixed-Center school, who maintain that a coordinate system with a moving origin is not a coordinate system but a mood.

The dispute was referred to the Subcommittee for Cartographic Anomalies in the winter of the year recorded as 3,308 by the Depth Ledger, which heard eleven days of testimony and issued no finding. Its secretary, Marn Elbick, wrote in the margin of the hearing register that "both parties are correct about the other." The margin note has been cited in subsequent proceedings more often than the proceedings themselves.

The practical consequence is that a plexocentric survey must declare its school in the header. Surveys that do not are returned unopened, and the returning clerk keeps the stamp.

Application in the Cartographic Department

The Department of Cartographic Anomalies employs plexocentric mathematics chiefly for internal navigational aids, where its indifference to absolute position is considered a virtue. The Department's corridor charts are drawn on a plexohedron of ninety-four facets, each facet corresponding to a survey station; a courier moving between stations is described as a projection and arrives, by definition, at the surface.

The Department's sister body, the Department of Cartographic Acclimation, declined to adopt the method, and the two offices have not co-signed a survey since. Acclimation's position, stated in a circular of the period, is that its couriers "do not require a theory of where they are in order to be there."

Criticism and Standing

Algernon's Theorem on Conceptual Plenitude is sometimes invoked against the discipline on the grounds that plexocentric mathematics describes more coordinate systems than can be distinguished, and therefore describes none. The counter-argument, advanced by the Directorate of Cartographic Review in its fourth ruling, is that the theorem concerns conceptual plenitude and not cartographic sufficiency, and that the distinction, while narrow, is load-bearing.

The Algernon Incompatibility Theorem is likewise cited, though incorrectly; the Directorate has asked that it stop being cited.

Plexocentric mathematics remains a required subject for any clerk seeking the survey seal, examined over two sessions, with the second session consisting entirely of Collapse problems. Candidates are advised to bring their own stamp pad. The discipline has never been suspended, though it came close in the year 3,341, when a single Rotation error propagated through the corridor charts of eleven floors and the Department was obliged to redraw them from the Centroid out, a task that took the better part of a season and produced a set of charts that those who used them agreed were no worse than the originals.