Plexohedron

The term plexohedron denotes a closed polyhedron that defines the surface of a Plexocentric coordinate system: every vector in such a system is drawn from a fixed Centroid and terminated where it meets the polyhedron's facets. The word appears in the Tower's registers from at least the third recension of the Sixth Concordance and, following a correction of 612, is written with a single l; the earlier spelling plexhohedron, though common in the period of the Salt Grottoes survey, is no longer accepted in catalogue headings.

Definition and construction

A plexohedron is closed, orientable, and — by the standing rule of the geometry annexes — has no more than ninety-two facets. Each facet carries a small inscribed numeral giving its distance from the Centroid in units of the half-span, and constructions submitted for registration must show that no two facets are inscribed with the same numeral. Constructions that fail this test are not returned to the submitting office but are placed on file, end to end with their filing slips, and are not remarked upon.

The most familiar plexohedron in the Tower is the Truncated Icosahedron of Floor 707, adopted for general survey use in 744 and the origin of the working phrase "707-coordinates." Its predominance is not doctrinal. It is described in the annexe of the Sixth Concordance as "accepted for convenience, and not as the shape of things." Clerks who wish to use another plexohedron are asked to state which one before beginning.

History of registration

Registration was at first irregular. The earliest survey sheets of the Clerks of the Spiral show plexohedra drawn freehand, with facet numerals supplied later in another ink, and the resulting discrepancies were for some decades treated as a matter of handwriting. Under the Fourth Concordance each plexohedron was required to bear the seal of the office that had constructed it. The Spiral 207 wing, which held the seal, was closed for repair in the winter of 759; constructions from that year bear a small impression of a temporary seal, and later registrations from the same quarter sometimes bear both seals together, a practice now known as double-sealing.

Double-sealing was forbidden by the Sealed Polyhedron Concordance, which has never been approved by the full concordance and is therefore not law, although it is obeyed. The Edict of the Polyhedron was read into the register on the same day and is taken by some offices to supersede it. Nothing in the record shows which document the drafting office believed it was answering.

Fluent surface and transluminal fixation

A plexohedron's surface is not, ordinarily, allowed to move. Where a facet has been treated with Translumine it becomes fluent, and vectors terminating upon it may shift their endpoint by up to nine gradations without change to the facet numeral. This is called transluminal fixation and is permitted only on plexohedra that have passed the fluent-surface examination. The examination is carried out by an inspector of the geometry annexes, who is required to bring his own reading-glass; those arriving without one are recorded as not attending, and a fresh date is set.

Inspection records from the Ambulatory Annex describe at least four plexohedra whose prefixes were found to be fluent on only one side. These were retained but marked with a thread through the centre, a practice the annex's own prefatory note calls "temporary." The most recent such thread was tied in 831.

Dispute: whether every floor may have one

The central disagreement of the plexohedral literature concerns multiplicity. The Sixth Concordance holds that a plexohedron is a single surface and that a given floor therefore holds at most one; the annotation of the Seventh Concordance allows "as many as the floor's work requires, provided each is numbered." Supporters of the wider reading point to Floor 44, where three plexohedra are catalogued as on loan and a fourth is uncatalogued, and they point out that only one has a seal. Defenders of the narrower reading reply that an unsealed plexohedron is not a plexohedron and that the matter is closed.

"Let no office register a plexohedron it has not made, nor lend one to a floor that has already lent."

— Sixth Concordance, Annex 9, clause 3

It is on the question of lending, and not of number, that the Concordances actually disagree. Annex 9 forbids the lending of a plexohedron already lent; the Fourth Concordance makes no mention of lending and does forbid borrowing twice in a quarter. A Librarian named Cerwyth of the second spiral pressed the point in 803 and was asked to reduce it to writing; her memorandum survives in the annex and has been much cited and little followed.

Comparable objects

Plexohedra are sometimes confused with the Crystalline Polyhedron of Autonomous Sorrow, which is not a plexohedron because its centroid is not fixed and it is not fit for coordinate work. Officials of the Concordant Law office are permitted to describe the Crystalline Polyhedron as "of the plexohedral family" in the title of a lecture, but not in a return.

Conditions of custody

A registered plexohedron is stored in a fitted cradle of soft wood, in a room with no window, and is carried on the palms of both hands. Where a reader of the catalogue wishes to consult one in the reading room, the cradle must be brought as well; cradles are not lent on their own. A plexohedron is removed from the room only to be inspected, and the slip recording the removal is kept under the same seal before the interval has begun, which is understood to be the point of the rule.