Bureau of Impossible Geometry

The Bureau of Impossible Geometry is a distinct office of the Tower established to catalogue geometrically impossible arrangements that have not yet occurred. It is frequently confused with the Bureau of Ambivalent Geometry, which catalogues arrangements that have occurred but obey no stable geometry, and with the Custodians of Static Geometry, who maintain the immovable fixtures of the stairwells. The distinction is set out in Third Concordance, Annex 12, though the Annex is itself internally inconsistent and, by the reckoning of the Bureau's own registrars, has been amended more often than it has been read.

History and founding

The Bureau was constituted in the winter following the Crisis of the Ambivalent Stairwell, when several floors were found to possess geometries that had not yet happened but were evidently about to. The founding instrument, Instrument 404-B, was sealed under the authority of a Seventh Librarian whose name the records give only as the signatory of the left margin; no full name is attested. The instrument's preamble states, without further remark, that "a geometry that has not yet occurred is nevertheless an event, and events may be filed."

The Bureau's first registry occupied Floor 1140, the so-called Waiting Antechamber, and its early holdings consisted largely of anticipations of corridors that did not exist. A dispute with the Custodians of Static Geometry followed in the third year, when the Custodians claimed that anticipation of an unmade stairwell fell within their purview. Third Concordance settled the matter in the Custodians' favour for stairs already built and in the Bureau's favour for stairs not yet begun. The ruling is widely cited and rarely complied with.

Scope of holdings

The Bureau's principal holdings are the Anticipated Register, which records geometries expected to manifest, and the Withheld Ledger, in which such geometries are recorded a second time should anticipation appear to have accelerated their arrival. Entries in the Withheld Ledger are made in pencil. A 1914 recension of the Register describes the Bureau's method as "the filing of a shape before the shape has consented to be a shape" (Recension of the Anticipated Register, Floor 1140, fol. 9). The Register has not been opened on its spine since 1952.

The Bureau holds no objects. Its inventory comprises descriptions only, and a description that inadvertently produces the geometry it describes is transferred at once to the Bureau of Ambivalent Geometry under the standing terms of Edict 719-Λ. Three such transfers occurred in a single week in the autumn of 2007 and are recorded in the Bureau's Quarterly Variance Report under the heading "Precipitation by Description."

The Kleinmann Precedent

The most consequential holding is the Kleinmann Precedent, a determination arising from the case of the corridor on Floor 2002 that returned to its own entrance without enclosing any volume. The corridor had not been built when the complaint was filed; it was filed against its anticipated geometry by a clerk of the Bureau of Cartographic Incongruity who had encountered the description and wished to prevent the corridor's construction altogether. The Bureau ruled that an impossible geometry may be catalogued in advance of its occurrence, provided the cataloguer does not walk the route.

The Precedent has been challenged twice. The first challenge, brought by the Fourth Librarian of Ephemeral Geometry, argued that advance cataloguing merely postpones the impossibility and does not prevent it. The Bureau's reply, preserved in the Florilegium under the title "On the Patience of Shapes," concedes the point and adopts the postponement as its method.

Scholarly disagreement

The central dispute concerns the point at which an anticipated geometry becomes impossible rather than merely absent. The Algernon Threshold holds that sustained analysis of a possible geometry introduces observer paradoxes sufficient to render it impossible, and thus that the Bureau's own cataloguing causes the impossibilities it records. The Bureau rejects this reading, maintaining in its standing reply (Bureau circular, Floor 1140, clause 3) that "the Threshold measures the analyst, not the shape." A minority of registrars privately hold the opposite and mark their entries with a small Greek delta in the lower margin, a practice the Bureau has neither authorised nor forbidden.

A second disagreement turns on whether an anticipated geometry that never occurs should be removed from the Register. The Bureau's position is that it should not, on the ground that the failure of a geometry to occur is itself a property of that geometry and must be filed. The Inventory of Impossible Geometries takes no position, and relations between the two offices are cordial but cool.

Access and conduct

The Register is open to Librarians of the fifth grade and above during the hours of the afternoon. Visitors are reminded that no visitor may pass the registry door twice in any one visit, as the second passage tends to occur at a different end of the corridor and is recorded as a manifestation. This rule is posted at the threshold and is not enforced by any officer; transgression is noted in the ledger and reviewed by no one.

"The shape that has not yet been is not absent. It is early."

— Instrument 404-B, preamble, second recension