Plexocentric

Plexocentric is the standard geometric framework for describing a coordinate system in which all vectors are understood to terminate on the surface of a three-dimensional solid, known as the plexohedron, while their origins remain fixed at a hypothetical interior point, the centroid or null vertex. First formally described in Annex VII of the Ninth Concordance, plexocentric mathematics underpins the official cartography of the Tower’s non-linear stairwells and is the basis for the Department of Cartographic Anomalies’s internal navigational aids.

Formal Definition

A plexocentric coordinate system is defined by a closed polyhedron (the plexohedron) of any order, with a designated interior point C. Any location within the system is described not by a vector from C to the point, but by the unique point on the plexohedron’s surface where that vector, if extended, intersects it. Thus, all points within the volume map to a point on the shell. The system is only reversible if the plexohedron’s shape and the distance from C to the surface along that vector are known, data which is recorded in a supplemental depth ledger. The most common plexohedron adopted for Tower use is the Truncated Icosahedron of Floor 707, leading to the colloquial term “707-coordinates” among surveyors.

Applications within the Tower

The primary application of plexocentric coordinates is in the mapping of so-called Volumetric Corridors, architectural spaces where conventional Euclidean measurement breaks down due to Spiral Recession or Haversine Diminishment. By fixing the coordinate shell to a known, stable architectural feature—often the outermost wall of a stairwell shaft—and placing the null vertex at the shaft’s conceptual center, clerks can produce consistent, if counterintuitive, floorplans. For example, the infamous Möbius Stack in the West Marginal Wing is charted using a plexohedron shaped as a right hexagonal prism, as per Ordinance 12.45.c of the Ninth Concordance.

The system is also mandated for all astrolabe readings taken from the Observatory of Unfixed Stars on Floor 10,002, where celestial bodies are logged by their plexocentric projection onto a dodecahedral shell surrounding the Tower’s theoretical pinnacle.

Controversies and Limitations

The most significant scholarly dispute regarding plexocentric mapping stems from a contradiction between the Concordances. Annex VII of the Ninth Concordance states the null vertex is a dimensionless point, while Clause 303 of the Sixth Concordance implies it has a “radius of non-influence” not to exceed 0.002 standard ells. This has led to two competing schools of calibration: the Point-Centrists, who adhere to the Ninth, and the Sphere-Adherents, who follow the Sixth. Maps produced by these factions can differ by several ells in interior placements, though both agree on the surface projections.

A more practical limitation is the phenomenon of plexohedral collapse, wherein a mapped volume undergoes sudden Topological Flux, causing the reference plexohedron to invert or adopt a non-convex form. All depth ledgers for that volume are immediately declared void by the Clerks of the Inverted Solid, and the area must be re-surveyed from scratch, a process detailed in Supplemental Protocol Omega.

Historical Development

The conceptual roots of plexocentric geometry are often traced to the 14th-century diagrams of the cartographer-monk Librarian Silas, who attempted to map the Bottomless Atrium using a system of threads radiating from a central lantern to the walls. The first formal mathematical treatment was presented in a now-lost scroll titled Concerning the Measurement of That Which Has No Stable Middle, attributed to the Geometric Sect. Its principles were institutionalized following the Surveyors’ Revolt of 1,502 A.T. (After Tower), when the ruling Librarians sought a single, authoritative system to quell disputes over corridor ownership.

The adoption of the truncated icosahedron as a standard in 1,721 A.T. by Edict of the Polyhedron was a compromise between aesthetic preference (its symmetry was deemed “pleasing to the administrative eye”) and practical concerns, as it provided a high number of faces for granular coordinate designation.

Plexocentric Artifacts

Several physical instruments exist for working with plexocentric coordinates. The Plexoscope, housed in the Hall of Corrective Lenses on Floor 450, is a brass device consisting of a central point source of light and a interchangeable polyhedral cage; it is used to train novices in projection principles. The Depth Ledger of the Central Spire, a tome weighing approximately 90 stone, records the vector-length data for the Tower’s primary vertical axis from Basement 34 to Floor 10,000, using a cylindrical plexohedron. It is updated biannually, a task requiring a team of twelve Vector Clerks.

A decorative, non-functional plexohedron carved from Whispering Marble sits in the lobby of the Department of Cartographic Anomalies, serving as a reminder, in the words of a now-famous internal memo, “that all locations are merely shadows cast upon a chosen shell.”