geometreiden

Geometreiden is a term used in speculative geometry to denote a class of abstract geometric configurations that generalize traditional figures by focusing on the relationships among incidences, distances, and dualities rather than on concrete coordinates. In this thought-experimental framework, geometreiden are defined up to isomorphism by a compact set of axioms describing how shapes relate under a duality operation and a metric assignment.

A defining feature of geometreiden is a duality operation that exchanges elements such as vertices and faces, along with a generalized metric that assigns a notion of length to relational edges rather than to Euclidean segments. They may also include a curvature concept that can take non-integer values, allowing a broader range of geometric behavior than classical figures. Representations of geometreiden often use combinatorial structures like incidence graphs or cell complexes, embedded in an abstract metric space, with symmetry groups consisting of transformations that preserve both incidence relations and metric properties.

Origins and usage: The concept has appeared in speculative geometry writings and science-fiction discourse as a vehicle for exploring how geometry changes when core axioms are altered. It is sometimes employed pedagogically to illustrate the dependence of geometric intuition on underlying assumptions, and in world-building to describe the geometry of fictional spaces. Geometreiden are not part of established mathematical practice and lack standard notation or formal status in mainstream geometry.

See also: geometry, non-Euclidean geometry, duality, metric geometry, incidence geometry.