planem

Planem is a theoretical construct in geometry and data visualization used to denote a canonical planar representation of higher-dimensional relationships. The term combines the idea of a plane with a formal object, and appears in introductory texts to illustrate projecting complex structures onto a two-dimensional surface while preserving chosen invariants.

Definition: A planem consists of a plane P and an incidence structure I, such as a graph or hypergraph, embedded in P. Two planems are equivalent if there exists a plane isometry that maps one embedding to the other while preserving I.

Construction and variants: Planems arise by selecting a set of invariants to preserve (for example vertex adjacencies or incidence relations) and constructing an embedding of the given structure into P that honors these invariants. Different invariant choices yield different planem variants.

Properties: Planems allow study of distortions introduced by projection, exhibit invariance under plane isometries, and facilitate comparison of different planar representations of the same relational data.

Applications: In graph drawing, planems serve as a pedagogical tool for illustrating planar representations of networks. In data visualization, they provide a framework for discussing projection effects. In education, planems help explain planarity, embeddings, and distortion trade-offs.

History: The term planem has emerged in modern geometry pedagogy and is used in textbooks and lecture notes as a teaching construct for planar embeddings.