Winged-edge
A classic polygon-mesh data structure in which each edge remembers its two vertices, its two adjacent faces and its four neighbouring edges.
A mesh stored as just a list of triangles makes "which faces share this edge?" expensive — you have to search everything. Winged-edge, which Bruce Baumgart introduced in his 1972 Stanford thesis, hangs all the topology on the edge instead: its two end vertices, the two faces it separates, and the four edges that "wing" out from its ends. Adjacency queries then become constant time.
The cost is memory and bookkeeping: every edit must update many cross-references consistently. Later structures like the half-edge (directed half-edges) and quad-edge simplified traversal and today dominate modelling tools and mesh processing, but winged-edge was the first to make arbitrary topology tractable.