Schwarz surfaces are triply periodic minimal surfaces constructed by Schwarz (Schoen 1970, pp. 1-6). They include the Schwarz H surface (Chen and Weber 2021), Schwarz P surface, and Schwarz D surface. The P and D surfaces have different structures despite being locally isometric (Schoen 1970, pp. 1-6).
The following table summarizes the P, D, H, and CLP types (Brakke, Weber). The H and CLP types each form a family of surfaces, rather than a single surface up to scaling.
| type | designation | distinguishing structure |
| Schwarz P surface | primitive | simple cubic labyrinth networks |
| Schwarz D surface | diamond | diamond labyrinth networks |
| Schwarz H surface | hexagonal | one-parameter family with hexagonal symmetry |
| CLP | crossed layers of parallels | two-parameter family joining parallel layers |
The CLP surfaces are obtained by continuing a solution of Plateau's problem on a six-edge contour along the edges of a rectangular box. Geometrically, two perpendicular sets of parallel layers are joined by alternating tunnels (Brakke, Weber).
The construction uses the following two continuation principles.
1. If part of the boundary of a minimal surface is a straight line, then the image of the surface under a half-turn rotation about the line, when added to the original surface, extends the minimal surface.
2. If a minimal surface has a boundary arc in a plane and meets the plane at right angles along that arc, then the reflection of the surface across the plane, when added to the original surface, extends the minimal surface.
These principles extend a suitable minimal surface patch to a periodic surface (Wells 1991, pp. 224-225).
Surfaces similar to Schwarz's minimal surfaces have been cited among architectural applications of minimal surfaces (Bock Hyeng et al. 2025).
