Schwarz P Surface

The Schwarz P surface is an embedded surface that is a triply periodic minimal surface and separates three-dimensional Euclidean space into two congruent, interwoven labyrinths with simple cubic lattice networks as their skeletons. The letter P stands for primitive, referring to its primitive cubic lattice of translations that preserve each labyrinth (Schoen 1970, pp. 1-6, 37-39).

A cube-shaped unit cell can be viewed as a chamber with six openings, directed toward the centers of its six faces. Repeating these cells joins the openings into a connected network (Brakke). Identifying opposite faces of this labyrinth-preserving unit cell gives a compact surface of genus 3 (Schoen 1970, p. 6).

The Schwarz D surface and the gyroid are locally isometric to the Schwarz P surface, although their labyrinths have different structures (Brakke).

The photograph above shows a model of Tony Smith's sculpture Fermi (1973) with a single repeating unit at right. The repeating unit adapts the Schwarz P surface by reshaping its six arms to end in circles. A monumental version planned as a 56-foot-tall inflatable vinyl structure was never realized (Lippincott 2024, p. 503; Tony Smith Estate).