In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in R3 that is invariant under a rank-3 lattice of translations.
These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries. Monoclinic and triclinic examples are certain to exist, but have proven hard to parametrise.
TPMS are of relevance in natural science. TPMS have been observed as biological membranes, as block copolymers, equipotential surfaces in crystals etc. They have also been of interest in architecture, design and art.
Nearly all studied TPMS are free of self-intersections (i.e. embedded in R3): from a mathematical standpoint they are the most interesting (since self-intersecting surfaces are trivially abundant).
All connected TPMS have genus ≥ 3, and in every lattice there exist orientable embedded TPMS of every genus ≥3.
Embedded TPMS are orientable and divide space into two disjoint sub-volumes (labyrinths). If they are congruent the surface is said to be a balance surface.
The first examples of TPMS were the surfaces described by Schwarz in 1865, followed by a surface described by his student E. R. Neovius in 1883.
In 1970 Alan Schoen came up with 12 new TPMS based on skeleton graphs spanning crystallographic cells.
While Schoen's surfaces became popular in natural science the construction did not lend itself to a mathematical existence proof and remained largely unknown in mathematics, until H. Karcher proved their existence in 1989.
Using conjugate surfaces many more surfaces were found. While Weierstrass representations are known for the simpler examples, they are not known for many surfaces. Instead methods from Discrete differential geometry are often used.
The classification of TPMS is an open problem.
TPMS often come in families that can be continuously deformed into each other. Meeks found an explicit 5-parameter family for genus 3 TPMS that contained all then known examples of genus 3 surfaces except the gyroid.
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In differential geometry, the associate family (or Bonnet family) of a minimal surface is a one-parameter family of minimal surfaces which share the same Weierstrass data. That is, if the surface has the representation the family is described by where indicates the real part of a complex number. For θ = π/2 the surface is called the conjugate of the θ = 0 surface. The transformation can be viewed as locally rotating the principal curvature directions.
A gyroid is an infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970. It arises naturally in polymer science and biology, as an interface with high surface area. The gyroid is the unique non-trivial embedded member of the associate family of the Schwarz P and D surfaces. Its angle of association with respect to the D surface is approximately 38.01°. The gyroid is similar to the lidinoid. The gyroid was discovered in 1970 by NASA scientist Alan Schoen.
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame.
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