differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, a triply periodic minimal surface (TPMS) is a

minimal surface 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 ...

in ℝ³ that is invariant under a rank-3

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an ornam ...

of translations. These surfaces have the symmetries of a

crystallographic group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unc ...

. Numerous examples are known with cubic,

tetragonal In crystallography, the tetragonal crystal system is one of the 7 crystal systems. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square ...

rhombohedral In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombus, rhombi. It is a special case of a parallelepiped where all edges are the same length. It c ...

, and

orthorhombic In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a r ...

symmetries.

Monoclinic In crystallography, the monoclinic crystal system is one of the seven crystal systems. A crystal system is described by three vectors. In the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic s ...

and

triclinic 180px, Triclinic (a ≠ b ≠ c and α ≠ β ≠ γ ) In crystallography, the triclinic (or anorthic) crystal system is one of the 7 crystal systems. A crystal system is described by three basis vectors. In the triclinic system, the crystal is ...

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 In polymer chemistry, a copolymer is a polymer derived from more than one species of monomer. The polymerization of monomers into copolymers is called copolymerization. Copolymers obtained from the copolymerization of two monomer species are some ...

, equipotential surfaces in crystals etc. They have also been of interest in architecture, design and art.

Properties

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 Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

≥ 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.

History

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 Alan Hugh Schoen (born December 11, 1924) is an American physicist and computer scientist best known for his discovery of the gyroid, an infinitely connected triply periodic minimal surface. Professional career Alan Schoen received his B.S. degre ...

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 Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, ge ...

are often used.

Families

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.William H. Meeks, III. The Geometry and the Conformal Structure of Triply Periodic Minimal Surfaces in R3. PhD thesis, University of California, Berkeley, 1975. Members of this family can be continuously deformed into each other, remaining embedded in the process (although the lattice may change). The

gyroid A gyroid is an infinitely connected Triply periodic minimal surface, triply periodic minimal surface discovered by Alan Schoen in 1970. History and properties The gyroid is the unique non-trivial embedded member of the associate family of the ...

and

lidinoid In differential geometry, the lidinoid is a triply periodic minimal surface. The name comes from its Swedish discoverer Sven Lidin (who called it the HG surface). It has many similarities to the gyroid, and just as the gyroid is the unique emb ...

are each inside a separate 1-parameter family. Another approach to classifying TPMS is to examine their space groups. For surfaces containing lines the possible boundary polygons can be enumerated, providing a classification.

Generalisations

Periodic minimal surfaces can be constructed in S³ and H³. It is possible to generalise the division of space into labyrinths to find triply periodic (but possibly branched) minimal surfaces that divide space into more than two sub-volumes.

Quasiperiodic Quasiperiodicity is the property of a system that displays irregular periodicity. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is a pattern of recurrence with a component of unpred ...

minimal surfaces have been constructed in ℝ²×S¹. It has been suggested but not been proven that minimal surfaces with a

quasicrystal A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical cr ...

line order in ℝ³ exist.

External galleries of images

* TPMS at the Minimal Surface Archiv

* Periodic minimal surfaces galler

References

{{Minimal surfaces Differential geometry Minimal surfaces