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.
He calculated the angle of association and gave a convincing demonstration of pictures of intricate plastic models, but did not provide a proof of embeddedness.
Schoen noted that the gyroid contains neither straight lines nor planar symmetries.
Karcher gave a different, more contemporary treatment of the surface in 1989 using conjugate surface construction.
In 1996 Große-Brauckmann and Wohlgemuth proved that it is embedded, and in 1997 Große-Brauckmann provided CMC (constant mean curvature) variants of the gyroid and made further numerical investigations about the volume fractions of the minimal and CMC gyroids.
The gyroid separates space into two oppositely congruent labyrinths of passages. The gyroid has space group I4132 (no. 214). Channels run through the gyroid labyrinths in the (100) and (111) directions; passages emerge at 70.5 degree angles to any given channel as it is traversed, the direction at which they do so gyrating down the channel, giving rise to the name "gyroid". One way to visualize the surface is to picture the "square catenoids" of the P surface (formed by two squares in parallel planes, with a nearly circular waist); rotation about the edges of the square generate the P surface. In the associate family, these square catenoids "open up" (similar to the way the catenoid "opens up" to a helicoid) to form gyrating ribbons, then finally become the Schwarz D surface. For one value of the associate family parameter the gyrating ribbons lie in precisely the locations required to have an embedded surface.
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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.
In differential geometry, the Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz. In the 1880s Schwarz and his student E. R. Neovius described periodic minimal surfaces. They were later named by Alan Schoen in his seminal report that described the gyroid and other triply periodic minimal surfaces. The surfaces were generated using symmetry arguments: given a solution to Plateau's problem for a polygon, reflections of the surface across the boundary lines also produce valid minimal surfaces that can be continuously joined to the original solution.
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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