A gyroid is an infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970. [1] [2] It arises naturally in polymer science and biology, as an interface with high surface area.
A gyroid is an infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970. [1] [2] 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 [3] gave a different, more contemporary treatment of the surface in 1989 using conjugate surface construction. In 1996 Große-Brauckmann and Wohlgemuth [4] 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). [5] 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.
The gyroid refers to the member that is in the associate family of the Schwarz P surface, but in fact the gyroid exists in several families that preserve various symmetries of the surface; a more complete discussion of families of these minimal surfaces appears in triply periodic minimal surfaces.
Curiously, like some other triply periodic minimal surfaces, the gyroid surface can be trigonometrically approximated by a short equation:
The gyroid structure is closely related to the K4 crystal (Laves' graph of girth ten). [6]
In nature, self-assembled gyroid structures are found in certain surfactant or lipid mesophases [7] and block copolymers. In a typical A-B diblock copolymer phase diagram, the gyroid phase can be formed at intermediate volume fractions between the lamellar and cylindrical phases. In A-B-C block copolymers, the double and alternating-gyroid phases can be formed. [8] Such self-assembled polymer structures have found applications in experimental supercapacitors, [9] solar cells [10] photocatalysts, [11] and nanoporous membranes. [12] Gyroid membrane structures are occasionally found inside cells. [13] Gyroid structures have photonic band gaps that make them potential photonic crystals. [14] Single gyroid photonic crystals have been observed in biological structural coloration such as butterfly wing scales and bird feathers, inspiring work on biomimetic materials. [15] [16] [17] The gyroid mitochondrial membranes found in the retinal cone cells of certain tree shrew species present a unique structure which may have an optical function. [18]
In 2017, MIT researchers studied the possibility of using the gyroid shape to turn bi-dimensional materials, such as graphene, into a three-dimensional structural material with low density, yet high tensile strength. [19]
Gyroid shows potential as electrodes in hydrogen production.
Researchers from Cambridge University have shown the controlled chemical vapor deposition of sub–60 nm graphene gyroids. These interwoven structures are one of the smallest free-standing graphene 3D structures. They are conductive, mechanically stable, and easily transferable, and are of interest for a wide range of applications. [20]
The gyroid pattern has also found use in 3D printing for lightweight internal structures, due to its high strength, combined with speed and ease of printing using an FDM 3D printer. [21] [22]
In an in silico study, researchers from the university hospital Charité in Berlin investigated the potential of gyroid architecture when used as a scaffold in a large bone defect in a rat femur. When comparing the regenerated bone within a gyroid scaffold compared to a traditional strut-like scaffold, they found that gyroid scaffolds led to less bone formation and attributed this reduced bone formation to the gyroid architecture hindering cell penetration. [23]
In mathematics, physics, and art, moiré patterns or moiré fringes are large-scale interference patterns that can be produced when a partially opaque ruled pattern with transparent gaps is overlaid on another similar pattern. For the moiré interference pattern to appear, the two patterns must not be completely identical, but rather displaced, rotated, or have slightly different pitch.
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis. It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally described in 1744 by the mathematician Leonhard Euler.
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature.
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Graphene is an allotrope of carbon consisting of a single layer of atoms arranged in a honeycomb nanostructure. The name is derived from "graphite" and the suffix -ene, reflecting the fact that the graphite allotrope of carbon contains numerous double bonds in a two-dimensional sheet.
A Euclidean graph is periodic if there exists a basis of that Euclidean space whose corresponding translations induce symmetries of that graph. Equivalently, a periodic Euclidean graph is a periodic realization of an abelian covering graph over a finite graph. A Euclidean graph is uniformly discrete if there is a minimal distance between any two vertices. Periodic graphs are closely related to tessellations of space and the geometry of their symmetry groups, hence to geometric group theory, as well as to discrete geometry and the theory of polytopes, and similar areas.
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In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in that is invariant under a rank-3 lattice of translations.
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature. This includes minimal surfaces as a subset, but typically they are treated as special case.
In differential geometry, the Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz.
In differential geometry, the lidinoid is a triply periodic minimal surface. The name comes from its Swedish discoverer Sven Lidin.
In differential geometry, the Neovius surface is a triply periodic minimal surface originally discovered by Finnish mathematician Edvard Rudolf Neovius.
In geometry and crystallography, the Laves graph is an infinite and highly symmetric system of points and line segments in three-dimensional Euclidean space, forming a periodic graph. Three equal-length segments meet at 120° angles at each point, and all cycles use ten or more segments. It is the shortest possible triply periodic graph, relative to the volume of its fundamental domain. One arrangement of the Laves graph uses one out of every eight of the points in the integer lattice as its points, and connects all pairs of these points that are nearest neighbors, at distance . It can also be defined, divorced from its geometry, as an abstract undirected graph, a covering graph of the complete graph on four vertices.
A nanophotonic resonator or nanocavity is an optical cavity which is on the order of tens to hundreds of nanometers in size. Optical cavities are a major component of all lasers, they are responsible for providing amplification of a light source via positive feedback, a process known as amplified spontaneous emission or ASE. Nanophotonic resonators offer inherently higher light energy confinement than ordinary cavities, which means stronger light-material interactions, and therefore lower lasing threshold provided the quality factor of the resonator is high. Nanophotonic resonators can be made with photonic crystals, silicon, diamond, or metals such as gold.
In physics, Dirac cones are features that occur in some electronic band structures that describe unusual electron transport properties of materials like graphene and topological insulators. In these materials, at energies near the Fermi level, the valence band and conduction band take the shape of the upper and lower halves of a conical surface, meeting at what are called Dirac points.
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Sternotomis callais, the African longhorn beetle, is a species of beetle in the family Cerambycidae. It was described by Léon Fairmaire in 1891. It is known from Cameroon, the Republic of the Congo, the Democratic Republic of the Congo, Equatorial Guinea, and Angola. It feeds on Coffea liberica var. dewevrei.
A graphene morphology is any of the structures related to, and formed from, single sheets of graphene. 'Graphene' is typically used to refer to the crystalline monolayer of the naturally occurring material graphite. Due to quantum confinement of electrons within the material at these low dimensions, small differences in graphene morphology can greatly impact the physical and chemical properties of these materials. Commonly studied graphene morphologies include the monolayer sheets, bilayer sheets, graphene nanoribbons and other 3D structures formed from stacking of the monolayer sheets.
Alan Hugh Schoen was an American physicist and computer scientist best known for his discovery of the gyroid, an infinitely connected triply periodic minimal surface.
Magalí Lingenfelder is an Argentinian chemist who is head of the Max Planck Laboratory for Molecular Nanoscience in École Polytechnique Fédérale de Lausanne. Her work looks to control atomic interfaces for energy conversion and antimicrobial surfaces. She was awarded the Max Planck Society Otto Hahn Medal in 2008.
