![Gyroid unit cell](https://upload.wikimedia.org/wikipedia/commons/b/b3/Gyroid_unit_cell.stl)
A gyroid is an infinitely connected
triply periodic 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 ...
discovered by
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 ...
in 1970.
History and properties
The gyroid is the unique non-trivial embedded member of the
associate family
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
:x_k(\zeta) = \Re \left\ ...
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
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 ...
. The gyroid was discovered in 1970 by NASA scientist
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 ...
. 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
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature.Carl Johan Lejdfors, Surfaces of Constant Mean Curvature. Master’s thesis Lund University, Centre for Mathematical Sciences Mathematics 2 ...
) 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
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 unchan ...
''I4
132'' (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.
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 surface
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 know ...
s.
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).
Applications
![TiO2 alternating gyroid and Ta2O5 double gyroid](https://upload.wikimedia.org/wikipedia/commons/7/71/TiO2_alternating_gyroid_and_Ta2O5_double_gyroid.png)
In nature, self-assembled gyroid structures are found in certain surfactant or lipid
mesophase In chemistry and chemical physics, a mesophase is a state of matter intermediate between liquid and solid. Gelatin is a common example of a partially ordered structure in a mesophase. Further, biological structures such as the lipid bilayers of cel ...
s and block
copolymer
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 ...
s. 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. Such self-assembled polymer structures have found applications in experimental
supercapacitors
A supercapacitor (SC), also called an ultracapacitor, is a high-capacity capacitor, with a capacitance value much higher than other capacitors but with lower voltage limits. It bridges the gap between electrolytic capacitors and Rechargeable ba ...
, solar cells photocatalysts, and nanoporous membranes.
Gyroid membrane structures are occasionally found inside cells.
Gyroid structures have photonic
band gap
In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (in ...
s that make them potential
photonic crystals
A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of natural crystals gives rise to X-ray diffraction and that the atomi ...
. Single gyroid photonic crystals have been observed in biological
structural coloration
Structural coloration in animals, and a few plants, is the production of colour by microscopically structured surfaces fine enough to interfere with visible light instead of pigments, although some structural coloration occurs in combination wit ...
such as butterfly wing scales and bird feathers, inspiring work on
biomimetic materials Biomimetic materials are materials developed using inspiration from nature. This may be useful in the design of composite materials. Natural structures have inspired and innovated human creations. Notable examples of these natural structures includ ...
. The gyroid mitochondrial membranes found in the retinal
cone cell
Cone cells, or cones, are photoreceptor cells in the retinas of vertebrate eyes including the human eye. They respond differently to light of different wavelengths, and the combination of their responses is responsible for color vision. Cone ...
s of certain
tree shrew
The treeshrews (or tree shrews or banxrings) are small mammals native to the tropical forests of South and Southeast Asia. They make up the entire order Scandentia, which split into two families: the Tupaiidae (19 species, "ordinary" treeshrews) ...
species present a unique structure which may have an optical function.
In 2017, MIT researchers studied the possibility of using the gyroid shape to turn bi-dimensional materials, such as
graphene
Graphene () is an allotrope of carbon consisting of a single layer of atoms arranged in a hexagonal lattice nanostructure. , into a three-dimensional structural material with low density, yet high
tensile strength
Ultimate tensile strength (UTS), often shortened to tensile strength (TS), ultimate strength, or F_\text within equations, is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle materials t ...
.
Researchers from
Cambridge University
, mottoeng = Literal: From here, light and sacred draughts.
Non literal: From this place, we gain enlightenment and precious knowledge.
, established =
, other_name = The Chancellor, Masters and Schola ...
have shown the controlled
chemical vapor deposition
Chemical vapor deposition (CVD) is a vacuum deposition method used to produce high quality, and high-performance, solid materials. The process is often used in the semiconductor industry to produce thin films.
In typical CVD, the wafer (substra ...
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.
The gyroid pattern has also found use in
3D printing
3D printing or additive manufacturing is the Manufacturing, construction of a three-dimensional object from a computer-aided design, CAD model or a digital 3D modeling, 3D model. It can be done in a variety of processes in which material is ...
for lightweight internal structures, due to its high strength, combined with speed and ease of printing using an
FDM 3D printer.
In an
in silico
In biology and other experimental sciences, an ''in silico'' experiment is one performed on computer or via computer simulation. The phrase is pseudo-Latin for 'in silicon' (correct la, in silicio), referring to silicon in computer chips. It ...
study, researchers from the university hospital
Charité
The Charité – Universitätsmedizin Berlin (Charité – Berlin University of Medicine) is one of Europe's largest university hospitals, affiliated with Humboldt University and Free University Berlin. With numerous Collaborative Research Cen ...
in
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
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.
References
External links
Triply Periodic Minimal Surfaces at schoengeometry.comat
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...
Rotatable picture of a gyroid's periodElectrochemical Nanofabrication: Principles and Applications
{{Minimal surfaces
Minimal surfaces