History and Properties 
The gyroid is the unique non-trivial embedded member of the associate family of the Schwarz P and D surfaces with angle of association approximately 38.01°. The gyroid is similar to the Lidinoid. The gyroid was discovered in 1970 by Alan Schoen, then a scientist at NASA. He calculated the angle of association in his NASA Technical Report and gave a convincing demonstration but did not provide a proof of embeddedness (although he did provide pictures of intricate plastic models). Schoen notes that the gyroid contains neither straight lines nor planar symmetries. Karcher  gave a different, more contemporary treatment of the surface in 1989 using the conjugate surface construction. In 1996 Große-Brauckmann and Wohlgemuth  proved that it is embedded, and in 1997 Große-Brauckmann provided CMC 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 identical labyrinths of passages. The gyroid has space group Ia3d. 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 visual the surface is to picture the “square catenoids” of the P surface (these are 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 which preserve various symmetries of the surface; a more complete discussion of families of these minimal surfaces appears in the entry on triply periodic minimal surfaces.
Curiously, like some other triply periodic minimal surfaces, the gyroid surface can be trigonometrically approximated by the equation:
In nature, self assembled gyroid structures are found in certain surfactant or lipid mesophases and block copolymers. In the polymer phase diagram, the gyroid phase is between the lamellar and cylindrical phases. Such self-assembled polymer structures have found applications in experimental supercapacitors, solar cells and nanoporous membranes.
Gyroid membrane structures are occasionally found inside cells.
Gyroid structures have photonic band gaps that make them potential photonic crystals. Gyroid structures have been observed in biological structural coloration such as butterfly wing scales, inspiring work on biomimetic materials. The gyroid mitochondrial membranes in tree shrew cones might have an optical function.
- Alan H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970).
- David Hoffman, Computing Minimal Surfaces. In Global Theory of Minimal Surfaces: Proceedings of the Clay Mathematics Institute 2001 Summer School, Mathematical Sciences Research Institute, Berkeley, California, June 25-July 27, 2001 American Mathematical Society 2005.
- Hermann Karcher's The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions, Manuscripta Math. 64, 291-357 (1989)
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- Saranathan V, Osuji CO, Mochrie SG, Noh H, Narayanan S, Sandy A, Dufresne ER, Prum RO. Structure, function, and self-assembly of single network gyroid (I4132) photonic crystals in butterfly wing scales. Proc Natl Acad Sci U S A. 2010 Jun 29;107(26):11676-81.
- Michielsen K, Stavenga DG. Jun 14. J R Soc Interface. 2008 Jan 6;5(18):85-94. Gyroid cuticular structures in butterfly wing scales: biological photonic crystals.
- Zakaria Almsherqi, Felix Margadant and Yuru Deng. A look through ‘lens’ cubic mitochondria. Interface Focus (2012) 2, 539–545