In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having a mean curvature of zero. 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. However the term is used for more general surfaces that may self-intersect or do not have constraints.
Gyroid & Geometry of interfaces: topological complexity in biology and materials: A gyroid is an infinitely connected triply periodicminimal surface discovered by Alan Schoen in 1970. 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”. Curiously, like some other triply periodic minimal surfaces, the gyroid surface can be trigonometrically approximated by the equation: sinX.cosY + sinY.cosZ + sinZ.cosX = 0
Applications: 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: biomimetic materials.
Article - Geometry of interfaces: topological complexity in biology and materials.
Figure 3: Topologically complex porous chitin structure in the wing scales of the butterfly Callophrys Rubi. Light microscopy of the ventral (upper) side of the wings illustrates the arrangement of individual scales on the wings (each several hundreds of micrometres in length). Electron microscopy reveals the structure of the wing-scales to consist of two layers of parallel ribs that cover a poly-crystalline porous chitin matrix structured according to a single srs net of symmetry I4132, commensurable with the lattice parameter of biological gyroid membranes.
The recognition of the chitin matrix as a consolidated three-dimensional cubic pattern rather than a two-dimensional film lining the gyroid is important. In particular, the spectacular optical features of these wing scales, certainly due in part to the cubic structure of the chitin, are dependent on this structural feature. Also, the porosity of the chitin matrices in the wing scales of this species is tuned to optimize optical effects in vivo, possibly an impressive example of the efficacy of natural selection, rather than simply a coincidental by-product of the self-assembly process.
Figure 6: Various views of the gyroid minimal surface. Within each of the two network domains bounded by the gyroid, there are two chiral elements of opposite rotation sense: threefold screw axes along the  and, of opposite rotation sense, fourfold screw axes along the  axes. The centred skeletal graph is the srs network with three-valent vertices.
The link between math and architecture goes back to ancient times, when the two disciplines were virtually indistinguishable. Pyramids and temples were some of the earliest examples of mathematical principles at work. Today, math continues to feature prominently in building design. We’re not just talking about mere measurements — though elements like that are integral to architecture. Thanks to modern technology, architects can explore a variety of exciting design options based on complex mathematical languages, allowing them to build groundbreaking forms. Take a look at several structures past the break that were modelled after mathematics.
1. Mobius Strip Temple, China 2. Tetrahedral-Shaped Church, Colorado 3. Pentagonal, Phyllotactic Greenhouse and Education Center. The Eden Project, Cornwall 4. A Mathematically-Inclined Cucumber in the Sky, London 5. Experimental Math-Music Pavilion, 1958 World’s Fair 6. Modern Music-Math Home, Toronto 7. Solar Algorithm Wizardry, Barcelona 8. Cube Village, Holland 9. Magic Square Cathedral, Gaudi Cathederal, Barcelona 10. Fractal Gas Station Makeover, Los Angeles
Your calls for the United States to get involved in this crisis undermines the democratic process in Nigeria and co-opts the growing movement against the inept and kleptocratic Jonathan administration. It was Nigerians who took their good for nothing President to task and challenged him to address the plight of the missing girls. It is in their hands to seek justice for these girls and to ensure that the Nigerian government is held accountable. Your emphasis on U.S. action does more harm to the people you are supposedly trying to help and it only expands and sustain U.S. military might. If you must do something, learn more about the amazing activists and journalists…who have risked arrests and their lives as they challenge the Nigerian government to do better for its people within the democratic process. If you must tweet, tweet to support and embolden them, don’t direct your calls to action to the United States government who seeks to only embolden American militarism. Don’t join the American government and military in co-opting this movement started and sustained by Nigerians.