Geometry 1 : "Gyroid"
A minimal surface is a mathematical surface that locally minimizes its area, such as helicoid, catenoid, and soap bubbles. A Gyroid is an infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970. An equations which describe a Gyroid is following:
sin (x) cos (y) + sin (y) cos (z) + sin (z) cos (x) = 0
This type of a structure can be found in wings of butterfly and bones of an echinoderm, which means Gyroid satisfies required stiffness and lightness in nature. Biomimetic design is a powerful way to apply principles in nature to design artificial objects. 3D printed Gyroid is one of new examples of biomimetic fabrication.
Geometry 2 : "Diamond"
Diamond structure is the crystal structure found in single crystal made by covalent bond. There are tin, silicon, cubic germanium besides diamonds. It contains 8 atoms in the unit lattice, and one atom has four tetrahedrons in the other four atoms.
This structure was elucidated by William Henry Bragg who is British physicist and William Lawrence Bragg in 1918.
In addition, the virtual crystal structure called "K4 lattice" which is the twin of the crystal structure of diamond is also famous for being closely related to the Gyroid structure.