Tetrakaidecahedron (Kelvin Cell)
The tetrakaidecahedron (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices .
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of the bubbles which make up a foam . Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which improved upon the typical flat-faced polyhedron pictured above .
Scientists still use packed tetrakaidecahedra as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use tetrakaidecahedra in their paper Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation.
The tetrakaidecahedron remained the best contender for a minimal surface area, space-filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example of a space-filling geometry with even less surface area in a Philosophical Magazine Letters article . Architects used the Weaire-Phalen geometry in the design of the Beijing National Aquatics Center, "the water cube" .
Whether or not anyone has discovered the space-filling shape with the absolute minimum surface area is still an open problem !
 Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series 158 (2009).
 Weisstein, Eric. "Kelvin's Conjecture." From MathWorld--A Wolfram Web Resource.
 Rogers, Peter. "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool," The Guardian (5 May 2004).