# Tetrakaidecahedron (Kelvin Cell)

## Definition

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 also thought about foam in his discussion of this problem. Kelvin's proposed tetrakaidecahedron actually had curved faces in constrast with the typical flat-faced polyhedron pictured above.

You can find other 14-sided poyhedra here: http://mathworld.wolfram.com/Tetradecahedron.html

And make your own here: http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells

## Applications

The tetrakaidecahedron is used to model foams. Koehler, Hilgenfeldt, and Stone use the tetrakaidecahedron in their paper Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation.

The tetrakaidecahedron filled space with the least amount of surface area from 1887 to 1994. In 1994, Weaire and Phalen presented a space-filling geometry with even less surface area in a Philosophical Magazine Letters article. The Weaire-Phalen geometry was used as the structure for the Beijing Aquatic Center "the water cube." [3]

## References

[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series **158** (2009).

[2] Weisstein, Eric. "Kelvin's Conjecture." From MathWorld--A Wolfram Web Resource.

[3] Rogers, Peter. "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool," The Guardian (5 May 2004).