# Difference between revisions of "Tetrakaidecahedron (Kelvin Cell)"

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== Definition == | == Definition == | ||

− | The '''tetrakaidecahedron''' 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. | + | 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 [1]. |

− | [[Image:Tetrakaidecahedron.png]] | + | |

+ | [[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]] | ||

+ | |||

+ | 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 [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which improved upon the typical flat-faced polyhedron pictured above [2]. | ||

+ | |||

+ | [http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.] | ||

+ | |||

+ | [http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!] | ||

+ | |||

+ | == Applications == | ||

+ | |||

+ | 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 [2]. Architects used the Weaire-Phalen geometry in the design of the Beijing National Aquatics Center, "the water cube" [3]. | ||

+ | |||

+ | Whether or not anyone has discovered the space-filling shape with the absolute minimum surface area is still an open question [2]! | ||

== References == | == References == | ||

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

− | + | ||

− | http://mathworld.wolfram.com/KelvinsConjecture.html | + | [2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource. |

− | http:// | + | |

+ | [3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004). |

## Latest revision as of 23:39, 30 November 2009

## 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 [1].

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 [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which improved upon the typical flat-faced polyhedron pictured above [2].

Click here for other 14-sided poyhedra.

Click here to print a template to make your own tetrakaidecahedron!

## Applications

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 [2]. Architects used the Weaire-Phalen geometry in the design of the Beijing National Aquatics Center, "the water cube" [3].

Whether or not anyone has discovered the space-filling shape with the absolute minimum surface area is still an open question [2]!

## 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).