The Amazing Tetrahedron

There are only 5 Platonic solids, and the simplest of the 5 is the 4-cornered (vertices), 4-faced, 4-sided shape known as the Tetrahedron.

So what's the big deal, right? It looks pretty simple, so it must be uninteresting, right?

I thought so, but for such a simple shape, it's actually much more interesting than it looks.

First, a cool explanation of it's name, from Pat Ballew's Math Words::

Tetrahedron A tetrahedron is the most simple of three space shapes since it consists of only four vertices (see figure below). The Greek tetra stands for four, and can still be found in some science words such as tetrachloride or tetravalent. The hedra is from the Greek for base, or seat. 

OK, now that we know what it is, what can we do with it? What properties does it have that one may call: Interesting?

For starters, check out the WolframMathWorld article on Dual Polyhedra, that is to say, if you put a plane on a polygon where a point exists, what new shape do you get, that is to say, what is its dual?

As it turns out, the dual of a cube is an octagon and vice versa, and the dual of an icosahedron is a dodecahedron and vice versa, but the dual of a tetrahedron is ... another tetrahedron!

In short, a tetrahedron is its own dual, as illustrated below:

"The process of forming duals is illustrated above for the Platonic solids. The top row shows the original solids. The middle row shows the vertex figures of the original solid as lines superposed on the tangential polygons forming the corresponding duals. Finally, the polyhedron compounds consisting of a polyhedron and its dual are illustrated in the bottom row." ... WolframMathWorld

The shape in the lower left-hand corner, being the polyhedron compound of a tetrahedron, has its own name: Stella Octangula, which frankly sounds like the name of a science-fiction heroine:

:

What interests me about that shape is it forms six crosses, and if the outermost points are connected, they form a cube.

A tetrahedron has 2 different nets, that is 2 different 2-dimensional shapes that can be folded to obtain a tetrahedron:

Origami tetrahedron, from WolframMathWorld

A tetrahedron is a three dimensional polytope, a polygon. A 4th-dimensional polytope is a polychoron.The 5-cell, or pentachoron, is the 4D analog of the tetrahedron.  Just as the tetrahedron has four triangular faces, the pentachoron has five tetrahedral faces.



Jason Hise developed that. Visit his webpage http://www.entropygames.net/index.php for more cool stuff.

Applications of tetrahedra, from the  Wikipedia article on Tetrahedron:

 

Numerical analysis

In numerical analysis, complicated three-dimensional shapes are commonly broken down into, or approximated by, a polygonal mesh of irregular tetrahedra in the process of setting up the equations for finite element analysis especially in the numerical solution of partial differential equations. These methods have wide applications in practical applications in computational fluid dynamics, aerodynamics, electromagnetic fields, civil engineering, chemical engineering, naval architecture and engineering, and related fields.

 

Chemistry

The tetrahedron shape is seen in nature in covalent bonds of molecules. All sp³-hybridized atoms are surrounded by atoms lying in each corner of a tetrahedron. For instance in a methane molecule (CH₄) or an ammonium ion (NH₄⁺), four hydrogen atoms surround a central carbon or nitrogen atom with tetrahedral symmetry. For this reason, one of the leading journals in organic chemistry is called Tetrahedron. See also tetrahedral molecular geometry. The central angle between any two vertices of a perfect tetrahedron is , or approximately 109.47°).
Water, H₂O, also has a tetrahedral structure, with two hydrogen atoms and two lone pairs of electrons around the central oxygen atoms. Its tetrahedral symmetry is not perfect, however, because the lone pairs repel due to their negative charges.
Quaternary phase diagrams in chemistry are represented graphically as tetrahedra.
However, quaternary phase diagrams in communication engineering are represented graphically on a two-dimensional plane.

 

Electricity and electronics

If six equal resistors are soldered together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.^[8][9]
Since silicon is the most common semiconductor used in solid-state electronics, and silicon has a valence of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on how crystals of silicon form and what shapes they assume.

 

Games

Especially in roleplaying, this solid is known as a 4-sided die, one of the more common polyhedral dice, with the number rolled appearing around the bottom or on the top vertex. Some Rubik's Cube-like puzzles are tetrahedral, such as the Pyraminx and Pyramorphix.

 

Color space

Tetrahedra are used in color space conversion algorithms specifically for cases in which the luminance axis diagonally segments the color space (e.g. RGB, CMY).^[10]

 

Contemporary Art

The Austrian artist Martina Schettina created a tetrahedron using fluorescent lamps. It was shown at the light art biennale Austria 2010.^[11]
It is used as album artwork, surrounded by black flames on The End of All Things to Come by Mudvayne.

 

Geology

The tetrahedral hypothesis, originally published by William Lowthian Green to explain the formation of the Earth,^[12] was popular through the early 20th century.^[13][14]

 

See also

• Boerdijk–Coxeter helix
• Caltrop
• Demihypercube
• Disphenoid — a tetrahedron with mirror symmetry
• Hill tetrahedron
• Simplex
• Tetra-Pak
• Tetrahedral kite
• Tetrahedral number
• Triangular dipyramid — constructed by joining two tetrahedra along one face
• Tetrahedron packing