Tetrahedron

The word “tetrahedron” is derived from the the two Greek words “tetra”, which means four, and “hedra”, which means area, or surface. That leads directly to the specified object: a geometrical body consisting of four conjoined areas.

It shows an enormously high symmetry, since it consists of four equal areas, each corresponding to an equilateral triangle. That triangle himself is the one of the highest possible symmetry (3 symmetry axis, point symmetry, axially symmetrical to multiples of 60°).

The high symmetry of the tetrahedron is for example visible by the fact, that it remains always exactly the same body, independent of the area on which it is positioned. But there is one aspect, where this body differs from the other platonic bodies, since it has no point symmetry as the others.

Looking closer to the tetrahedron reveals the following, geometrical characteristics – it has 4 vertices, 6 edges, and as well 3 edges as 3 areas joined to a vertex.

The following parameters are effective:

  • the height, when positioned on an area, comes up to the square root of 2/3, multiplied with the length of an edge (resulting in about 81,6 % of the length of an edge)
  • the height relative to an edge comes up to the reciprocal of the square root of 2, multiplied with the length of an edge (resulting in about 70,7 % of the length of an edge)
  • the surface area comes up to the square root of 3, multiplied with the square of the length of an edge
  • the volume comes up to the twelfth of the square root of 2, multiplied with the cube of the length of an edge
  • the angle between two areas is about 70,5°
  • the angle between an area and an edge is about 54,7°

The one who wants to deal more profoundly with the geometrical characteristics of a tetrahedron will find a lot of material quite suitable for self studies on Internet. There can be found a lot on topics as in-circle, edge-circle, circumcircle, symmetry axis, symmetry areas, measures, and so on.

But the direct experimentation on the object itself is more valuable in my point of view, to really grasp and conceive in detail what you are investigating – whatever the concrete manner for that may be. The resulting perceptions are not so formal and theoretical, more apparent, and may be directly experienced in a playful examination of the object.

So you have for example always a body that is dual to a Platonic body, resulting again in a Platonic body. Through a geometrical transformation a Platonic body is converted into another one, thus revealing a kind of relatedness among them. In case of the tetrahedron his dual partner is himself, since the transformation results in another tetrahedron (of half of the size). So the tetrahedron is dual to himself. A short plausibility check gives rise to that relation: it has four areas, so that the individual center points will result in four vertices, which is only the case for the tetrahedron within the Platonic bodies.

Practical Example

Another way of experimenting with this shape is the tesselation, i.e. the identification of a way how to consistently fill space with a Platonic body. How does it look like for the tetrahedron – is it feasible? One possibility of determination is the construction of a tetrahedron of twice the size, or duplicated length of an edge, out of tetrahedrons of original size.

First step: a tetrahedron.

Next step: two tetrahedrons.

Next step: three tetrahedrons – looks good, since the resulting ground area between the small tetrahedrons is again an equilateral triangle.

Last step: a fourth tetrahedron that is placed on the top. The new shape now is clearly a tetrahedron again, and with twice the length of the edge of the original one. But what kind of body arose in the middle of the four tetrahedrons?

Looking more precisely in that region reveals an area in shape of a square, what is not fitting to the tetrahedron. But it can neither be a cube (hexahedron), since the outer surface areas are still equilateral triangles. So, only the octahedron is remaining. That means, for the construction of a tetrahedron of twice the length of an edge four tetrahedrons and one octahedron of original size are required. In this way the consistent space filling becomes feasible, however not with the tetrahedron alone…

Arrangement systems…

Mathematics, and solid-state physics, are dealing for quite some time with the topic of close-packing of equal spheres in space, to get a better understanding of minerals and crystallization.

Thereby the hexagonal close-packing (hcp) was already proven to be the most dense solution in 1831 by Carl Friedrich Gauß.

Having a deeper look at this arrangement discloses exactly the same structure as deduced above from the tesselation. The basic element of this arrangement is the tetrahedron, where the vertices are represented by the centers of the spheres.

This arrangement can be found for example at hydrogen that is cooled down below 14,02 K, respectively -259,2 °C. Hydrogen transitions then from a liquid state to a solid state and takes a crystalline structure, where the hydrogen atoms arrange themselves in a hexagonal close-packing.

The atomic nucleus of helium consists of 4 nucleons (2 protons and 2 neutrons) that are hold together by the strong interaction. That force does not differentiate between protons and neutrons and interconnects the nucleons to the most tight arrangement possible, thus resulting in the shape of a tetrahedron again.

Methane is a native gas and essential part of natural gas. His molecular formula is CH4, so it consists of one carbon atom and four hydrogen atoms. The hydrogen atoms of that molecule are equally distributed around the carbon atom, thus taking exactly the vertex positions of a tetrahedron.

Calcium fluoride is a natural mineral, better known under the name fluorite, respectively fluorspar. The elements calcium and fluorine, that are hold together by ionic bond, arrange themselves in a regular crystal lattice. In that joining the element calcium takes exactly the positions that are corresponding to the vertices of a tetrahedron.

Hereon related hexagonal structures can be found at diverging locations, as for example in honeycombs, basalt prisms and columns in Iceland or Romania, in the atmospheric build of water crystals (snowflakes), at the formation of cloud cluster, as well as in the upper atmosphere at the North Pole of Saturn.

A stable structure, that can be rediscovered on many planes…