The word “octahedron” is derived from the the two Greek words “octo”, which means eight, and “hedra”, which means area, or surface. That leads directly to the specified object: a geometrical body consisting of eight conjoined areas.
It shows an enormously high symmetry, since it consists of eight 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 octahedron is for example visible by the fact, that it remains always exactly the same body, independent of the area on which it is positioned. Furthermore it has point symmetry.
Looking closer to the octahedron reveals the following, geometrical characteristics – it has 6 vertices, 12 edges, and as well 4 edges as 4 areas joined to a vertex.
The following parameters are effective:
- the height, when positioned on a vertex, comes up to the square root of 2, multiplied with the length of an edge (resulting in about 141,4 % of the length of an edge). This is twice the height of the tetrahedron, when positioned on an edge.
- 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). This is the height of the tetrahedron, when positioned on an area.
- the surface area comes up to twice the square root of 3, multiplied with the square of the length of an edge. This is twice the surface of the tetrahedron.
- the volume comes up to one-third of the square root of 2, multiplied with the cube of the length of an edge. This is four times the volume of the tetrahedron.
- the angle between two areas at a vertex is about 70,5°. The tetrahedron has the same angle.
- the angle between two areas at an edge is about 109,5°. That is twice the value of the second tetrahedron angle.
- the angle between two edges is the right angle. This implies a relation to the square, and thus to the cube (hexahedron).
The one who wants to deal more profoundly with the geometrical characteristics of an octahedron 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 octahedron his dual partner is the cube (hexahedron). A short plausibility check gives rise to that relation: it has eight areas, so that the individual center points will result in eight vertices, which is only the case for the hexahedron within the Platonic bodies.
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 octahedron – is it feasible? One possibility of determination is the construction of a octahedron of twice the size, or duplicated length of an edge, out of octahedrons of original size.
First step: an octahedron.
Next step: two octahedrons.
Next step: four octahedrons – looks good, since the resulting form in the middle of the small octahedrons looks like a vertex of an octahedron and is again based on equilateral triangles.
Last step: another octahedron is placed on the top and on the bottom. 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 bodies arose in the middle of the six octahedrons?
Having a patient look in the leading area, in the middle of the last picture it becomes visible – it is a tetrahedron. That reveals the same relation as the tesselation of the tetrahedron. Counting now carefully will give the following result: for the construction of an octahedron of twice the length of an edge six octahedrons and eight tetrahedrons of original size are required. In this way the consistent space filling becomes feasible, however not with the octahedron alone…