Rhombic dodecahedron made by a cube and six pyramids
We already know that the Rhombic Dodecahedron is related with honeycombs:
And that you can build a Rhombic Dodecahedron addign six pyramids to a cube:
Cundy and Rollet wrote: "The rhombic dodecahedron and the cube. If a cube is divided by the six diametral planes which pass through pairs of opposite edges, it breaks up into six square pyramids. If these pyramids are assembled outwards on the faces of another cube, the result is a rhombic dodecahedron." (Cundy and Rollet, pag. 122)
The cube is dissected into six congruent square-based pyramids. Four edges of one pyramid are half the space diagonals of the cube.
This construction has several interesting consequences.
First, you can see in the interactive application that this six pyramids fill a cube. The volume of one of this pyramids is one-sixth of the volume of the cube.
Then you do not need the formula for the volume of a pyramid to calculate the volume of this particular pyramid.
By the way, the volume of a pyramid is one third of the base area times the perpendicular height. We can check that these pyramids verify the formula: their base area is 1 and the height is 1/2.
This is a very simple and important example of pyramid because after this example we can deduce the volume of pyramids with rectangular bases (expanding the cube in three directions).
The second consequence of this construction is that we can calculate the volume of a Rhombic Dodecahedron.
Remember that if inside the Rhombic Dodecahedron there is a unit cube, then the length of the edge of this Rhombic Dodecahedron is:
Then the volume of this rhombic dodecahedron is twice the volume of the inside cube:
To calculate the volume if the edge length is 1:
The general formula for the volume of a rhombic dodecahedron of edge length a is:
You can play with the interactive application and see beautiful symmetries like these:
The third consequence is that the Rhombic Dodecahedron fills the space.
We already know that we could build a beautiful box inspired in honeycombs.
Now we can build a rhombic dodecahedron that is also a box.
You can download the template and build a model with cardboard. The designer of this box was John Edminster and you can find references in Beach Packaging Design.
The rhombic dodecahedron is related with the cube. You can build a cube to put inside the rhombic dodecahedron. Inside the cube you can put a cuboctahedron (in this case, a omega star, a beautiful origami construction. Six pieces of paper are needed).
The rhombic dodecahedron is also related with the octahedron (you know that the cube and the octahedron are dual polyhedra). And you can build an octahedron and put it inside the dodecahedron.
Johannes Kepler - The Six Cornered Snowflake: a New Year's gif - Paul Dry Books, Philadelphia, Pennsylvania, 2010. English translation of Kepler's book 'De Nive Sexangula'. With notes by Owen Gingerich and Guillermo Bleichmar and illustrations by the spanish mathematician Capi Corrales Rodrigáñez.
D'Arcy Thompson - On Growth And Form - Cambridge University Press, 1942.
Hugo Steinhaus - Mathematical Snapshots - Oxford University Press - Third Edition.
Magnus Wenninger - 'Polyhedron Models', Cambridge University Press.
Peter R. Cromwell - 'Polyhedra', Cambridge University Press, 1999.
H.Martin Cundy and A.P. Rollet, 'Mathematical Models', Oxford University Press, Second Edition, 1961.
W.W. Rouse Ball and H.S.M. Coxeter - 'Matematical Recreations & Essays', The MacMillan Company, 1947.