matematicas visuales visual math

We already know that a rhombic dodecahedron could be made by a cube and six pyramids. And we had studied several consequences of that fact.

Rhombic Dodecahedron (4): Rhombic Dodecahedron made of a cube and six sixth of a cube
You can build a Rhombic Dodecahedron adding six pyramids to a cube. This fact has several interesting consequences.

In this page we are going to see another very interesting property: Rombic dodecahedra fills or tessellates space, with no gaps between.

Hugo Steinhaus wrote: "We have already mentioned tilling the whole of space with cubes. We can get another filling of this sort by the following procedure: we make the cubes alternately black and white to get a kind of chessboard and then we remove the black ones. We decompose each void space into 6 pyramids on square bases with a common vertex in the certer of the void space. If we consider a single white cube with 6 pyramids based on it, we see a rhombi dodecahedron with a cube inscribed in it. It is obvious that by our procedure we have filled the whole of space with congruent rhombic dodecahedra.(...) The vertices are of two kinds: (1) where 4 solids meet; (2) where 6 solids meet." (Steinhaus, pag. 185)

Rhombic dodecahedron is a space-filling polyhedron, tessellation | matematicasvisuales
Rhombic dodecahedron is a space-filling polyhedron, tessellation | matematicasvisuales
Rhombic dodecahedron is a space-filling polyhedron, tessellation | matematicasvisuales
Rhombic dodecahedron is a space-filling polyhedron, tessellation | matematicasvisuales
Rhombic dodecahedron is a space-filling polyhedron, tessellation | matematicasvisuales
Rhombic dodecahedron is a space-filling polyhedron, tessellation | matematicasvisuales

Another interesting example of a space-filling polyhedron is the truncated octahedron.

The volume of a truncated octahedron
The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces and 6 square faces. Its volume can be calculated knowing the volume of an octahedron.
The truncated octahedron is a space-filling polyhedron
These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.

Zome is a perfect tool to build this tessellation: rhombic dodecahedra in yellow, cubes in blue and octahedra in green:

Rhombic dodecahedron is a space-filling polyhedron, tessellation, construcción con Zome | matematicasvisuales
Rhombic dodecahedron is a space-filling polyhedron, tessellation, construcción con Zome | matematicasvisuales
Rhombic dodecahedron is a space-filling polyhedron, tessellation, construcción con Zome | matematicasvisuales

You can play with another version of this interactive application to see that the rhombic dodecahedron is a space-filling polyhedra.

Rhombic dodecahedron is a space-filling polyhedron, tessellation | matematicasvisuales
Rhombic dodecahedron is a space-filling polyhedron, tessellation | matematicasvisuales
Rhombic dodecahedron is a space-filling polyhedron, tessellation | matematicasvisuales
Rhombic dodecahedron is a space-filling polyhedron, tessellation | matematicasvisuales
Rhombic dodecahedron is a space-filling polyhedron, tessellation | matematicasvisuales
Rhombic dodecahedron is a space-filling polyhedron, tessellation | matematicasvisuales

REFERENCES

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.

MORE LINKS

Rhombic Dodecahedron (1): honeycombs
Humankind has always been fascinated by how bees build their honeycombs. Kepler related honeycombs with a polyhedron called Rhombic Dodecahedron.
Rhombic Dodecahedron (2): honeycomb minima property
We want to close a hexagonal prism as bees do, using three rhombi. Then, which is the shape of these three rhombi that closes the prism with the minimum surface area?.
Rhombic Dodecahedron (3): Pyramidated cube
Adding six pyramids to a cube you can build new polyhedra with twenty four triangular faces. For specific pyramids you get a Rhombic Dodecahedron that has twelve rhombic faces.
Construcción de poliedros. Cuboctaedro y dodecaedro rómbico: Taller de Talento Matemático de Zaragoza 2014 (Spanish)
Material for a session about polyhedra (Zaragoza, 9th May 2014). Simple techniques to build polyhedra like the tetrahedron, octahedron, the cuboctahedron and the rhombic dodecahedron. We can build a box that is a rhombic dodecahedron.
Leonardo da Vinci: Drawing of a truncated octahedron made to Luca Pacioli's De divina proportione.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the truncated octahedron.
Leonardo da Vinci: Drawing of a cuboctahedron made to Luca Pacioli's De divina proportione.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the cuboctahedron.
Standard Paper Size DIN A
There is a standarization of the size of the paper that is called DIN A. Successive paper sizes in the series A1, A2, A3, A4, and so forth, are defined by halving the preceding paper size along the larger dimension.
The volume of a cuboctahedron
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of a cube.
The volume of a cuboctahedron (II)
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an octahedron.
Volume of an octahedron
The volume of an octahedron is four times the volume of a tetrahedron. It is easy to calculate and then we can get the volume of a tetrahedron.
The volume of a truncated octahedron
The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces and 6 square faces. Its volume can be calculated knowing the volume of an octahedron.
The truncated octahedron is a space-filling polyhedron
These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.
Chamfered Cube
You can chamfer a cube and then you get a polyhedron similar (but not equal) to a truncated octahedron. You can get also a rhombic dodecahedron.
The Dodecahedron and the Cube
A Cube can be inscribed in a Dodecahedron. A Dodecahedron can be seen as a cube with six 'roofs'. You can fold a dodecahedron into a cube.
Pyritohedron
If you fold the six roofs of a regular dodecahedron into a cube there is an empty space. This space can be filled with an irregular dodecahedron composed of identical irregular pentagons (a kind of pyritohedron).