matematicas visuales visual math

We know that that the rhombic dodecahedron is a polyhedron related with how bees build honeycombs. The bottom of a cell (the keel as Kepler called it) is made by three rhombuses. And Kepler knew that with twelve of these rhombi we can build a polyhedron called rhombic dodecahedron.

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?.

In this page we are going to take another point of view.

One way to build new polyhedra is start with one known polyhedra and add pyramids in each of their faces.

This is a very well known technique. For example, Luca Pacioli studied several of these polyhedra in his book "De Divine Proportione" (around 1500).

Pacioli wrote about the "pyramidated cube" and he called it "exacedron elevatus" (Spanish translation):

"[El hexaedro elevado sólido o hueco] está compuesto por seis pirámides laterales cuadriláteras exteriores, que se ofrecen a nuestra vista según la situación del cuerpo. En cuanto al cubo interior sobre el que se apoyan dichas pirámides, y que sólo puede ser imaginado por el intelecto pues se oculta a nuestra vista por la superposición de dichas pirámides, sus seis superficies cuadradas son bases de las mencionadas seis pirámides, que son todas de la misma altura, y se ocultan a la vista y circundan a dicho cubo."
('La divina proporción' de Luca Pacioli, page 92, Spanish translation by Juan Calatrava, Editorial Akal, 4th edition, 2008)

These drawings were made by Leonardo da Vinci:

Leonardo da Vinci: pyramidated cube. Editorial Akal | matematicasvisuales
Leonardo da Vinci: pyramidated cube. Editorial Akal | matematicasvisuales

Playing with transprency we can see the interior cube:

Pyramidated cube and Rhombic Dodecahedron: transparency, you can see the interior cube | matematicasvisuales

You can separate the pyramids:

Pyramidated cube and Rhombic Dodecahedron: you can separate pyramids | matematicasvisuales

One interesting fact is that you can make that these twenty four triangular faces become twelve rhombuses and then you get a new polyhedron called Rhombic Docecahedron:

Pyramidated cube and Rhombic Dodecahedron: only twelve faces that are rhombuses | matematicasvisuales

Pyramidated cube and Rhombic Dodecahedron: only twelve faces that are rhombuses | matematicasvisuales

We are going to calculate the diagonal D of one of these rhombuses:

Pyramidated cube and Rhombic Dodecahedron: calculating the diagonal of one rhombi | matematicasvisuales
Pyramidated cube and Rhombic Dodecahedron: calculating the diagonal of one rhombi | matematicasvisuales
Pyramidated cube and Rhombic Dodecahedron: calculating the diagonal of one rhombi | matematicasvisuales

These are the same rhombuses that we found studying honeycombs.

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?.


The length of the edge of this rhombic dodecahedron is:

Pyramidated cube and Rhombic Dodecahedron: the length of the side of a Rhombic Dodecahedron | matematicasVisuales

These rhombuses are related with the proportions of the standard size of paper DinA.

Standar 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.


You can use trigonometry to calculate the angles of one rhombi:

Pyramidated cube and Rhombic Dodecahedron: calculating angles using trigonometry | matematicasVisuales
Pyramidated cube and Rhombic Dodecahedron: calculating angles using trigonometry | matematicasVisuales

Johannes Kepler was the first mathematician to write about the rhombic dodecahedron. For example, this is a drawing in his book "Harmonices Mundi":

Pyramidated cube and Rhombic Dodecahedron: Kepler in his book Harmonices Mundi | matematicasvisuales

Some garnet cristals have this shape:

Pyramidated cube and Rhombic Dodecahedron: Garnet cristals with the shape of a rhombic dodecahedron | 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 (5): Rhombic Dodecahedron is a space filling polyhedron
The Rhombic Dodecahedron fills the space without gaps.
Construcción de poliedros. Cuboctaedro y dodecaedro rómbico: Taller de Talento Matemático de Zaragoza 2014 (Spanish)
Material for a sesion 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.
Standar 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).