matematicas visuales visual math

A rhombic dodecahedron could be made by a cube and six pyramids. Then it is easy to calculate the volumen of this polyhedron.

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.

And we know that the rhombic dodecahedron is a space-filling polyhedron.

Rhombic Dodecahedron (5): Rhombic Dodecahedron is a space filling polyhedron
The Rhombic Dodecahedron fills the space without gaps.

In this page we are going to see from another point of view the relation between the cube and the rhombic dodecahedron.

Augmented cube | matematicasvisuales

You can put these six pyramids in a chain like this:

Cube and rhombicdodecahedron are 'reversibles' | matematicasvisuales

Cundy and Rollet wrote: "The resulting chain of pyramids can be turned inwards to form a cube, or, turned outwards, placed as a jacket over another cube to form the rhombic dodecahedron." (Cundy and Rollet, pag. 122.)

Cube and rhombicdodecahedron are 'reversibles' | matematicasvisuales
Cube and rhombicdodecahedron are 'reversibles' | matematicasvisuales

Cube and rhombicdodecahedron are 'reversibles' | matematicasvisuales
Cube and rhombicdodecahedron are 'reversibles' | matematicasvisuales

Turning inwards you get a cube:

Cube and rhombicdodecahedron are 'reversibles' | matematicasvisuales
Cube and rhombicdodecahedron are 'reversibles' | matematicasvisuales
Cube and rhombicdodecahedron are 'reversibles' | matematicasvisuales
Cube and rhombicdodecahedron are 'reversibles' | matematicasvisuales

This is a model made by cardboard:

Cube and rhombic dodecahedron, construcción con cartulina | matematicasvisuales


We already know that there is a relation between the Din A paper size and the rhombic dodecahedron.

Rhombic Dodecahedron (3): Augmented 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.
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.

Michael Grodzins has explored the possibilities of folding dipyramids using Din A paper size. Then he link six of these dipyramids and he build a rhombic dodecahedron. He published several pages about that: Instructables: (Un)Folding the Mysteries of A4 Paper, Instructables: Mysteries of the (Di)pyramids and more.

This is my model, following his instructions:

Cube and rhombic dodecahedron, Din A dipyramid, Michael Grodzins | matematicasvisuales
Cube and rhombic dodecahedron, Din A dipyramid, Michael Grodzins | matematicasvisuales
Cube and rhombic dodecahedron, Din A dipyramid, Michael Grodzins | matematicasvisuales
Cube and rhombic dodecahedron, Din A dipyramid, Michael Grodzins | matematicasvisuales
Cube and rhombic dodecahedron, Din A dipyramid, Michael Grodzins | matematicasvisuales
Cube and rhombic dodecahedron, Din A dipyramid, Michael Grodzins | 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): Augmented 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.
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.
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.
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.
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.
Leonardo da Vinci:Drawing of an augmented rhombicuboctahedron 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 augmented rhombicuboctahedron.
Leonardo da Vinci:Drawing of an augmented rhombicuboctahedron made to Luca Pacioli's De divina proportione (2).
We can see the interior of the augmented rhombicuboctahedron. Luca Pacioli wrote that you 'can see the interior only with your imagination'.
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.
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.
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).