matematicas visuales visual math

This interactive mathlet is an adaptation of the drawing that Leonardo da Vinci made of the rhombicuboctahedron or small rhombicuboctahedron (Vigintisex basium planus vacuus) for Luca Pacioli's book 'De Divina Proportione'.

Pacioli wrote about this polyhedron (Spanish translation):

"Otro cuerpo muy distinto de los ya nombrados, Excelso Duque, es el llamado cuerpo de veintiséis bases, de hermosísimo principio y origen derivado. De sus veintiséis bases, dieciocho son cuadradas, equiláteras y rectángulas, y ocho triangulares, igualmente equiláteras y equiángulas. Tiene cuarenta y ocho lados o líneas y noventa y seis ángulos superficiales, setenta y dos de los cuales son rectos -los de sus dieciocho bases cuadradas- y veinticuatro agudos -los de sus ocho triángulos equiláteros-. Estos noventa y seis ángulos superficiales determinan la formación de dicho cuerpo de veinticuatro ángulos sólidos, cada uno de los cuales consta de un ángulo superficial del triángulo y tres ángulos rectos de tres cuadrados. Y de sus cuarenta y ocho líneas, veinticuatro son comunes a los triángulos y a los cuadrados, ya que de sus dieciocho cuadrados, debidamente unidos entre sí, resultan necesariamente los ocho triángulos, como se ha dicho a proposito de los otros abcisos. El origen de este cuerpo es el hexaedro uniformemente cortado en todas sus partes, como nos muestra a la vista su propia forma material. Su conocimiento resulta utilísimo por muchas consideraciones para quien bien lo sepa aplicar, sobre todo en arquitectura. Y esto por lo que respecta al conocimiento de su sólido plano o hueco."
('La divina proporción' de Luca Pacioli, page 97, Spanish translation by Juan Calatrava, Editorial Akal, 4th edition, 2008)
Leonardo da Vinci: rhombicuboctahedron. Editorial Akal | matematicasvisuales
Leonardo da Vinci's drawing of the rhombicuboctahedron or small rhombicuboctahedron (Vigintisex basium planus vacuus) for Luca Pacioli's book 'De divina proportione'. (There is an Spanish version, 'La divina proporción' Editorial Akal. Image used with permission of Editorial Akal).
Leonardo da Vinci: rhombicuboctahedron. Editorial Akal | matematicasvisuales
Leonardo da Vinci's drawing of the rhombicuboctahedron or small rhombicuboctahedron (Vigintisex basium planus solidus) for Luca Pacioli's book 'De divina proportione'. (There is an Spanish version, 'La divina proporción' Editorial Akal. Image used with permission of Editorial Akal).

In the interactive application you can rotate the polyhedron and play with the "hole" in each face:

Rhombicuboctahedron or small rhombicuboctahedron | matematicasVisuales
Rhombicuboctahedron or small rhombicuboctahedron | matematicasVisuales
Rhombicuboctahedron or small rhombicuboctahedron | matematicasVisuales

It is easy to build a rhombicuboctahedron with cardboard:

Resources: Building polyhedra gluing faces
Using cardboard you can build beautiful polyhedra cutting polygons and glue them toghether. This is a very simple and effective technique. You can download several templates. Then print, cut and glue: very easy!
Rhombicuboctahedron or small rhombicuboctahedron | matematicasVisuales
Rhombicuboctahedron or small rhombicuboctahedron | matematicasVisuales
Rhombicuboctahedron or small rhombicuboctahedron | matematicasVisuales

Rhombicuboctahedron or small rhombicuboctahedron | matematicasVisuales
Edam (Holland), 2016

This polyhedron can be augmented with pyramids and you get a beautiful star:

Rhombicuboctahedron or small rhombicuboctahedron pyramidated, augmented, Ulm, Germany | matematicasVisuales
Ulm (Germany), 2013
Rhombicuboctahedron or small rhombicuboctahedron pyramidated, augmented, Rothemburg, Germany | matematicasVisuales
Rothenburg ob der Tauber (Germany), 2013
Rhombicuboctahedron or small rhombicuboctahedron pyramidated, augmented, Rothemburg, Germany | matematicasVisuales
Rothenburg ob der Tauber (Germany), 2013
Rhombicuboctahedron or small rhombicuboctahedron pyramidated, augmented, Rothemburg, Germany | matematicasVisuales
Rothenburg ob der Tauber (Germany), 2013
Rhombicuboctahedron or small rhombicuboctahedron pyramidated, augmented, Rothemburg, Germany | matematicasVisuales
Rothenburg ob der Tauber (Germany), 2013
Rhombicuboctahedron or small rhombicuboctahedron pyramidated, augmented, Haarlem, Holland | matematicasVisuales
Haarlem (Holland), 2016
Rhombicuboctahedron or small rhombicuboctahedron pyramidated, augmented | matematicasVisuales

Sundial in the Deutsches Museum in Munich. Its shape is a rhombicuboctahedron.

Rhombicuboctahedron or small rhombicuboctahedron: Sundial in the Deutsches Museum (Munich, Germany) | matematicasVisuales
Sundial in the Deutsches Museum (Munich, Germany), 2016
Rhombicuboctahedron or small rhombicuboctahedron: Sundial in the Deutsches Museum (Munich, Germany) | matematicasVisuales
Sundial in the Deutsches Museum (Munich, Germany), 2016
Rhombicuboctahedron or small rhombicuboctahedron: Sundial in the Deutsches Museum (Munich, Germany) | matematicasVisuales
Sundial in the Deutsches Museum (Munich, Germany), 2016

REFERENCES

Luca Pacioli - La divina proporción - Ediciones Akal, 4th edition, 2004. Spanish edition of 'De divina proportione'. Translation by Juan Calatrava.
Leonardo da Vinci's Geometric Sketches Frank J. Swetz's article in MathDl, Loci:Convergence.
Leonardo da Vinci's Polyhedra George Hart's excellent website about polyhedra.
Dan Pedoe - Geometry and the Liberal Arts - St. Martin's Press.
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

Leonardo da Vinci: Drawing of a dodecahedron 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 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.
Leonardo da Vinci: Drawing of an stellated octahedron (stella octangula) 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 stellated octahedron (stella octangula).
Leonardo da Vinci:Drawing of an 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 octahedron.
Plane developments of geometric bodies: Octahedron
The first drawing of a plane net of a regular octahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Resources: Building polyhedra gluing faces
Using cardboard you can build beautiful polyhedra cutting polygons and glue them toghether. This is a very simple and effective technique. You can download several templates. Then print, cut and glue: very easy!
Resources: How to build polyhedra using paper and rubber bands
A very simple technique to build complex and colorful polyhedra.
Cube, octahedron, tetrahedron and other polyhedra: Taller de Talento Matemático Zaragoza,Spain, 2014-2015 (Spanish)
Material for a sesion about polyhedra (Zaragoza, 7th November 2014). We study the octahedron and the tetrahedron and their volumes. The truncated octahedron helps us to this task. We build a cubic box with cardboard and an origami tetrahedron.
Duality: cube and octahedron. Taller de Talento Matemático de Zaragoza, Spain. 2015-2016 XII edition (Spanish)
Material for a sesion about polyhedra (Zaragoza, 23rd Octuber 2015) . Building a cube with cardboard and an origami octahedron.
Building polyhedra. Basic techniques: Taller de Talento Matemático de Zaragoza (Spanish)
Material for a sesion about polyhedra (Zaragoza, 13th Abril 2012).
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.
Plane developments of geometric bodies: Tetrahedron
The first drawing of a plane net of a regular tetrahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
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 cuboctahedron (II)
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an octahedron.
Stellated cuboctahedron
The compound polyhedron of a cube and an octahedron is an stellated cuboctahedron.It is the same to say that the cuboctahedron is the solid common to the cube and the octahedron in this polyhedron.
Truncated tetrahedron
The truncated tetrahedron is an Archimedean solid made by 4 triangles and 4 hexagons.
Truncations of the cube and octahedron
When you truncate a cube you get a truncated cube and a cuboctahedron. If you truncate an octahedron you get a truncated octahedron and a cuboctahedron.