matematicas visuales visual math

The rhombicuboctahedron is an Archimedean polyhedron. Its faces consist of 8 triangles and 12+6 squares. The polyhedron that we are going to study in this page is very similar to the rhombicuboctahedron.

Leonardo da Vinci:Drawing of a 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 rhombicuboctahedron.


"In attempting to make a model of his polyhedron, J.C.P. Miller accidentally discovered a 'pseudo-rhombicuboctahedron', bounded likewise by 8 triangles and 18 squares, and isogonal in the loose or 'local' sense (each vertex being surrounded by one triangle and three squares), but not in the strict sense (which implies that the apearance of the solid as a whole must remain the same when viewed from the direction of each vertex in turn)." (Ball and Coxeter, p. 137)

This polyhedron is one of the Johnson solids. It has been rediscovered many times. "It has been called by a variety of names: the pseudo rhomb-cub-octahedron, Miller's solid, and elongated square gyro bicupola. The last of these names is Johnson's and it indicates how the solid is constructed from elementary polyhedra: take two square cupolas (square bicupola) rotated relative to one another (gyro) and separate by a prism (elongated)." (Cromwell, p.89)

Pseudo Rhombicuboctahedron or Elongated Square gyro bicupola | matematicasVisuales

You can compare these two polyhedra:

Pseudo Rhombicuboctahedron or Elongated Square gyro bicupola | matematicasVisuales

Playing with the interactive application we can rotate the polyhedron and change the "hole" and transparency of each face. We can compare these images which the images of a rhombicuboctahedron.

Pseudo Rhombicuboctahedron or Elongated Square gyro bicupola | matematicasVisuales
Pseudo Rhombicuboctahedron or Elongated Square gyro bicupola | matematicasVisuales
Pseudo Rhombicuboctahedron or Elongated Square gyro bicupola | matematicasVisuales


These two polyhedra have the same faces but they cannot be made from the same net.

It is easy to build a pseudo rhombicuboctahedron with cardboard. The rhombicuboctahedron is on the left and the pseudo rhombicuboctahedron is on the right.

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!
Pseudo Rhombicuboctahedron or Elongated Square gyro bicupola, cardboard model | matematicasVisuales
Pseudo Rhombicuboctahedron or Elongated Square gyro bicupola, cardboard model | matematicasVisuales
Pseudo Rhombicuboctahedron or Elongated Square gyro bicupola, cardboard model | matematicasVisuales
Pseudo Rhombicuboctahedron or Elongated Square gyro bicupola, cardboard model | matematicasVisuales
Pseudo Rhombicuboctahedron or Elongated Square gyro bicupola, cardboard model | matematicasVisuales
Pseudo Rhombicuboctahedron or Elongated Square gyro bicupola, cardboard model | matematicasVisuales

With this two parts we can build a rhombicuboctahedron ...

Rhombicuboctahedron or Elongated Square gyro bicupola, cardboard model | matematicasVisuales

... or a pseudo rhombicuboctahedron.

Pseudo Rhombicuboctahedron or Elongated Square gyro bicupola, cardboard model | matematicasVisuales

REFERENCES

W.W.W. Rouse Ball and H.S.M. Coxeter, 'Mathematical Recreatins & Essays', The Macmillan Company, New York, 1947.
Peter R. Cromwell - 'Polyhedra', Cambridge University Press, 1999. (pp. 89 and 366-369)
Pseudo Rhombicuboctahedra in George Hart's excellent website about polyhedra.

MORE LINKS

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'.
Augmented Rhombicuboctahedron
Starting with a Rhombicubotahedron we can add pyramids over each face. The we get a beautiful polyhedron that it is like a star.
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.
Leonardo da Vinci:Drawing of a SEPTUAGINTA 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 Campanus' sphere.
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 session 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 session 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 session about polyhedra (Zaragoza, 13th Abril 2012).
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 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.