matematicas visuales visual math

A cuboctahedron is an Archimedean solid. It is generated by truncating the vertices of a cube or of an octahedron at 1/2 edge-length. There are 6 square faces on the cuboctahedron, one for each face of the cube. There are 8 equilateral triangular faces, one for each vertex of the cube.

We are going to calculate the volume of an octahedron of edge-length 1 starting from the volume of a cube.

If a cuboctahedron has edge-length 1, the cube that contains it is:

Volume of a cuboctahedron: Cuboctahedron inside a cube | matematicasvisuales

The volume of this cube is:

To calculate the volume of a cuboctahedron we have to subtract from the volume of the cube the volume of the 8 pyramids that we cut off.

The volume of each of these 8 pyramids is:

Volume of a cuboctahedron: the volume of a pyramid that we cut off from the cube | matematicasvisuales

Now, we can calculate the volume of the cuboctahedron (what we subtract, 8 pyramids, may be reasssembled into an octahedron of edge-length 1)

Volume of a cuboctahedron: The volume of a cuboctahedron is the volume of a cube minus the volume of an octahedron | matematicasvisuales
Volume of a cuboctahedron | matematicasvisuales

Then the volume of a cuboctahedron of edge-length a is:

Now we are going to see one interesting property of the cuboctahedron. If we think in this cube as made for eight small cubes (with the center of the cube as a vertex shared by the small cubes) we can see that the distance from the center of the cuboctahedron (its center of gravity) to any vertex is the edge length (it is equal to a diagonal of a side of one small cube).

Volume of a cuboctahedron: Cuboctahedron in a cube to see that the distance from each vertice to the center is the same as the edge-length | matematicasvisuales

Then a cuboctahedron is made of six half octahedra and eight tetrahedra, all of these pyramids sharing one vertex in the center of gravity of the cuboctahedron.

The cuboctahedron is the only spatial configuration in wich the length of each polyhedral edge is equal to that the distance from its center of gravity to any vertex.

More than that, its 24 edges lies in four hexagons centered in the center of gravity of the cuboctahedron.

In this origami cuboctahedron, can you see the hexagon (paralel to the floor)?

 Cuboctahedron: origami following instructions from Tomoko Fusê's book 'Unit Origami', one hexagon| matematicasvisuales
Cuboctahedron: origami following instructions from Tomoko Fusê's book 'Unit Origami'| matematicasvisuales
I made this origami cuboctahdron following the instructions in Tomoko Fusè's book 'Unit Origami' (Japan Publications, Inc. 1990)

Can you see here four hexagons?

Volume of a cuboctahedron: Skeleton of a cuboctahedron to see four hexagons | matematicasvisuales
Volume of a cuboctahedron: Another skeleton of a cuboctahedron to see four hexagons | matematicasvisuales

Vertices of this beautiful modular origami model, the Omega Star, are vertices of a cuboctahedron:

Volume of a cuboctahedron: Omega Star, modular origami model. Its vertices are the vertices of a cuboctahedron | matematicasvisuales

One interesting property of the cuboctahedron is that the distance from the solid's center to each vertice is equal to its edge length:

Volume of a cuboctahedron: building a cuboctahedron with zome | matematicasvisuales
Volume of a cuboctahedron: zome, the distance from the center to each vertice is the edge length  | matematicasvisuales
Volume of a cuboctahedron: tubes,  the distance from the center to each vertice is the edge length   | matematicasvisuales

Volume of a cuboctahedron: A cuboctahedron on a wall in Rothenburg ob der Tauber (Germany)  | matematicasvisuales
A cuboctahedron on a wall in Rothenburg ob der Tauber (Germany, 2013)
Volume of a cuboctahedron: A cuboctahedron on a door in the Schottenkirche St. Jakob (Scots Church of St. James) in Regensburg (Germany)  | matematicasvisuales
A cuboctahedron on a door in the Schottenkirche St. Jakob (Scots Church of St. James) in Regensburg (Germany, 2014)

Cuboctahedrons in the Schöner Brunnen (The Beautiful Fountain) in Nuremberg (Germany, 2014).

In he Beautiful Fountain you can see several images of scientis of the Antiquity:

Volume of a cuboctahedron: Schöner Brunnen (The Beautiful Fountain) in Nuremberg (Germany, 2014) | matematicasvisuales
Schöner Brunnen (The Beautiful Fountain) in Nuremberg (Germany, 2014, RCR)
Volume of a cuboctahedron: Pythagoras in Schöner Brunnen (The Beautiful Fountain) in Nuremberg (Germany, 2014) | matematicasvisuales
Pythagoras in Schöner Brunnen (The Beautiful Fountain) in Nuremberg (Germany, 2014, RCR)
Volume of a cuboctahedron: Euclid in Schöner Brunnen (The Beautiful Fountain) in Nuremberg (Germany, 2014) | matematicasvisuales
Euclid in Schöner Brunnen (The Beautiful Fountain) in Nuremberg (Germany, 2014, RCR)
Volume of a cuboctahedron: Ptolemy in Schöner Brunnen (The Beautiful Fountain) in Nuremberg (Germany, 2014) | matematicasvisuales
Ptolemy in Schöner Brunnen (The Beautiful Fountain) in Nuremberg (Germany, 2014, RCR)

There are several polyhedra (cuboctahedron, icosidodecahedron and more) in some intarsia panels in El Escorial (Madrid, Spain). I think that this intarsia panels are not very well known although Cromwell wrote: "There are also examples of polyhedra in the royal palace at El Escorial just outside Madrid. The palace was erected by Philip II (1527-1598), who is said to have excelled in mathematical studies as a young prince. The doors to the throne-room at the palace were a gift from his father-in-law, Maximilian II. They were ornately carved and inlaid b German craftsmen. The intarsia panels contain some of the typical elements (lutes, books) and some polyhedra". (Cromwell, p. 117)

El Escorial: intarsia with several polyhedra  | matematicasvisuales
El Escorial: intarsia with several polyhedra  | matematicasvisuales
El Escorial: intarsia with several polyhedra: cuboctahedron  | matematicasvisuales
El Escorial: intarsia with several polyhedra: cuboctahedron  | matematicasvisuales

REFERENCES

Hugo Steinhaus - Mathematical Snapshots - Oxford University Press - Third Edition (p. 197)
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 (p. 87).
W.W. Rouse Ball and H.S.M. Coxeter - 'Matematical Recreations & Essays', The MacMillan Company, 1947.
Luca Pacioli - De divina proportione - (La divina proporción) Ediciones Akal, 4ª edición, 2004. Spanish translation by Juan Calatrava.
Mª Paz Aguiló, La ebanistería alemana en el Monasterio de El Escorial.

MORE LINKS

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.
The volume of an stellated octahedron (stella octangula)
The stellated octahedron was drawn by Leonardo for Luca Pacioli's book 'De Divina Proportione'. A hundred years later, Kepler named it stella octangula.
The volume of the tetrahedron
The volume of a tetrahedron is one third of the prism that contains it.
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 .
Plane developments of geometric bodies (1): Nets of prisms
We study different prisms and we can see how they develop into a plane net. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (2): Prisms cut by an oblique plane
Plane nets of prisms with a regular base with different side number cut by an oblique plane.
Plane developments of geometric bodies (3): Cylinders
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (4): Cylinders cut by an oblique plane
We study different cylinders cut by an oblique plane. The section that we get is an ellipse.
Plane developments of geometric bodies (5): Pyramid and pyramidal frustrum
Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.
Plane developments of geometric bodies (7): Cone and conical frustrum
Plane developments of cones and conical frustum. How to calculate the lateral surface area.
Plane developments of geometric bodies (8): Cones cut by an oblique plane
Plane developments of cones cut by an oblique plane. The section is an ellipse.
Plane developments of geometric bodies: Dodecahedron
The first drawing of a plane net of a regular dodecahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
The icosahedron and its volume
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
Regular dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
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.
Hexagonal section of a cube
We can cut in half a cube by a plane and get a section that is a regular hexagon. Using eight of this pieces we can made a truncated octahedron.
A truncated octahedron made by eight half cubes
Using eight half cubes we can make a truncated octahedron. The cube tesselate the space an so do the truncated octahedron. We can calculate the volume of a truncated octahedron.
The truncated octahedron is a space-filling polyhedron
These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.
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.
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 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 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).
Resources: Modular Origami
Modular Origami is a nice technique to build polyhedra.
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.
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.