matematicas visuales visual math

Sabemos que este poliedro puede verse como formado por ocho medios cubos. Entonces sabemos calcular su volumen de un modo muy sencillo.

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.

En esta página vamos a calcular ese volumen de otro modo.

The volume of a truncated octahedron can be easily calculated if we know the volume of an octahedron.

The volume of an octahedron of edge length 1 is:

We can calculate the volume of an octahedron of edge length 3:

Octahedron of edge length 3 | matematicasvisuales

To calculate the volume of a truncated octahedron we have to remove to an octahedron of edge length 3 six pyramids. These six pyramids make 3 octahedra of edge length 1.

The volume of a truncated octahedron of edge length 1 can be calculated using the volume of an octahedron | matematicasvisuales

Además este poliedro es un sólido arquimediano. Pertenece a la segunda familia de poliedros más famosa. Sus caras son poliedros regulares no todos iguales. En este caso vemos que tiene 6 cuadrados y 8 hexágonos. Todos sus vértices tienen que ser iguales.

Truncated Octahedron: | matematicasVisuales

Este es el desarrollo plano de esta figura:

Truncated Octahedron: | matematicasVisuales

Construcción de poliedros: solapas | matematicasVisuales
Construcción de poliedros: solapas para recortar | matematicasVisuales

Es un poliedro arquimediano que podemos ver como un octaedro al que se le han quitado seis pirámides en sus vértices. Por lo tanto, su nombre es octaedro truncado.

 | matematicasVisuales
 | matematicasVisuales
 | matematicasVisuales

El volumen del octaedro se puede calcular a partir del octaedro truncado (que ya sabemos que tiene un volumen muy sencillo de calcular).

Por otra parte, sabemos que hay una relación entre el volumen del octaedro de lado 3 y el de lado 1:

 | matematicasVisuales
 | matematicasVisuales

El octaedro de lado 3 está formado por un octaedro truncado de lado 1 y 3 pares de pirámides. Cada par forma un octaedro de lado 1.

El volumen del octaedro truncado es igual al volumen de un octaedro de lado 3 menos 3 octaedros de lado 1 | matematicasvisuales

Por lo tanto, podemos escribir:

El volumen del octaedro de lado 1 es:

Si sabemos el volumen del octaedro ya podemos calcular el volumen del tetraedro pues sabemos que es la cuarta parte.

Hemos podido calcular el volumen del octaedro y el del tetraedro sin recurrir a la fórmula del volumen de la pirámide. Manipulando las construcciones de los poliedros y viendo sus propiedades. También hemos usado la relación entre los volúmenes de figuras semejantes.

REFERENCES

Hugo Steinhaus, Mathematical Snapshots, Dover Publications (third edition, 1999)
We can read some pages of this book in Google Books: Mathematical Snapshots by Hugo Steinhaus.
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.

MORE LINKS

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.
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.
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 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 .
Regular dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
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).
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 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.
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.
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.