matematicas visuales visual math

A regular octahedron is a polyhedron with eight equilateral triangular faces, six vertices and twelve edges. It is a Platonic solid well known from Antiquity.

Octahedron plane net: an octahedron | matematicasVisuales
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.

Leonardo da Vinci drew two octahedra for Luca Pacioli's book 'De Divina Proportione' (published in 1509).

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.

It was Durer the first to publish plane nets of polyhedra. In his book 'Underweysung der Messung' ('Four Books of Measurement', published in 1525) the author draw plane developments of several Platonic and Archimedean solids, for example, this regular tetrahedron:

Octahedron plane net: plane net of an octahedron by Durer | matematicasVisuales

"Whether or not Dürer was acquainted with the work of the two Italian specialists in this fiel, Luca Pacioli and Piero della Francesca, is an open question. Certain it is that (...) he tackled the problem in an entirely independent way. Pacioli (...) illustrates them [several polyhedra] in perspective or stereographic images. Dürer treats (...)[more polyhedra] and instead of representing the solids in perspective or stereographic images, he devised the apparently original and, if one may say so, proto-topological method of developing them on the plane surface in such a way that the facets form a coherent 'net' which, when cut out of paper and properly folded where two facets adjoin, will form an actual, three-dimensional model of the solid in question." (Panofsky, p.259)

Playing with the application you can see how an octahedron develops into a plane net.

Octahedron plane net: developing octahedron | matematicasVisuales
Octahedron plane net: developing octahedron | matematicasVisuales
Octahedron plane net: developing octahedron | matematicasVisuales
Octahedron plane net: develope octahedron | matematicasVisuales

You can download the plane net of this octahedron and build your own.

Octahedron plane net: cardboard octahedron | matematicasVisuales

INVESTIGATE

Some relations between the octahedron and other polyhedra:

Octahedra and cubes are dual polyhedra:

Octahedron plane net: octahedron and cube are dual polyhedra | matematicasVisuales

Kepler's drawing of an octahedron inside a cube:

Octahedron plane net: Kepler drawing duality octahedron and cube | matematicasVisuales

Cube and octahedron in reciprocal position (Stellation of a cuboctahedron)

Octahedron plane net: cube and octahedron in reciprocal position. Stellation of an cuboctahedron | matematicasVisuales
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.

Stella Octangula is the name that Kepler used to one stellation of an octahedron:

Octahedron plane net: Stella octangula, stellation of an octahedron, inside a cube | matematicasVisuales

Octahedron plane net: Stella octangula, stellation of an octahedron | matematicasVisuales
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 truncated octahedron is an Archimedean solid that has very interesting properties:

Octahedron plane net: truncated octahedron | matematicasVisuales
Octahedron plane net: truncated octahedron | matematicasVisuales
Octahedron plane net: cube and octahedron truncations | matematicasVisuales
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.

We can get an octahedron truncating a tetrahedron:

Octahedron plane net: octahedron as a tetrahedron truncation | matematicasVisuales
Octahedron plane net: octahedron as a tetrahedron truncation | matematicasVisuales
Octahedron plane net: octahedron as a tetrahedron truncation | matematicasVisuales

REFERENCES

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.
Erwin Panofsky - The Life and Art of Albrecht Dürer - Princeton University Press
Dan Pedoe - Geometry and the Liberal Arts - St. Martin's Press (p. 76)
Hugo Steinhaus - Mathematical Snapshots - Oxford University Press - Third Edition (p. 197)
W.W. Rouse Ball and H.S.M. Coxeter - 'Matematical Recreations & Essays', The MacMillan Company, 1947.

MORE LINKS

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: 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 .
Resources: Building Polyhedra with cardboard (Plane Nets)
Using cardboard you can draw plane nets and build polyhedra.
Resources: How to build polyhedra using paper and rubber bands
A very simple technique to build complex and colorful polyhedra.
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: Tensegrity
Examples of polyhedra built using tensegrity.
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.
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.
Resources: Building polyhedra using Zome
Examples of polyhedra built using Zome.
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 (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 (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 (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 (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 (8): Cones cut by an oblique plane
Plane developments of cones cut by an oblique plane. The section is an ellipse.
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.
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.
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.
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 a truncated tetrahedron 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 tetrahedron.