matematicas visuales visual math

It is a very interesting experience to build and touch a model of an octahedron.

We can use cardboard (the octahedron consists of eight equilateral triangles):

Octahedron: built using cardboard | matematicasvisuales

Very basic origami (the six vertices are in three squares in three orthogonal planes):

Octahedron: built using modular origami | matematicasvisuales
Octahedron: built using modular origami | matematicasvisuales

Or you can use twelve plastic tubes:

Octahedron: built plastic tubes | matematicasvisuales



An octahedron is composed by two pyramids of square base.

Octahedron: an octahedron is composed by two pyramids | matematicasvisuales

We can see the height of these two pyramides as the diagonal of a square.

Octahedron: an octahedron diagonal | matematicasvisuales
Octahedron: built plastic tubes | matematicasvisuales
The height of an octahedron is the diagonal of a square | matematicasvisuales

The diagonal of a square of edge length 1 is:

Therefore, the volume of an octahedron of edge length 1 is (remember that the volume of a pyramid is one third of the base area times the perpendicular height):

And the volume of an octahedron of edge length a is:

Using that we can calculate the volume of a tetrahedron. We can consider a tetrahedron of edge length 2:

Octahedron: A tetrahedron of edge length 2 is made of one octahedron and four tetrahedra of edge length 1 | matematicasvisuales
Octahedron: A tetrahedron of edge length 2 is made of one octahedron and four tetrahedra of edge length 1 | matematicasvisuales

We can write a relation:

Volume of a tetrahedron | matematicasvisuales

A tetrahedron of edge length 2 is made of one octahedron and four tetrahedra of edge length 1:

Octahedron: A tetrahedron of edge length 2 is made of one octahedron and four tetrahedra of edge length 1 | matematicasvisuales
A tetrahedron of edge length 2 is made of one octahedron and four tetrahedra of edge length 1 | matematicasvisuales
A formula that relates the volume of a tetrahedron and a octahedron | matematicasvisuales
Octahedron: A tetrahedron of edge length 2 is made of one octahedron and four tetrahedra of edge length 1 | matematicasvisuales
Octahedron: A tetrahedron of edge length 2 is made of one octahedron and four tetrahedra of edge length 1 | matematicasvisuales

Then, the volume of an octahedron is four times the volume of a tetrahedron and we can recalculate the volume of a tetrahedron.

Origami: octahedron and tetrahedron.

Volume of a octahedron: origami, octahedra inside a tetrathedron  | matematicasVisuales
Volume of a octahedron origami, octahedra inside a tetrahedron | 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.

MORE LINKS

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.
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: Building polyhedra using tubes
Examples of polyhedra built using tubes.
Resources: Modular Origami
Modular Origami is a nice technique to build polyhedra.
Resources: Tensegrity
Examples of polyhedra built using tensegrity.
Resources: Building polyhedra using Zome
Examples of polyhedra built using Zome.
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).
Kepler: The Area of a Circle
Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
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: 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 .
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.