matematicas visuales visual math

The experience of building an octahedron with paper or using 12 plastic tubes forming 4 squares in three orthogonal planes, or just piling 6 oranges is not replaced with a two dimensional representation.



One octahedron made using 12 plastic tubes | matematicasvisuales

An interactive application can remind us how can we take a look at a octahedron.

One way to handle an octahedron to calculate its volume | matematicasvisuales

An octahedron is composed by two pyramids of square base. We can see the height of these two pyramides as the diagonal of a square.

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:

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:

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:

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

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



It is easy to build with plastic tubes a figure formed by four tetrahedra and one octahedron.



Octahedron and tetrahedra built using plastic tubes | matematicasvisuales

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

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.
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.
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).