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. An interactive application can remind us how can we take a look at a octahedron. 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 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: We can write a relation: A tetrahedron of edge length 2 is made of one octahedron and four tetrahedra of edge length 1: 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. Origami: octahedron and tetrahedron. 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.
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A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of a cube.
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 stellated octahedron was drawn by Leonardo for Luca Pacioli's book 'De Divina Proportione'. A hundred years later, Kepler named it stella octangula.
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.
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
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 twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
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.
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 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 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 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 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 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).
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.
