The stellated octahedron is the only stellation of the octahedron. It was depicted by Leonardo da Vinci in Pacioli's book 'De Divina Proportione' in 1509. A hundred years later was named 'Stella octangula' by Johannes Kepler.
It can be seen as the compound of two tetrahedra (one tetrahedron an its dual, a second tetrahedron). The eight vertices are the vertices of a cube.
As an stellation is an octahedron with tetrahedral pyramids on each face.
Then is very easy to calculate the volume of an stellated octahedron if we know the volume of an octahedron and the volume of a tetrahedron.
Then the volume of an stellated octahedron of side length a is:
REFERENCES
Luca Pacioli - La divina proporción - Ediciones Akal, 4ª edición, 2004. Traducción de Juan Calatrava.
Leonardo da Vinci's Polyhedra página de George Hart en su excelente sitio sobre poliedros.
LINKS
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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).
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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.
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The volume of the tetrahedron
The volume of a tetrahedron is one third of the prism that contains it.
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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.
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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.
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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.
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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.
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