matematicas visuales visual math

This kind of polyhedra that are inscribed in a sphere was well know by Euclid. In proposition 17 of book XII Euclid shows how to construct a polyhedron sandwiched between two concentric spheres.

During the Middle Ages, Campanus of Novara (1220-1296) wrote a very popular version of Euclid's Elements. Campanus used this construction to describe a 72-faced polyhedron. This polyhedron was very popular during the Renaissance and was known as Campanus´ sphere.

Leonardo da Vinci draw this polyhedron to Luca Pacioli's book 'De Divina Proportione'.

Leonardo da Vinci:Drawing of a SEPTUAGINTA 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 Campanus' sphere.


If you play with the interactive application you can get images like these:

Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere | Images manipulating the interactive application | matematicasvisuales

With more and more faces this polyhedron is more and more like a sphere:

Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere | Images manipulating the interactive application | matematicasvisuales
Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere | Images manipulating the interactive application | matematicasvisuales

One way to construct a polyhedron inside a sphere is as follows:

Consider a sphere and let N and S be the North ans South poles of this sphere and let C be its equatorial circle. Inscribe in C a regular polygon with a number of sides which is a multiple of 4, say 4n. Then construct 2n 4n-gons, each of which has a vertex at N, another at S and two vertices passing through two opposite corners of the equatorial polygon (like longitude lines). Then connect the vertices of these polygons with 'latitude lines'. The result is a polyhedron with 8n^2 faces.

Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere | Images manipulating the interactive application | matematicasvisuales

If n=1 the result is the octahedron.

Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere: octahedron | Images manipulating the interactive application | matematicasvisuales
Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere: octahedron | Images manipulating the interactive application | matematicasvisuales

If n=2 the result is a polyhedron with 32 faces.

Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere: polyhedron with 32 faces | Images manipulating the interactive application | matematicasvisuales
Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere: polyhedron with 32 faces | Images manipulating the interactive application | matematicasvisuales
Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere: polyhedron with 32 faces | Images manipulating the interactive application | matematicasvisuales

If n=3 the result is a polyhedron with 72 faces, that was known as Campanus' sphere and Luca Pacioli called it Septuaginta.

Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere: polyhedron with 72 faces | Images manipulating the interactive application | matematicasvisuales
Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere: polyhedron with 72 faces | Images manipulating the interactive application | matematicasvisuales
Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere: polyhedron with 72 faces | Images manipulating the interactive application | matematicasvisuales

Using the interactive application you can draw polyhedra that the polygon in the equatorial circle have an ever or odd number of sides and that the 'longitude polygons' have a different number of sides than the 'equatorial polygon'.

Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere | Images manipulating the interactive application | matematicasvisuales
Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere | Images manipulating the interactive application | matematicasvisuales
Leonardo da Vinci: Septuaginta. Campanus' sphere. Polyhedra inscribed in a sphere | Images manipulating the interactive application | matematicasvisuales

REFERENCES

Luca Pacioli - La divina proporción - Ediciones Akal, 4ª edición, 2004. Traducción de Juan Calatrava.
Peter R. Cromwell - 'Polyhedra', Cambridge University Press, 1999.
Dan Pedoe - Geometry and the Liberal Arts - St. Martin's Press (p. 76)
Leonardo da Vinci's Polyhedra and Fra Giovanni's Intarsia Polyhedra en el excelente sitio web sobre poliedros de George Hart.

MORE LINKS

The World
Basic world map in a sphere. Latitude and longitude
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.
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.
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 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).
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.
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.
Kepler: The Area of a Circle
Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
Cavalieri: The volume of a sphere
Using Cavalieri's Principle we can calculate the volume of a sphere.
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
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of a cube.