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This interactive mathlet is an adaptation of the drawing that Leonardo da Vinci made of a polyhedron with 72 faces like a sphere (septuaginta duarum basium vacuus) for Luca Pacioli's book 'De Divina Proportione'.

Pacioli wrote about this polyhedron (Spanish translation):

"Entre estos cuerpos conviene colocar, Excelso Duque, el llamado cuerpo de setenta y dos bases, del que da una plena descripción nuestro filósofo megarense en la decimocuarta de su duodécimo. Dicho cuerpo, aunque tiene sus bases planas laterales y angulares y deformes, no se puede decir que dependa o derive de alguno de los cuerpos regulares, sino que se forma y se crea, según lo que en dicho lugar demuestra nuestro filósofo, mediante la figura dodecagonal, es decir, de doce lados iguales. Cuarenta y dos de sus bases son cuadrangulares, no equiláteras ni equiángulas, y tienen sus dos lados opuestos dirigidos hacia uno y otro polo o cono e iguales entre sí; sus otras veinticuatro bases son triangulares, igualmente no equiláteras; doce de ellas están en torno a uno de los conos y doce en torno al otro, y cada una de ellas tiene dos lados iguales, los que tienden hacia el punto del polo inferior y superior. (...) Este cuerpo de setenta y dos bases es muy frecuentemente empleado por los arquitectos en sus disposiciones de edificios, por ser forma bastante útil, sobre todo cuando hay que hacer tribunas y otras bóvedas o cielos."
('La divina proporción' de Luca Pacioli, page 99, Spanish translation by Juan Calatrava, Editorial Akal, 4th edition, 2008)
Leonardo da Vinci: Septuaginta. Editorial Akal | matematicasvisuales
Leonardo da Vinci's drawing of a polyhedron with 72 faces like a sphere (septuaginta duarum basium vacuus) for Luca Pacioli's book 'De divina proportione'. (There is an Spanish version, 'La divina proporción' Editorial Akal. Image used with permission of Editorial Akal).
Leonardo da Vinci: Septuaginta. Editorial Akal | matematicasvisuales
Leonardo da Vinci's drawing of a polyhedron with 72 faces like a sphere (septuaginta duarum basium solidus) for Luca Pacioli's book 'De divina proportione'. (There is an Spanish version, 'La divina proporción' Editorial Akal. Image used with permission of Editorial Akal).


Polyhedra of this kind are useful in Architecture and art. One example in Madrid (Spain):

Leonardo da Vinci: Septuaginta. Madrid (Spain) | matematicasvisuales
Leonardo da Vinci: Septuaginta. Madrid (Spain) | matematicasvisuales

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

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

These polyhedra (with some restrictions) are called Campanus'sphere. There were very popular during the Reinaissance.



REFERENCES

Luca Pacioli - La divina proporción - Ediciones Akal, 4th edition, 2004. Spanish edition of 'De divina proportione'. Translation by Juan Calatrava.
Peter R. Cromwell - 'Polyhedra', Cambridge University Press, 1999.
Fra' Gioavanni`s Intarsias in Verona, article by H. Thomas and K. Williams (The Mathematical Tourist).
Some images of a beautiful intarsia in the Monastery of El Escorial (Spain).
Leonardo da Vinci's Geometric Sketches Frank J. Swetz's article in MathDl, Loci:Convergence.

MORE LINKS

Sections in the sphere
We want to calculate the surface area of sections of a sphere using the Pythagorean Theorem. We also study the relation with the Geometric Mean and the Right Triangle Altitude Theorem.
Cavalieri: The volume of a sphere
Using Cavalieri's Principle we can calculate the volume of a sphere.
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.
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.