This interactive mathlet is an adaptation of the drawing that Leonardo da Vinci made of the truncated octahedron (octocedron abscisus vacuus) for Luca Pacioli's book 'De Divina Proportione'.
Pacioli wrote about the truncated octahedron (Spanish translation):
"El octaedro abciso o cortado plano sólido o hueco tiene treinta y seis líneas que forman setenta y dos ángulos superficiales,
cuarenta y ocho de los cuales pertenecen a los hexágonos y veinticuatro a los cuadrados, y contiene veinticuatro ángulos sólidos y
catorce bases, ocho de las cuales son hexagonales, o sea, de seis lados, y seis tetragonales, o sea cuadradas. Pero veinticuatro
de las mencionadas líneas son comunes a los cuadrados y a los hexágonos. Los cuadrados están formados a partir de los hexágonos que,
en número de ocho, se tocan de modo uniforme, como nos hace ver claramente el intelecto en su forma material."
('La divina proporción' de Luca Pacioli, page 93, Spanish translation by Juan Calatrava, Editorial Akal, 4th edition, 2008)
Leonardo da Vinci's drawing of the truncated octahedron (octocedron abscisus 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's drawing of the truncated octahedron (octocedron abscisus 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).
REFERENCES
Luca Pacioli - La divina proporción - Ediciones Akal, 4th edition, 2004. Spanish edition of 'De divina proportione'. Translation by Juan Calatrava.
Leonardo da Vinci's Geometric Sketches artículo de Frank J. Swetz en MathDl, Loci:Convergence.
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 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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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 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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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.
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The truncated octahedron is a space-filling polyhedron
These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.
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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.
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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.
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