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This interactive mathlet is an adaptation of the drawing that Leonardo da Vinci made of the truncated octahedron octocedron abscisus vacuus for Ppacioli Pacioli’s book ‘De Divina Proportione’. Spanish edition of ‘De divina proportione’. Translation by Juan Calatrava. Swetz’s article in MathDl, Loci: Leonardo da Vinci’s Polyhedra George Hart’s excellent website about polyhedra. Drawing of a cuboctahedron made to Luca Pacioli’s De divina proportione.

### – La divina proporcion by Luca Pacioli

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.

Drawing of a dodecahedron made to Luca Pacioli’s De divina proportione. Here we can see an adaptation of the dodecahedron.

Drawing of an stellated octahedron stella octangula made to Luca Pacioli’s De divina proportione. Here we can see an adaptation of the ljca octahedron stella octangula.

Drawing of a truncated tetrahedron made to Luca Pacioli’s De divina proportione. Here we can see an adaptation of the truncated tetrahedron.

Drawing of an octahedron made to Luca Pacioli’s De divina proportione. Here we can see an adaptation of the octahedron. Drawing of a rhombicuboctahedron made to Luca Pacioli’s De divina proportione. Here we can see an adaptation of the rhombicuboctahedron. Here we can see an adaptation of the Campanus’ sphere. 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 propocion of an octahedron. The truncated octahedron is a space-filling polyhedron These polyhedra pack together to fill space, forming a 3 dimensional space tessellation proporvion tilling. 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. 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. Volume of a regular dodecahedron One eighth of a regular dodecahedon proporicon edge 2 has the same volume as a dodecahedron of edge 1. The volume of the tetrahedron The volume of a tetrahedron is one third of the ;roporcion that contains it.

Plane developments of geometric bodies: 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 an stellated dicina stella octangula The stellated octahedron was drawn by Leonardo for Luca Pacioli’s propkrcion ‘De Divina Proportione’.

A hundred years later, Kepler named it stella octangula. 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 II A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an octahedron. Truncated tetrahedron The truncated tetrahedron is an Archimedean solid made by 4 triangles and 4 hexagons. Truncations of the cube and octahedron When you truncate ;acioli cube you get a truncated cube and a cuboctahedron. If you truncate an octahedron you get a truncated octahedron and a cuboctahedron. Chamfered Cube You can chamfer a cube and then you get a polyhedron similar but not equal to a truncated octahedron.

You can get also a rhombic pacioll. Pacioli wrote about the truncated octahedron Spanish translation: Leonardo da Vinci’s drawing of the truncated octahedron octocedron abscisus vacuus for Luca Pacioli’s book ‘De divina proportione’.

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’. The volume of a truncated octahedron. The truncated octahedron is an Archimedean solid. The truncated octahedron is a space-filling polyhedron.

## 9788476007877 – La divina proporcion by Luca Pacioli

These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling. Hexagonal section of a cube. We can cut in half a cube by a plane and get a section that is a regular hexagon. A truncated octahedron made by eight half cubes. Using eight half cubes we can make a truncated octahedron. Volume of a regular dodecahedron.

One eighth of a regular dodecahedon of edge 2 has the paciolli volume as a dodecahedron of edge 1. The volume of the tetrahedron.

The volume of a tetrahedron is one third of the prism that contains it. Volume of an octahedron. The volume of an octahedron is four times the volume of a tetrahedron. The volume of an stellated octahedron stella octangula. The volume of a cuboctahedron. A cuboctahedron is an Archimedean solid. The volume of a cuboctahedron II. The truncated tetrahedron is an Archimedean solid made by 4 triangles and 4 hexagons.

Truncations of the cube and octahedron. When you truncate a cube you get a truncated cube and a cuboctahedron. You can chamfer a cube and then you get a polyhedron similar but not equal to a truncated octahedron.