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One eighth of a dodecahedron of side length 2 has the same volume of a dodecahedron of side length 1.

Volume of a Dodecahedron: the volume of one eight of a dodecahedron of side length 2 has the same volume of a dodecahedron of side length 1 | matematicasVisuales

We are going to study the volume of a dodecahedron looking at this figure.

Volume of a Dodecahedron: pieces to calculate the volume of a dodecahderon | matematicasVisuales
Volume of a Dodecahedron: Zome pieces to calculate the volume of a dodecahedron | matematicasVisuales
Volume of a Dodecahedron: Zome pieces to calculate the volume of a dodecahedron | matematicasVisuales

We can decompose one eight of a dodecahedron of side length 2 in several pieces and calculate the volume of these pieces.

Volume of a Dodecahedron: pieces to calculate the volume of a dodecahderon | matematicasVisuales

There are one cube, three wedges and three pyramids:

Volume of a Dodecahedron: pieces to calculate the volume of a dodecahderon, There are one cube, three wedges and three pyramids | matematicasVisuales

These are important measures in this figure, related with the golden section:

Volume of a Dodecahedron: some connexions between the dodecahedron and the golden ratio | matematicasVisuales

Remember that

Volume of a Dodecahedron: Zome, the golden ratio and the dodecahedron | matematicasVisuales

There is one cube and its volume is:

Volume of a Dodecahedron: the volume of a cube | matematicasVisuales

There are three wedges and the volume of one wedge is:

Volume of a Dodecahedron: the volume of a wedge | matematicasVisuales

There are three pyramids and the volumen of a pyramid is:

Volume of a Dodecahedron: the volume of a pyramid | matematicasVisuales

Then the volume of a dodecahedron of side length 1 is:

The volume of a dodecahedron of side length a is:

Some minerals, like pyrite, cristalize in dodecahedra (non-regular, it is sometimes called pyritohedron)

Volume of a Dodecahedron: Some minerals, like pyrite cristalize in dodecahedra (non-regular, it is sometimes called pyritohedron) | matematicasVisuales

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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.
The Dodecahedron and the Cube
A Cube can be inscribed in a Dodecahedron. A Dodecahedron can be seen as a cube with six 'roofs'. You can fold a dodecahedron into a cube.
Pyritohedron
If you fold the six roofs of a regular dodecahedron into a cube there is an empty space. This space can be filled with an irregular dodecahedron composed of identical irregular pentagons (a kind of pyritohedron).
The Diagonal of a Regular Pentagon and the Golden Ratio
The diagonal of a regular pentagon are in golden ratio to its sides and the point of intersection of two diagonals of a regular pentagon are said to divide each other in the golden ratio or 'in extreme and mean ratio'.
Drawing a regular pentagon with ruler and compass
You can draw a regular pentagon given one of its sides constructing the golden ratio with ruler and compass.
Durer's approximation of a Regular Pentagon
In his book 'Underweysung der Messung' Durer draw a non-regular pentagon with ruler and a fixed compass. It is a simple construction and a very good approximation of a regular pentagon.
The icosahedron and its volume
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
The golden ratio
From Euclid's definition of the division of a segment into its extreme and mean ratio we introduce a property of golden rectangles and we deduce the equation and the value of the golden ratio.
The golden rectangle
A golden rectangle is made of an square and another golden rectangle.
The golden spiral
The golden spiral is a good approximation of an equiangular spiral.
The golden rectangle and two equiangular spirals
Two equiangular spirals contains all vertices of golden rectangles.
The golden rectangle and the dilative rotation
A golden rectangle is made of an square an another golden rectangle. These rectangles are related through an dilative rotation.
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 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.
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).
The volume of the tetrahedron
The volume of a tetrahedron is one third of the prism that contains it.
Plane developments of geometric bodies: Tetrahedron
The first drawing of a plane net of a regular tetrahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
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 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.
The volume of an stellated octahedron (stella octangula)
The stellated octahedron was drawn by Leonardo for Luca Pacioli's book '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.
The truncated octahedron is a space-filling polyhedron
These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.
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 dodecahedron.