matematicas visuales visual math

When you fold a dodecahedron into a cube, inside the cube there is an empty space.

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: | matematicasVisuales

We can fill this empty space with a polyhedron that is a kind of pyritohedron, that is to say, this polyhedron is an irregular dodecahedron composed of identical irregular pentagons. In this case is a concave pyritohedron an is called concave pyritohedral dodecahedron.

Pyritohedron: | matematicasVisuales
Pyritohedron: | matematicasVisuales
Pyritohedron: | matematicasVisuales
Pyritohedron: | matematicasVisuales
Pyritohedron: | matematicasVisuales
Pyritohedron: | matematicasVisuales
Pyritohedron: | matematicasVisuales
Pyritohedron: | matematicasVisuales

We encourage you to build your own pyritohedron:

Pyritohedron: | matematicasVisuales
Pyritohedron: | matematicasVisuales
Pyritohedron: | matematicasVisuales
Pyritohedron: | matematicasVisuales

We can build this pyritohedron using Zome:

Pyritohedron: | matematicasVisuales

Eight vertices are the vertices of a cube:

Pyritohedron: | matematicasVisuales
Pyritohedron: | matematicasVisuales

In the interior, twelve vertices are in three golden rectangles:

Pyritohedron: | matematicasVisuales

The three golden rectangles in three different colors:

Pyritohedron: | matematicasVisuales

These twelve vertices are the vertices of an icosahedron:

Pyritohedron: | matematicasVisuales
Pyritohedron: | matematicasVisuales

To calculate the volume of this pyritohedron of side length 1 you should remember some properties of the golden ratio, and it is very similar to the calculation of the volume of a dodecahedron. The roof that was outside the cube it is now inside the cube.

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'.
Volume of a regular dodecahedron
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.

The volume of this pyritohedron is the volume of a cube minus six times the volume of a roof.

Remember the volume of the cube:

And the volume of one roof is:

Then the volume of this pyritohedron is:

MORE LINKS

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.
Regular dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
Volume of a regular dodecahedron
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
Plane developments of geometric bodies: Dodecahedron
The first drawing of a plane net of a regular dodecahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
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 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.