matematicas visuales visual math

Leonardo da Vinci drew two dodecahedra for Luca Pacioli's book 'The Divine Proportione'.

Dodecahedron: Leonardo da Vinci's dodecahedron drawing in Pacioli's book 'The Divine Proportione' | matematicasVisuales

Kepler was interested in this wonderful geometric body (For example, you can see this drawing in his book 'Hamonices Mundi - The Harmony of the World', (1619) (Read the original book at Posner Memorial Collection):

Dodecahedron: Kepler drawing of a dodecahedron in 'Hamonices Mundi - The Harmony of the World' | matematicasVisuales

In Dürer's book 'Underweysung der Messung' the author published the first plane net of a regular dodecahedron.

A regular dodecahedron is a platonic polyhedron made by 12 regular pentagons. Then there is a deep relation between the dodecahedron and the golden ratio.

Points of coordinates:

Dodecahedron: vertices | matematicasVisuales

are the 20 vertices of a regular dodecahedron of edge 2.

Where is the golden ratio we've seen in the golden rectangle.

We can calculate the distance between pairs of adjacents vertices to convince ourselves.

We can build this figure using Zome:

Dodecahedron: zome model | matematicasVisuales
Dodecahedron: zome model | matematicasVisuales

Kepler showed us how to put a cube inside a regular dodecahedron:

Dodecahedron: Kepler drawing of a cube inside a dodecahedron | matematicasVisuales

A cube can be inscribed in a dodecahedron so that each edge of the cube lies in a face of the dodecahedron and joins two alternate vertices of that face. (Ball and Coxeter, p. 131)

Dodecahedron:  | matematicasVisuales

A regular dodecahedron can be inscribed into an sphere. To calculate the radius of this sphere we can think that its center is the center of the cube, and the cube has side length equal to phi if the side lenght of the dodecahedron is 1. Or you can see this figure. Then

Dodecahedron: radius of the circunscribed sphere, zome model | matematicasVisuales
Dodecahedron: radius of the circunscribed sphere, zome model | matematicasVisuales
Dodecahedron:  radius of the circunscribed sphere, zome model| matematicasVisuales

To calculate the volume of a dodecahedron, notice that the volume of a dodecahedron of side length 1 is one eighth the volume of a dodecahedron of side length 2, it is to say, this body:

Dodecahedron: the volume of a dodecahedron of side length 1 is one eighth the volume of a dodecahedron of side length 2 | matematicasVisuales

With the help of Zome we can study the volume of a regular dodecahedron.

Origami: building a dodecahedron with the Tom Hull's PHiZZ module:

Dodecahedron: origami, Tom Hull module | matematicasVisuales

Origami: building a dodecahedron with the penultimate module (learn more about the penultimate module):

Dodecahedron: origami, penultimate module | matematicasVisuales

Dodecahedron and icosahedron are dual polyhedra.

Dodecahedron: dodecahedron and icosahedron duality, zome model | matematicasVisuales

You can build a dodecahedron and an icosahedron with corresponding edges perpendicularly bisecting each other. Then, the common part is an icosidodecahedron. (Ball and Coxeter, 136)

Dodecahedron: dodecahedron and icosahedron duality, zome model  | matematicasVisuales
Dodecahedron: Icosidodecahedron, paper model with rubber bands | matematicasVisuales

If you want to learn how to build this polyhedron or to download templates go to Resources: how to build polyhedra with paper and rubber bands.

Icosidodecahedron with twenty triangular faces and twelve pentagonal faces it is a toy for your pet:

Dodecahedron: Icosidodecahedron, a toy for your pet | matematicasVisuales

Vertices of the dodecahedron (20) and vertices of the icosahedron (12) are the vertices of a rhombic triacontahedron:

Dodecahedron: dodecahedron and icosahedron duality, triacontahedron, zome model | matematicasVisuales

Triacontahedron and icosidodecahedron are dual polyhedra just as the dodecahedron and the icosahedron are dual polyhedra.

Five tetrahedra inside a dodecahedron. One of the most wonderful constructions I have ever made. The first time I read about this construction was in Magnus J. Wenninger's book 'Polyhedron models for the classroom' and I made this model more than twenty years ago. You can download a template from Korthalsates site.

Dodecahedron: five tetrahedra inside a dodecahedron, paper model | matematicasVisuales

Origami: Five tetrahedra inside a dodecahedron. See instructions on how to buil this origami model:

Dodecahedron: five tetrahedra inside a dodecahedron, origami | matematicasVisuales

Zome: Five tetrahedra inside a dodecahedron:

Dodecahedron: five tetrahedra inside a dodecahedron, Zome model | matematicasVisuales
Dodecahedron: A tetrahedron inside a dodecahedron, paper model, Zome model | matematicasVisuales

The side length of the tetrahedron inside a dodecahedron of side length 1 is:

Dodecahedron: A tetrahedron inside a cube inside a dodecahedron, Zome model | matematicasVisuales

Tubes: Five tetrahedra inside a dodecahedron:

Five cubes inside a dodecahedron:

Dodecahedron: five cubes inside a dodecahedron, paper model | matematicasVisuales
Dodecahedron: five cubes inside a dodecahedron, paper model | matematicasVisuales

More models based in the dodecahedron, a great dodecahedron, a Kepler-Poinsot polyhedron:

Dodecahedron: a great dodecahedron, a Kepler-Poinsot polyhedron, paper model | matematicasVisuales

REFERENCES

W.W. Rouse Ball and H.S.M. Coxeter - 'Matematical Recreations & Essays', The MacMillan Company, 1947.
Magnus Wenninger - 'Polyhedron Models', Cambridge University Press.
Peter R. Cromwell - 'Polyhedra', Cambridge University Press, 1999.
Origami: dodecahedron with the Tom Hull's PHiZZ module.
Origami: Penultimate module.
Kepler, 'Hamonices Mundi - The Harmony of the World', (1619) (Read the original book at Posner Memorial Collection).

MORE LINKS

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 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.