matematicas visuales visual math

The regular dodecahedron is a platonic polyhedron very well known from Antiquity.

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

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.

Albert Durer published a plane net of a dodecahedron for the first time in 1525.

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 .

Kepler was interested in this wonderful geometric body (For example, you can see this drawing in his book 'Harmonices 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

We have already studied a lot of properties of this beautiful polyhedron in matematicasVisuales:

Regular dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.

And we know how to calculate its volume:

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

In this page we are going to study some relations between the dodecahedron and the cube.

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: a cube inside a dodecahedron | matematicasVisuales

Kepler showed us this construction and saw the dodecahedron as a cube with six roofs added:

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

In the interactive application in this page you can see how a dodecahedron can be fold into a cube:

Dodecahedron and cube: dodecahedron folding into a cube | matematicasVisuales
Dodecahedron and cube: dodecahedron folding into a cube | matematicasVisuales
Dodecahedron and cube: dodecahedron folding into a cube | matematicasVisuales
Dodecahedron and cube: dodecahedron folding into a cube | matematicasVisuales
Dodecahedron and cube: dodecahedron folding into a cube | matematicasVisuales

I encourage you to build your own polyhedron:

Dodecahedron and cube: building | matematicasVisuales
Dodecahedron and cube: building | matematicasVisuales
Dodecahedron and cube: building | matematicasVisuales
Dodecahedron and cube: building | matematicasVisuales
Dodecahedron and cube: building | matematicasVisuales
Dodecahedron and cube: building | matematicasVisuales
Dodecahedron and cube: building | matematicasVisuales

To calculate the volume of a dodecahedron of side length 1 you can remember some properties of the golden ratio:

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 a dodecahedron is the volume of a cube plus six times the volume of a roof.

The side length of the cube is equal to a diagonal of the pentagon, then:

Dodecahedron and cube: cube inside | matematicasVisuales

Now we are going to calculate the volume of one roof.

Dodecahedron and cube: to calculate the volume of one roof | matematicasVisuales

One property of this roof is that its height is 1/2.

We can build a roof with Zome:

Dodecahedron and cube: roof built with zome | matematicasVisuales
Dodecahedron and cube: roof built with zome | matematicasVisuales
Dodecahedron and cube: roof built with zome | matematicasVisuales

We can consider two parts of the roof:

Dodecahedron and cube: to calculate the volume of one roof | matematicasVisuales
Dodecahedron and cube: to calculate the volume of one roof | matematicasVisuales
Dodecahedron and cube: to calculate the volume of one roof | matematicasVisuales
Dodecahedron and cube: to calculate the volume of one roof | matematicasVisuales

Then the volume of one roof is:

And the volume of a dodecahedron of side lenght 1 is

Five cubes inside a dodecahedron:

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

REFERENCES

Hugo Steinhaus - 'Mathematical Snapshots' - Oxford University Press - Third Edition.
Magnus Wenninger - 'Polyhedron Models', Cambridge University Press.
Peter R. Cromwell - 'Polyhedra', Cambridge University Press, 1999.
H.Martin Cundy and A.P. Rollet, 'Mathematical Models', Oxford University Press, Second Edition, 1961.
W.W. Rouse Ball and H.S.M. Coxeter - 'Matematical Recreations & Essays', The MacMillan Company, 1947.

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.
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.