matematicas visuales visual math

An icosahedron is a platonic solid. Its faces, like the thetrahedron and the octahedron, are equilateral triangles.

It is very easy to build one using its plane net, as Dürer taught us in his book 'Underweysung der Messung' ('Four Books on Measurement') published in 1525:

icosahedron: plane net of an icosahedron by Durer | matematicasVisuales

Or we can make this beautiful lamp:

icosahedron: beautiful lamp | matematicasVisuales
icosahedron: how to make a beautiful lamp | matematicasVisuales

In this page we are going to study the relation between the golden ratio and the icosahedron.

The golden rectangle is a beautiful construction also related with some spirals and the dodecahedron.

We can put three golden rectangles (in three mutually orthogonal planes) and make a well known construction. These rectangles have 12 vertices. The distance between any pair of neighbouring points is equal to the short side of one golden rectangle. Then, these 12 points coincide with the 12 vertices of an icosahedron.

icosahedron: Three golden rectangles and the icosahedron | matematicasVisuales

By symmetry, these twelve triangles are isosceles triangles but, they are equilateral?

icosahedron: three golden rectangles; are the triangles equilateral? | matematicasVisuales

One easy way to convince yourself about that is calculating the height of one of these triangles:

icosahedron: three golden rectangles; are the triangles equilateral?, some calculations | matematicasVisuales

Remember that

Then

And h is the height of a equilateral triangle of side length 2:

Three planes contains the 20 vertices of an icosahedron. Their edges form Borromean rings.

You can made the Borromean rings using three balloons. This constrution is inspired in a video about the logo of the International Mathematical Union (IMU) designed by John Sullivan (The Borromean Rings: a new logo for the IMU).

icosahedron: Borromean Rings with balloons | matematicasVisuales

We are going to calculate the volumen of an icosahedron of side length 2:

An icosahedron has twenty equilateral triangles.

icosahedron: calculating its volume; one face area | matematicasVisuales

The area of one of the twenty equilateral triangles that the icosahedron of side 2 have is:

The volume of an icosahedron of side 1 is one eigth of the volume of an icosahedron of side 2.

The volume of the icosahedron of side 1 is the same as the volume of two and a half pyramids. We need to calculate the height of one of this pyramids: the distance between the center and one triangular face.

icosahedron: calculating its volume; the distance between de center and one face | matematicasVisuales

There are two similar triangles and we can write:

icosahedron: calculating its volume;  a formula to calculate the height of a pyramid inside the icosahedron| matematicasVisuales

Then the volume of one pyramid is:

And the volume of an icosahedron of side 1 is:

icosahedron:  | matematicasVisuales

The six vertices of an octahedron lies in three mutually ortogonal squares. We can put inside each square a golden rectangle (for example, using tubes):

icosahedron: a square and a golden rectangle with tubes | matematicasVisuales

The same construction with Zome:

icosahedron: a square and a golden rectangle with zome | matematicasVisuales

Then, if the twelve edges of an octahedron are divided in the golden ratio (in some order) these vertices are the vertices of an icosahedron.

icosahedron: octahedron and icosahedron | matematicasVisuales

It is not difficult to calculate the volume of the octahedron and the volume of the six bipyramids and get the volume of the icosahadron.

icosahedron: octahedron and icosahedron, six bipyramids | matematicasVisuales

Icosahedron and octahedron, one inside each other, with Zome: .

An icosahedron, an octahedron and a tetrahedron, one inside each other, made using using tubes:

We can build the structure of an icosahedron using six sticks and six elastic bands. It is one of the simplest examples of using "tensegrity".

icosahedron: tensegrity | matematicasVisuales
icosahedron: tensegrity | matematicasVisuales
icosahedron: tensegrity | matematicasVisuales

REFERENCES

Coxeter - Introduction to Geometry (John Whiley and sons)

MORE LINKS

Pythagoras Theorem: Euclid's demonstration
Demonstration of Pythagoras Theorem inspired in Euclid.
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.
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.
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 .
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.
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 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.
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).
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.
Dilation and rotation in an equiangular spiral
Two transformations of an equiangular spiral with the same general efect.
Equiangular spiral
In an equiangular spiral the angle between the position vector and the tangent is constant.
Plane developments of geometric bodies (1): Nets of prisms
We study different prisms and we can see how they develop into a plane net. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (3): Cylinders
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (5): Pyramid and pyramidal frustrum
Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.
Plane developments of geometric bodies (7): Cone and conical frustrum
Plane developments of cones and conical frustum. How to calculate the lateral surface area.
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 .