Very nice construction of an icosahedron. You only need three golden rectangles.

To learn how to draw a golden rectangle is a beautiful experience. The golden number is related with the regular pentagon:

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.

I think that the best material to build this model is wood. But it is very simple to build one using cardboard. You can download and print this drawing of several golden rectangles:

Now we can build one model using wood or DM. This example has detachable parts:

We start drawing:

We can use different materials:

If you have access to a laser cutting machine you can reproduce the model as many times as you want. The result is very accurate. Sara San Gregorio, who develops the proyect Microarquitectura in MediaLab-Prado in Madrid, produced a digital file with the drawing and cut this model:

You can se the relation with the diagonal of a pentagon.

This construction is related with the Borromean Rings.

Borromean Rings made using rubber balloons. Inspired in the logo of the International Mathematical Union (IMU) designed by John Sullivan (The Borromean Rings: a new logo for the IMU)

It is very easy to build an icosahedron using cardboard. Durer showed us how to draw its plane net.

MORE LINKS

Resources: Building Polyhedra with cardboard (Plane Nets)

Using cardboard you can draw plane nets and build polyhedra.

Resources: How to build polyhedra using paper and rubber bands

A very simple technique to build complex and colorful polyhedra.

Resources: Building polyhedra gluing discs

Simple technique to build polyhedra gluing discs made of cardboard or paper.

Resources: Acona Biconbi, designed by Bruno Munari

Italian designer Bruno Munari conceived 'Acona Biconbi' as a work of sculpture. It is also a beautiful game to play with colors and shapes.

Resources: Modular Origami

Modular Origami is a nice technique to build polyhedra.

Resources: Building polyhedra using tubes

Examples of polyhedra built using tubes.

Resources: Tensegrity

Examples of polyhedra built using tensegrity.

Resources: Building polyhedra using Zome

Examples of polyhedra built using Zome.

Building polyhedra. Basic techniques: Taller de Talento Matemático de Zaragoza (Spanish)

Material for a session about polyhedra (Zaragoza, 13th Abril 2012).

Construcción de poliedros. Cuboctaedro y dodecaedro rómbico: Taller de Talento Matemático de Zaragoza 2014 (Spanish)

Material for a session about polyhedra (Zaragoza, 9th May 2014). Simple techniques to build polyhedra like the tetrahedron, octahedron, the cuboctahedron and the rhombic dodecahedron. We can build a box that is a rhombic dodecahedron.

Cube, octahedron, tetrahedron and other polyhedra: Taller de Talento Matemático Zaragoza,Spain, 2014-2015 (Spanish)

Material for a session about polyhedra (Zaragoza, 7th November 2014). We study the octahedron and the tetrahedron and their volumes. The truncated octahedron helps us to this task. We build a cubic box with cardboard and an origami tetrahedron.

Duality: cube and octahedron. Taller de Talento Matemático de Zaragoza, Spain. 2015-2016 XII edition (Spanish)

Material for a session about polyhedra (Zaragoza, 23rd Octuber 2015) . Building a cube with cardboard and an origami octahedron.

The Cuboctahedron and the truncated octahedron. Taller de Talento Matemático de Zaragoza, Spain. 2016-2017 XIII edition (Spanish)

Material for a session about polyhedra (Zaragoza, 21st October 2016). Instructions to build several geometric bodies.

Microarquitectura and polyhedra (Spanish)

Microarquitectura is a construction game developed by Sara San Gregorio. You can play and build a lot of structures modelled on polyhedra.

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 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 .

Regular dodecahedron

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

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 spiral

The golden spiral is a good approximation of an equiangular spiral.

Hexagonal section of a cube

We can cut in half a cube by a plane and get a section that is a regular hexagon. Using eight of this pieces we can made a truncated octahedron.

A truncated octahedron made by eight half cubes

Using eight half cubes we can make a truncated octahedron. The cube tesselate the space an so do the truncated octahedron. We can calculate the volume of a truncated octahedron.

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.

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.

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.