matematicas visuales visual math
The Golden Rectangle

A rectangle can be divided in two pieces: a square with the smaller side and another rectangle.

For a certain proportion of the sides of the initial rectangle, by that procedure we obtain a similar rectangle to the previous one.

Then we have a golden rectangle.

Golden Rectangle: golden ratio | matematicasVisuales
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.

If we start from a golden rectangle that procedure can be repeated indefinitely.

Golden Rectangle: infinite golden rectangles | matematicasVisuales

This golden or divine proportion can be expressed in this way: the ratio of the larger part to the smaller is equal to the ratio of the whole to the larger part.

Then we obtain the golden number

This infinite process suggests that the sides of a golden rectangle are incommensurable or, otherwise, that the golden number is irrational.

We can see that rectangle ABDF and rectangle CDFH are similar and that CDFH is rotated (a quarter-turn)

Golden Rectangle: A pair of ortogonal lines| matematicasVisuales


We call O the intersection point.

Now we can consider the line OC:

Golden Rectangle: the line OC bisects | matematicasVisuales


Then OC bisects the (right) angle BOD


O is in the line CG. The same for AE and then

Golden Rectangle: A second pair of orthogonal lines | matematicasVisuales

These four straight lines, orthogonal in pairs, they contain all the vertices of those infinite rectangles. Each one of these pairs of straight lines bisects the other pair.

Golden Rectangle: four lines, orthogonal in pairs, that they contain all the vertices of those infinite rectangles | matematicasVisuales


Coxeter H. S. M. - Introduction to Geometry (John Whiley and Sons, Second Edition, 1969)


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 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.
Resources: The golden rectangle and the icosahedron
With three golden rectangles you can build an icosahedron.
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
Regular dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
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 .
Durer and transformations
He studied transformations of images, for example, faces.
Volume of a regular dodecahedron
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
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.
Equiangular spiral
In an equiangular spiral the angle between the position vector and the tangent is constant.
Dilation and rotation in an equiangular spiral
Two transformations of an equiangular spiral with the same general efect.
Dilative rotation
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
Standard Paper Size DIN A
There is a standarization of the size of the paper that is called DIN A. Successive paper sizes in the series A1, A2, A3, A4, and so forth, are defined by halving the preceding paper size along the larger dimension.