matematicas visuales visual math

If we start with a golden rectangle we can cut a square and we get a small golden rectangle.

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.

We can repeat this process and we have this construction:

Golden Rectangle: infinite golden rectangles | matematicasVisuales

Drawing up an arc of circumference in each square we get a golden spiral (also called Durer's Spiral).

Golden Rectangle: Durer spiral, golden spiral | matematicasVisuales
Golden Ratio: golden spiral, Drawing up an arc of circumference in each square we get a golden spiral | matematicasVisuales

We have already calculate the polar coordinates of several points and we got a sequence of points.

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.
Golden Rectangle: a sequence of points | matematicasVisuales

These vertices also belong to an equiangular spiral

Equiangular spiral
In an equiangular spiral the angle between the position vector and the tangent is constant.

We are going to calculate the equation of this equiangular spiral.

The general formula for an equiangular spiral is:

If n=1 then

We can write for point C

The equation of this equiangular spiral is

Golden Rectangle: Durer Spiral or Golden Spiral and Equiangular Golden Spiral | matematicasVisuales

Golden Ratio: Durer's Spiral, golden spiral, vertices belong to a equiangular spiral | matematicasVisuales

"This true spiral is closely approximated by the artificial spiral formed by circular quadrants inscribed in the successive squares. (But the true spiral cuts the sides of the squares at very small angles, instead of touching them)" Coxeter.

Golden Rectangle: The Golden Equiangular Spiral cut the sides of the squares | matematicasVisuales
Golden Rectangle: The Golden Equiangular Spiral cut the sides of the squares | matematicasVisuales
Golden Rectangle: The Golden Equiangular Spiral cut the sides of the squares | matematicasVisuales
Golden Ratio: golden spiral, The true equiangular spiral cuts the sides of the squares at very small angles, instead of touching them | matematicasVisuales

In the next applet you can have another look about the relation between the golden rectangle and the dilative rotation. You can see how several points follow the path of an equiangular spiral.

Golden Spiral in a furniture designed and made by Roberto Cardil (pine wood).

Golden Ratio: golden spiral, Furniture with a golden spiral | matematicasVisuales
Golden Ratio: golden spiral, Furniture with a golden spiral | matematicasVisuales

REFERENCES

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

MORE LINKS

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.
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.
The golden spiral
The golden spiral is a good approximation of an equiangular spiral.
The golden rectangle
A golden rectangle is made of an square and another golden rectangle.
Dilative rotation
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
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.
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
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.