matematicas visuales visual math

We can see that some vertices of the golden rectangles of this construction are in an equiangular spiral.

Golden Ratio: The golden rectangle and one equiangular spirals | matematicasVisuales

The golden spiral
The golden spiral is a good approximation of an equiangular spiral.
Coxeter ask to prove ("Introduction to Geometry" p 196) that the other vertices also are in another equiangular spiral.

Golden Ratio: The golden rectangle and two equiangular spirals | matematicasVisuales
Golden Ratio: The golden rectangle and two equiangular spirals | matematicasVisuales

This second golden spiral is homotetic to the initial golden spiral.

This second golden spiral is congruent by rotation to the initial golden spiral.

In the next animation we can see the rotation that transforms one spiral into the other.

Golden Ratio: The golden rectangle and two equiangular spirals, rotation | 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.
Resources: The golden rectangle and the icosahedron
With three golden rectangles you can build an icosahedron.
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.
The golden rectangle
A golden rectangle is made of an square and another golden rectangle.
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.
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 .
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.
Dilative rotation
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.