matematicas visuales visual math
Equiangular Spiral. Rotation

Coxeter wrote about the Equiangular Spiral (Logarithmic Spiral):

"This curve was first recognized by Descartes. Jacob Bernouilli found it so fascinating that he arranged to have it engraved on his tombstone with the inscription

Eadem mutata resurgo

These words (which E.T. Bell translates as "Though changed I shall arise the same") express a remarkable consequence of the way the curve can be shifted along itself by a dilative rotation: any dilatation has the same effect on it as a rotation, and vice versa." (Coxeter, p. 125)

Coxeter said that Steinhaus describes this property as an optical illusion: "If we turn the book about the vertex, the spiral seems to grow larger or smaller. Two spirals having the same constant angle with the radii are congruent." (Steinhaus, p. 132). Steinhaus wrote about the relation between Logarithmic Spirals and pursuing paths.

Dilatation and rotation of an Equiangular Spiral: Using position vectors you can change the spiral | matematicasVisuales
Using position vectors you can change the spiral.
Dilatation and rotation of an Equiangular Spiral: another example | matematicasVisuales
This is another example of equiangular spiral.


Coxeter - Introduction to Geometry (John Whiley and sons)
Hugo Steinhaus, Mathematical Snapshots, Dover Publications (third edition, 1999)
We can read some pages of this book in Google Books: Mathematical Snapshots by Hugo Steinhaus.
D'Arcy Thompson - On Growth and Form. (Cambridge University Press)


Dilative rotation
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
The golden rectangle and two equiangular spirals
Two equiangular spirals contains all vertices of golden rectangles.
The golden spiral
The golden spiral is a good approximation of an equiangular spiral.
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.
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.
Durer and transformations
He studied transformations of images, for example, faces.
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.
Multiplying two complex numbers
We can see it as a dilatative rotation.