matematicas visuales visual math

We have already study the definition and several properties of equiangular spirals (or logarithmic spirals).

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.

In this page we are going to see that there are infinitely many equiangular spirals through two points.

Some are anticlockwise:

Equiangular Spiral through two points: anticlockwise, counterclockwise | matematicasVisuales

Or clockwise:

Equiangular Spiral through two points: clockwise | matematicasVisuales

We can consider how many turns the spiral draw from one point to the other. In the next applet you can change how many rounds the spiral turns.

Equiangular Spiral through two points: anticlockwise, round several times| matematicasVisuales
Equiangular Spiral through two points: clockwise, round several times | matematicasVisuales

Equiangular spirals are connected with natural growth. Some examples of spirals:

You can play with this colorful spiral.

Equiangular Spiral through two points: colorful spiral | matematicasVisuales

REFERENCES

Coxeter - Introduction to Geometry (John Whiley and sons)
Steinhaus - Mathematical Snapshots.
D'Arcy Thompson - On Growth and Form. (Cambridge University Press)

MORE LINKS

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