matematicas visuales visual math

We have already study the golden ratio and several properties of the 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.

When we divide a golden rectangle in a square and another golden rectangle this new rectangle is similar to the initial one. We can repeat this procedure infinitely and we get a geometric sequence of rectangles.

There is a special point O

Golden Rectangle: | matematicasVisuales

From one rectangle to get the next one we can consider the transformation that is the product of a dilatation of center O

and a rotation (a negative quater-turn about O)

Golden Rectangle: | matematicasVisuales

This transformation is a dilative rotation.

Dilative rotation
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.

In the next applet you can see a continuous dilative rotation:

Now we are going to calculate polar coordinates of several points.

The pole is O

Taking OE as the initial line and the length OE as the unit of measurement, so that E is

From E to C the dilative rotation about O has ratio of magnification and it is a positive quarter-turn.

Golden Rectangle: | matematicasVisuales

In general, we can calculate a sequence of points

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.
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.
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.
Multiplying two complex numbers
We can see it as a dilatative rotation.
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.
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.