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
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.
Using position vectors you can change the spiral.
This is another example of equiangular spiral.
REFERENCES
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)
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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.
A golden rectangle is made of an square an another golden rectangle. These rectangles are related through an dilative rotation.
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
