FIBONACCI_SPIRAL
The Fibonacci Spiral

FIBONACCI_SPIRAL is a MATLAB library which displays points on a Fibonacci spiral, suggesting the arrangement of seeds in a sunflower, for instance.

The spirals exhibited in nature can be modeled by a simple spiral of points generated, in polar coordinates, by starting at R = 0, A = 0, and then repeated incrementing R by dR = 1, and A by an angle dA of about 137.5 degrees, or, more precisely, by 2 * PI * ( PHI - 1 ) / PHI radians, where PHI is the Golden Ratio, equal to (1+sqrt(5))/2.

The spiral along which the points occur sequentially is not, in fact, what the viewer first notices. Instead, an image of the spiral will seem to exhibit two families of spirals emerging at a sharp angle from the center. One family rotates clockwise, the other counter-clockwise.

The program fibonacci_spiral simply displays the blue dots representing the points; in this image, the Fibonacci spirals are evident. The program fibonacci_spiral_connected includes a red spiral line that connects the dots in the sequence in which they are generated. Plots of this kind are not very legible after 100 or so points.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

LORENZ_SIMULATION, a MATLAB program which solves the Lorenz equations and displays the solution, for various starting conditions.

MANDELBROT, a MATLAB program which generates an ASCII PPM image of the Mandelbrot set;

Reference:

1. Ian Stewart,
   Nature's Numbers: The Unreal Reality of Mathematics,
   Oxford, 2004,
   ISBN: 0198613369,
   LC: QA93.S737.

Source Code:

• fibonacci_spiral.m, draws blue dots representing points on a Fibonacci spiral.
• fibonacci_spiral_connected.m, draws blue dots on a Fibonacci spiral, and a red spiral line to indicate the order in which the dots were generated.

Examples and Tests:

• spiral_50.png, a spiral of 50 blue dots;
• spiral_50_connected.png, a spiral of 50 blue dots, with a red connecting line;
• spiral_100.png, a spiral of 100 blue dots;
• spiral_100_connected.png, a spiral of 100 blue dots with a red connecting line;
• spiral_500.png, a spiral of 500 blue dots;
• spiral_1000.png, a spiral of 1000 blue dots;

You can go up one level to the MATLAB source codes.