The **golden spiral** is a very special geometric shape; a logarithmic spiral with the golden ratio φ for its growth factor.

The golden ratio is the number:

The golden spiral gets get wider by a factor of φ every quarter turn it makes; the ratio by which it gets wider is always a golden ratio. The image below illustrates how these spirals are self-similar, which means that the shape is repeated—infinitely— when magnified.

## Applications of and Approximations of the Golden Spiral

There are many spirals in nature—the nautilus’ shell, the arms of spiral galaxies—that appear to be logarithmic with ratios close to the golden ratio. None of them, though, are exact golden spirals; Some are approximations.

## Writing an Equation for the Golden Spiral

The easiest way to write an equation for these spirals is with polar coordinates. That way, the equation for a golden spiral with an initial radius of one will be:

A more general formula, where *a* is the initial radius of the spiral, is the following:

The growth factor b is defined as b = (ln φ) / Θ_{right}, where Θ_{right} is a right angle. If we’re working with degrees, Θ_{right} will be 90, and the absolute value of b will be 0.0053468. If we’re working with radians, Θ_{right} will be 0.3063489.

## References

Weisstein, Eric W. Golden Logarithmic Spirals. From MathWorld–A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/GoldenSpiral.html on January 19, 2018

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