Golden spiral

Approximate and true golden spirals: the green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a golden spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of a larger square to the next smaller square is in the golden ratio.

In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio.^[1] That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.

Formula

The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of the growth factor b:^[2]

$r = ae^{b\theta}\,$

or

$\theta = \frac{1}{b} \ln(r/a),$

with e being the base of Natural Logarithms, a being an arbitrary positive real constant, and b such that when θ is a right angle (a quarter turn in either direction):

$e^{b\theta_\mathrm{right}}\, = \varphi$

Therefore, b is given by

$b = {\ln{\varphi} \over \theta_\mathrm{right}}.$

The numerical value of b depends on whether the right angle is measured as 90 degrees or as $\textstyle\frac{\pi}{2}$ radians; and since the angle can be in either direction, it is easiest to write the formula for the absolute value of $b$ (that is, b can also be the negative of this value):

A Fibonacci spiral approximates the golden spiral using quarter-circle arcs inscribed in squares of integer Fibonacci-number side, shown for square sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.

$|b| = {\ln{\varphi} \over 90} \doteq 0.0053468\,$ for θ in degrees;

$|b| = {\ln{\varphi} \over \pi/2} \doteq 0.3063489\,$ for θ in radians  A212225.

An alternate formula for a logarithmic and golden spiral is:^[3]

$r = ac^{\theta}\,$

where the constant c is given by:

$c = e^b\,$

which for the golden spiral gives c values of:

$c = \varphi ^ \frac{1}{90} \doteq 1.0053611$

if θ is measured in degrees, and

$c = \varphi ^ \frac{2}{\pi} \doteq 1.358456.$  A212224

if θ is measured in radians.

Approximations of the golden spiral

Lithuanian coin

There are several similar spirals that approximate, but do not exactly equal, a golden spiral.^[4] These are often confused with the golden spiral.

For example, a golden spiral can be approximated by first starting with a rectangle for which the ratio between its length and width is the golden ratio. This rectangle can then be partitioned into a square and a similar rectangle and the rectangle can then be split in the same way. After continuing this process for an arbitrary amount of steps, the result will be an almost complete partitioning of the rectangle into squares. The corners of these squares can be connected by quarter-circles. The result, though not a true logarithmic spiral, approximates a golden spiral (See the first image).

Another approximation is a Fibonacci spiral, which is constructed similarly to the above method except that you start with a rectangle partitioned into 2 squares and then in each step add to the rectangle's longest side a square of the same length. Since the ratio between consecutive Fibonacci numbers approaches the golden ratio as the Fibonacci numbers approach infinity, so too does this spiral get more similar to the previous approximation the more squares are added. (See the second image).

Spirals in nature

Approximate logarithmic spirals can occur in nature (for example, the arms of spiral galaxies^[5] or phyllotaxis of leaves); golden spirals are one special case of these logarithmic spirals. A recent in-depth analysis of spirals observed in mouse corneal epithelial cells indicated that some can be characterized by the golden spiral, and others by the logarithmic spiral governing the M51 Galaxy.^[6] It is sometimes stated that spiral galaxies and nautilus shells get wider in the pattern of a golden spiral, and hence are related to both φ and the Fibonacci series.^[7] In truth, spiral galaxies and nautilus shells (and many mollusk shells) exhibit logarithmic spiral growth, but at a variety of angles usually distinctly different from that of the golden spiral.^[8][9][10] This pattern allows the organism to grow without changing shape.

See also

• Golden angle
• Golden ratio
• Golden rectangle
• Logarithmic spiral

References

1. Chang, Yu-sung, "Golden Spiral", The Wolfram Demonstrations Project.
2. Priya Hemenway (2005). Divine Proportion: Φ Phi in Art, Nature, and Science. Sterling Publishing Co. pp. 127–129. ISBN 1-4027-3522-7.
3. Klaus Mainzer (1996). Symmetries of Nature: A Handbook for Philosophy of Nature and Science. Walter de Gruyter. pp. 45, 199–200. ISBN 3-11-012990-6.
4. Charles B. Madden (1999). Fractals in Music: introductory mathematics for musical analysis. High Art Press. pp. 14–16. ISBN 0-9671727-6-4.
5. Midhat Gazale (1999). Gnomon: From Pharaohs to Fractals. Princeton University Press. p. 3. ISBN 9780691005140.
6. Rhee, Jerry; Nejad, Talisa Mohammad; Comets, Olivier; Flannery, Sean; Gulsoy, Eine Begum; Iannaccone, Philip; Foster, Craig (2 January 2015). "Promoting convergence: The Phi spiral in abduction of mouse corneal behaviors". Complexity 20 (3): 22–38. doi:10.1002/cplx.21562.
7. For example, these books: Jan C. A. Boeyens (2009). Chemistry from First Principles. Springer. p. 261. ISBN 9781402085451., P D Frey (2011). Borderlines of Identity: A Psychologist's Personal Exploration. Xlibris Corporation. ISBN 9781465355850., Russell Howell and James Bradley (2011). Mathematics Through the Eyes of Faith. HarperCollins. p. 162. ISBN 978-0062024473., Charles Seife (2000). Zero: The Biography of a Dangerous Idea. Penguin. p. 40. ISBN 978-0140296471., Sandra Kynes (2008). Sea Magic: Connecting With the Ocean's Energy. Llewellyn Worldwide. p. 100. ISBN 9780738713533., Bruce Burger (1998). Esoteric Anatomy: The Body as Consciousness. North Atlantic Books. p. 144. ISBN 9781556432248.
8. David Darling (2004). The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes. John Wiley & Sons. p. 188. ISBN 9780471270478.
9. Devlin, Keith (May 2007). "The myth that will not go away".
10. Peterson, Ivars (2005-04-01). "Sea Shell Spirals". Science News. Society for Science & the Public.