The Golden Ratio and Leaves

written by Deborah S.

In our previous article, we learned about the Fibonacci Sequence.  Today, let’s discuss another mathematical concept that is lesser known, yet also intricately linked: the Golden Ratio.  

Just as you may have heard of π (pi), which is equal to 3.14159…, and perhaps another constant, e, which is equal to 2.1718…, the Golden Ratio is an irrational number given the name φ (phi) and is equal to 1.618… You might wonder: why is it called the Golden Ratio if it’s actually a number? An astute observation, dear reader!  Phi is the result of a ratio. Mathematically stated, the Golden Ratio is a/b for any numbers “a” and “b” where (a+b)/a = a/b.  No matter what numbers “a” and “b” you find that satisfy this equation, a/b always equals 1.618…

This relationship can be difficult to grasp in text, but might be easier to understand visually:

graphic by Stannered
graphic by Kyle Stout

This unique value is related to the Fibonacci Sequence in that the result of dividing each Fibonacci number by its predecessor gets closer and closer to φ as one goes deeper in the sequence. Here are the first several examples:

Fib Sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 

The ratio of Fib Numbers:

  • 1/1 = 1
  • 2/1 = 2
  • 3/2 = 1.5
  • 5/3 = 1.66…
  • 8/5 = 1.6
  • 13/8 = 1.625
  • 21/13 = 1.615…
  • 34/21 = 1.619… 
  • 55/34 = 1.618… 
  • 89/55 = 1.618…

And so on.

Given the visual representation of the Golden Ratio, it is perhaps not surprising that other Golden visualizations stem from this ratio, such as the Golden Rectangle and Golden Spiral. The Golden Rectangle is a figure whose long side is of length (a+b) and the shorter side is length a.

A golden rectangle - graphic by Ahecht

Putting a line inside the rectangle perpendicular to the (a+b) side at length a, creates another Golden Rectangle within the larger one.  Then, connecting diagonal corners of the “a by a” squares created by this recursion with quarter circles creates the Golden Spiral.

The Golden Spiral created inside Golden Rectangles - photo by Jahbor

Cool math, but what does any of this have to do with nature?

You may have heard claims that the Golden Ratio is prevalent in natural spirals, such as seashells, or various lengths of fingers, or that faces whose measurements fit the spiral are more beautiful than others.

Unfortunately, these claims seem to be largely unsubstantiated, or, at the very least, difficult to definitively verify since the measurements of real objects vary from subject to subject and rarely adhere to clean numbers.

Of course, just because an object does not meet strict Golden Ratio specifics does not mean observations about that object are not mathematical.  Mathematicians love to claim beautifully symmetric and geometrically suggestive shapes and solids as mathematical objects. So, continue to think of math when you see a spiral or a pretty face, but maybe they’re just not as “Golden” as we once thought.

However, one concept connected to the Golden Ratio does seem to appear in nature: the Golden Angle. The measurement is the angle needed to cut the circumference of a circle into lengths a and b for the Golden Ratio.  Such an angle is about 137.5°.

A representation of the Golden Angle - graphic by Dicklyon

Furthering this notion are Golden Sections. This term refers to the angles applied to just 90° of a circle into Golden Ratio segments. The angles suss out to be 55.6° and 34.4°. Together, they add to 90° but cut that right angle into an a and b section that fit the Golden Ratio.

Visualization of a Golden Section - graphic by Deborah S.

Golden Angles and Golden Sections allow us to understand how tree branches and leaves grow.  

Recall that, in the previous article, we discussed how branches or leaves space themselves around trunks or branches based on ratios of Fibonacci numbers.  If we closely examine the angles that these branches make with trunks and these leaves make with branches, Golden Angles and Sections pop up in many trees!  

Though a study showed that some species only found correspondence between Golden Angles/Sections in  58%  of branches (the pear tree), observations for other tree species were much higher, often over 80%!

Golden Angles and Sections showing up in tree branches - graphic from Progress in Natural Science

The observers hypothesize these angles might aid in airflow and sunlight exposure.  Unlike Fibonacci numbers in branch spiraling, no enterprising seventh grader has yet figured out the reason trees display these angles and sections. No matter the purpose, we have a new meaning for the Golden Rule!

While some popular Golden perceptions may have been overblown or untrue, perhaps we look for too strict an adherence to these Golden Rules.  After all, the “perfect” triangles, cubes, and circles of our computers and graph paper do not exist in the real world. Is 80% adherence to the Golden Angle and Section enough to link the phenomenon to actual tree growth? Perhaps the Golden Ratios represent perfect efficiency, and various pieces of nature “strive” via evolution toward a perfection they can only approximate.

Further Reading and Exploration

Modeling golden section in plants – Progress in Natural Science

Golden Ratio – MathWorld

Golden Angle – MathWorld

The Myth That Will Not Go Away – Keith Devlin

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