Kate Whittington, kate.whittington@bgci.org |
12/07/13 | London
**The Golden Ratio & the Fibonacci Sequence**

**PHOTO CREDITS:**

#### Cauliflower: from Flickr by Amber Case

#### Golden Angle: from Wikimedia Commons

#### Sunflower: from Wikimedia Commons by Dr. Helmut Haß, Koblenz

#### Cactus images: from Wikimedia Commons 8 spirals & 13 spirals by Marozols

#### Pine cone: from Wikimedia Commons

#### Arabidopsis: from Flickr by Tico

Always looking to increase and inspire the up-take of STEM (science, technology, engineering and maths) subjects in schools, botanic gardens provide great outdoor learning sites to use plants to encourage inquiry-based learning for a wide range of subjects. The relevance of plants to the study of biology on the school curriculum is obvious, but plants also have surprising links to other STEM subjects.

Following on from last week’s post on technology and engineering, this week we’re looking at the mathematical aspects of plants.

Have you ever noticed subtle patterns in the plant world? Perhaps the arrangement of leaves up a stem, or the spiralling centre of a sunflower? Well it turns out there’s a lot more to these aesthetically pleasing patterns than meets the eye, in that many of them exhibit consistent mathematical properties…

If you were to get out you protractor and measure the angle between two florets, seeds, leaves or other structures of a flower with a spiralling pattern, you will likely find that each successive one is arranged at an angle of approximately 137.5° relative to the previous.

This is called the **Golden Angle**.

This is closely related to the **golden ratio **which, in mathematics, is when the ratio of two quantities is the same as the ratio of their *sum* to their *maximum. *The golden angle, therefore, is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the golden ratio (i.e the ratio of the length of the larger arc to the length of the smaller arc is the same as the ratio of the full circumference to the length of the larger arc).

The result is a pattern of interconnecting spirals – some clockwise, some counterclockwise. If you were to count how many there are of each, you would find that the number of left spirals and the number of right spirals are successive **Fibonacci numbers.**

What are Fibonacci numbers? These are a mathematical sequence whereby each successive number is the sum of the previous two, thereby forming the following series:

**1**, **1**, **2** (1+1), **3** (2+1), **5** (3+2), **8** (5+3), **13** (5+8), **21** (13+8), **34** (21+13), **55** (34+21) and so on…

On a typical sunflower head you would find 34 spirals in one direction and 55 in the other.

Wherever the golden ratio or angle appears, Fibonacci numbers tend to be found also because the ratio between two successive numbers is close to the golden ratio (which is about **1.61803**). The larger the two Fibonacci numbers, the closer their ratio is to the golden ratio, for example:

5 ÷ 3 = 1.66666 (recurring), but further down the line 377 ÷ 233 = 1.618025751

In plants, this produces the most efficient packing of seeds within the flower head. Mathsisfun.com has a great little interactive animation where you can play around with some different values to see what (less effective) patterns other ratios would produce in a sunflower head.

First studied by ancient Greek mathematicians, the golden ratio crops up all over the place in both nature and art and, as a result, has been believed by some to have aesthetic, mystical or even divine properties!

But it’s not just some magical number that plants choose to use – patterns of leaves around a stem (termed “phyllotaxis” (from Ancient Greek *phýllon* "leaf" and *táxis* "arrangement") for example are arranged in order to maximise access to resources (in particular sunlight) by placing leaves as far apart as possible in the given space. Sometimes this means a spiralling pattern is produced (in which case you may observe the golden angle in action), but as you will be aware, sometimes this means leaves grow in pairs opposite one another, so the golden ratio does not always reign supreme. If you’re one for models and formulae there is a whole site dedicated to the mathematical study of plant pattern formation here.

For an amusing, fast-paced introduction for school kids to the Fibonacci sequence in plants, the Khan Academy has three great videos demonstrating the concept.

It’s still not agreed upon exactly how or *why *it came to be that plants grow this way, but some thoughts/explanation are given here.

There was also an exhibition held on this topic at the Botanical Garden of Smith College entitled “Plant Spirals: Beauty You Can Count On” which is now available to view online.

** **

**Do plants divide to survive?**

A couple of weeks ago a paper was released which, in the subsequent media coverage, claimed that plants “do math” in order to regulate their starch reserves.

Overnight, when plants are unable to obtain energy from sunlight to convert carbon dioxide into sugars and starch, they must ensure that the starch they have stored up during the daylight will last until dawn. A team from the John Innes Centre reported that the amount of starch consumed by the plant *Arabidopsis* overnight is being calculated by molecular level arithmetic division, in a process involving leaf chemicals.

Of course the plants aren’t *actually* “doing maths” in a conscious way like we do, rather, the model is an automatic mechanism for the optimum consumption rate of starch stores. There is also some debate over the mathematical models proposed but, nonetheless, the concept of regulating the breakdown of starch in proportion to the remaining hours of darkness is still a handy case study via which to introduce concepts of division in relation to a tangible plant-based problem.

**Resources**

By looking at plants, or gardens as a whole, there are lots of ways to use IBSE to encourage students to explore and consider mathematical patterns and concepts. Maths is often a difficult subject to engage children with, so learning outside the classroom provides a great opportunity to demonstrate the relevance and importance of mathematics in analysing and explaining natural phenomena using a hands-on approach and tangible everyday examples.

The Natural Curiosity Manual has some great ideas for teaching maths in gardens including introducing geometry by sorting leaves according to shape and size, measuring volumes of pumpkins using blocks, and learning to calculate area, perimeter and volume by getting children to design their own garden.

KidsGardening.org also has some fantastic examples of how to teach basic mathematical functions such as calculations, comparisons, and measurements using hands-on activities, most of which are from the book Math in the Garden.

If anyone has any other examples, resources, or personal experiences to share please leave a comment below!

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