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Fibonacci and Golden Ratio

Neon sign depicting the golden spiral mounted on a dark background.

Golden spiral neon sign (Alina Kurianova, iStockphoto)

Neon sign depicting the golden spiral mounted on a dark background.

Golden spiral neon sign (Alina Kurianova, iStockphoto)

Subjects
Art Math Science
Readability
5.89

Learn about the Fibonacci sequence and its “golden” relationship to shapes in the world of beauty and nature.

Beauty is in the Eye of the Beholder

What makes something “beautiful”? Is beauty totally subjective? If it is, then why have some things been considered beautiful for a long time, even though trends have changed? Could there be some pattern in nature that humans respond to, subconsciously?

The Greeks said that all beauty boils down to math. Philosophers, sociologists, biologists and mathematicians have all searched for a common quality that might explain our perception of beauty.

One of these qualities is . Scientists have proven that humans are attracted to symmetrical patterns. Even babies prefer to look at pictures of symmetrical faces than non-symmetrical ones! But symmetry alone can’t explain why humans find some things beautiful. So what else could be involved in our perception of beauty?

Blue morpho butterfly on a pink flower
Blue morpho butterfly on a flower (Source: M W via Pixabay). 
Image - Text Version

Shown is a colour photograph of an insect with two wings patterned in shades of blue. The pattern on the left and right wings are mirrors of each other.

The butterfly's wings are dark blue at the top , near its head, brightening to deep blue in the centre, near its torso, then bright blue. The lower edges of the wings have a thick stripe of black, with rows of white dots. 

Between its wings, the butterfly has a slim black torso and long antennae. It is perched on a glossy red flower with a large yellow stamen.

It turns out the Greeks were right about beauty and math. There is a common element in many of the things humans describe as beautiful. According to mathematicians as far back as the ancient Greeks and Egyptians, this element is a ratio of 1:1.618. This is called the Golden Ratio.

Fibonacci Numbers

So where does this “golden” number come from? The ratio is based on a sequence of numbers known as Fibonacci numbers, or the Fibonacci sequence. This was identified by mathematician Leonardo Bonacci around the year 1202.

Did you know?

Leonardo Bonacci became known as Leonardo of Pisa because he was from Pisa. He was even better known as Fibonacci, which means “son of Bonacci” in Italian. He was not called Fibonacci during his lifetime.

Fibonacci was one of the most important mathematicians in the . He introduced the Arabic numeral system to the Western world in his book Liber Abaci. This is the same number system we still use today. 

In the same book, Fibonacci introduced his famous rabbit problem:

A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair from which the second month on becomes productive?

( pp. 283-284, translated from original Latin)

His solution to this problem led to a series of numbers. The first two numbers are 0 and 1, and each following number is the sum of the two numbers before it:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, …

This is the Fibonacci sequence. The individual numbers within this sequence are called Fibonacci numbers.

The Fibonacci sequence can also be expressed using this equation:

Fn = F(n-1) + F(n-2)

Where n is greater than 1 (n>1).

This sequence of numbers may not seem like much. But it gets more interesting when we divide each number by the one that comes before it.

For example: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13

For these examples, the answers would be: 1.000, 2.000, 1.500, 1.667, 1.625, and 1.615.

Ratios for the first seven pairs of Fibonacci numbers
Ratios for the first seven pairs of Fibonacci numbers (©2022 Let’s Talk Science).
Images - Text Version 

Shown is a colour bar graph with 0 - 2.0 on the y axis, and 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13 on the x axis.

From left to right: the bar labelled 1/1 is pale purple and reaches up to 1.0. The bar labelled 2/1 is gold and reaches up to 2.0. The bar labelled 3/2 is bright purple and reaches up to 1.5. The bar labelled 5/3 is dark purple and reaches up to 1.667. The bar labelled 8/5 is orange and reaches up to 1.6. The bar labelled 13/8 is turquoise and reaches to 1.625. The last bar, labelled 21/13 is bright blue, and reaches up to 1.615. These ratios are written in the centre of each bar.

A dotted line stretches across the graph, at the level of 1.618033988749895. This is labelled with the symbol for Phi. A horizontal oval with a vertical line through the centre.

Did you know?

When students look at the relationship between one term and the next, they are doing a type of thinking called recursive thinking.

The Golden Ratio 

As the Fibonacci numbers get bigger, the ratio between each pair of numbers gets closer to 1.618033988749895. This number is called Phi. It can also be represented by the symbol Φ, the 21st letter of the Greek alphabet.

Phi is the Golden Ratio. It also has other unusual mathematical properties.

Did you know?

The Golden Ratio is also known as the Golden Section, the Golden Mean and the Divine Proportion.

The Golden Ratio can also be found using two quantities, like the lengths of two line segments.

Two quantities have the Golden Ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities:

Ratios of line segments in the Golden Ratio
Ratios of line segments in the Golden Ratio (©2022 Let’s Talk Science).
Image - Text Version

Shown is a colour illustration of line segments and a mathematical equation. 

At the top is a long black stripe labelled Line Segment. Below is a shorter blue stripe labelled Long Segment. To the right is an even shorter green stripe, labelled Short Segment. When placed end-to-end, the blue and green stripes equal the length of the black stripe.

The equation is below this illustration. On the left, the words Long Segment in blue is divided by the words Short Segment in green. To the right is an equals sign. Next, the words Line Segment in black is divided by the words Long Segment in blue. This is followed by another equals sign. Next, 1+ the square root of 5 is divided by two. This is followed by another equals sign. To the right, the answer is 1.61803...

A Golden Rectangle works in a similar way.

A golden rectangle is a rectangle with a long side of a + b and a short side of a. This is the entire coloured area of the diagram.

If we cut off a square section so that each side is equal to the shortest side, this is the pale blue square on the left. The piece that remains has the same ratio of side lengths as the original rectangle. This is the pink area on the right.

The ratio between sides and b is Φ or 1.618…, as you can see in the equation below:


Diagram of a golden rectangle
Golden rectangle (Source: Ahecht [public domain] via Wikimedia Commons). 
Image - Text Version

Shown is a colour diagram of a rectangle divided into a pale purple square and a smaller pink rectangle.

The top and left edges of the square are each labelled with a blue, lower case, italic a. The top edge of the smaller rectangle is labelled with a red, lower case, italic b. The entire bottom edge of the larger rectangle is labelled with a + b, in green italics.

We can grow this pattern by adding a larger square to the long side (a + b) of a rectangle. Then we can do the same thing to the rectangle that results from that combination.

You can see the pattern of squares and rectangles in this diagram.

Multiple sets of golden rectangles
Multiple Golden rectangles (Let’s Talk Science using an image by primo-piano via iStockphoto).
Image - Text Version

Shown is a black and white illustration of a rectangle divided into smaller squares and rectangles that get smaller as they move around the page, towards a spot in the bottom right quadrant. 

The largest rectangle is divided into many smaller shapes. The largest, on the left, is a square labelled with the number 34. 

To the right of the square is a vertical rectangle. This is divided into a square, labelled 21, and another, smaller, horizontal rectangle.

The third rectangle is divided again. The square in the right is labelled 13. The vertical rectangle is further divided into a square labelled 8, and a  horizontal rectangle that is divided again. The next square is labelled 5. Next to it, another vertical rectangle contains a square labelled 3 and a smaller horizontal rectangle, which, in turn, contains a square labelled 2. The smallest square is not labelled, but it looks like this pattern could continue, becoming smaller and smaller with each iteration.

We can take the Golden Rectangle one step further by adding a line that forms a quarter circle in each square.

This diagram shows that the lines connect to form a spiral.

Each square is also labelled with a number that indicates the length of its sides. They are the same as the numbers as in the Fibonacci sequence! We call this a Fibonacci Spiral.

Fibonacci sequence forming a spiral
The Golden ratio can also form a spiral (©2022 Let’s Talk Science).
Image - Text Version

Shown is a black and white illustration of a rectangle divided into smaller squares and rectangles, overlaid with a blue spiral line. 

The largest rectangle is divided into many smaller shapes. The largest, on the left, is a square labelled with the number 34. The blue line over it curves from the bottom left to the top right corner, in a quarter circle. 

To the right of the square is a vertical rectangle. This is divided into a square, labelled 21, and another, smaller, horizontal rectangle. The square labelled 21 is overlaid with another quarter circle, from the top left, to the bottom right corner.

The third rectangle is divided again. The square on the right is labelled 13. It is overlaid with a curved blue line from the top right to the bottom left. 

The vertical rectangle is further divided into a square labelled 8, and a horizontal rectangle that is divided again. The blue line continues to curl across these shapes.

The next square is labelled 5. Next to it, another vertical rectangle contains a square labelled 3 and a smaller horizontal rectangle, which, in turn, contains a square labelled 2. The blue line continues to curl smaller across these shapes.

The smallest square is not labelled, but this is the point where the blue spiral ends in a tight curl. The pattern looks as if it could continue, dividing into smaller and smaller shapes, with the spiral becoming tighter and tighter.

 

The Fibonacci Spiral Versus the Golden Spiral

The terms Fibonacci Spiral and Golden Spiral are often used interchangeably. But these two spirals are slightly different.

A Fibonacci spiral is made by creating a spiral of squares that increase in size by the numbers of the Fibonacci sequence. So: 1, 1, 2, 3, 5, 8, 13, 21, etc.. You can see this in the animated GIF below.

Animation of a Fibonacci sequence forming a spiral

How a Fibonacci Spiral could be made (©2022 Let’s Talk Science)

Image - Text Version

Shown is an animated gif of a spiral growing with larger and larger squares of different colours.

The first square tiny and blue, with a curved white line from the bottom left to the top right corner. The second square appears above. It is much larger and the line curves from the bottom right to top left. The third square is larger again. It appears left of the rest, and the line curves from the top right to the bottom left. The forth square appears below the others, with a line from top left to bottom right. The fifth square is orange, and appear on the right, with a line from the bottom left to the top right. The fifth square appears on top of the rest, in pink, with a line from the bottom right to the top left. The final square is so large it takes up more than half the page, and fills in all the space to the left of the rest. It is blue with a line curving from the top right to the bottom left.

When all the squares are put together, the curved lines across them form a spiral. This spiral grows out from a tiny blank square in the bottom right corner of the page.

A Golden Spiral, though, is created by nesting smaller and smaller Golden Rectangles within a large Golden Rectangle. Look at the difference in the GIF below:

Animation of a Fibonacci sequence forming a spiral

How a Golden Spiral could be made (©2022 Let’s Talk Science)

Image - Text Version

Shown is an animated gif of smaller and smaller rectangles and squares, appearing on top of each other, following the pattern of a spiral.

The first rectangle is blue and takes up the whole panel. It has a curved white line across the top left corner. The second shape is a pink rectangle that covers the right side of the blue, so that the blue portion is now a square shape. It has a curved white line across the top right corner. The third is an orange rectangle that covers the bottom portion of the pink, so the pink area forms a square. It has a curved white line across the bottom right corner. The fourth is a purple rectangle that covers the left portion of the orange, creating an orange square. Next is a red rectangle forming a purple square and a green rectangle forming a red square. Finally a tiny blue square appears next to a white one of the same size.

When the gif is complete, all the curved white lines form a spiral curling in to the final white square in the bottom right quadrant of the panel.

Do these spirals look familiar? Well they should! We can see the same pattern in nature.

Misconception Alert

It is worth noting that the spirals we see in natural objects do not match the Golden Ratio exactly, but they come close. 

Fibonacci spiral outlined on a sunflower
Fibonacci spiral on a sunflower (Let’s Talk Science using an image by Damian Pawlos via iStockphoto).
Image - Text Version

Shown is a colour photograph of the centre of a sunflower, with a blue spiral superimposed on it.

The flower has bright yellow petals. Its centre consists of tiny, pointed, deep yellow structures, densely packed into a circle. The spiral demonstrates that the tiny pointed structures are laid out in a spiral pattern.

Golden spiral outlined on a Nautilus shell
Fibonacci spiral on the interior of a nautilus shell (Let’s Talk Science using an image by duncan1890 via iStockphoto).
Image - Text Version

Shown is a colour photograph of a shell sliced in half to show the inside, with a red spiral superimposed on it.

The walls of the shell curl around from a central point, becoming wider and wider to the outside edge. This structure is divided into curved, wedge-shaped sections that also grow in size from the central point. The spiral demonstrates that the shell forms a Fibonacci spiral.

Fibonacci spiral on Galaxy Messier 101 nicknamed the Pinwheel Galaxy
Fibonacci spiral on Galaxy Messier 101 nicknamed the Pinwheel Galaxy (Let’s Talk Science using an image by European Space Agency & NASA [CC BY 3.0] via Wikimedia Commons).
Image - Text Version

Shown is a colour photograph of a wispy swirl shape in outer space, with a red spiral superimposed over it.

The background is black and scattered with dots of white and gold. The centre of the swirl is pale gold with thin lines of darker gold throughout. Wispy white tails curl out from the centre, curving into space. These bands of translucent white are punctuated by clumps of bright white. 

The red spiral layered over the photograph demonstrates that this galaxy forms a Fibonacci spiral.

The Golden Ratio can be used with other shapes as well. It is possible to find golden ratios in patterns involving circles, triangles, pentagons and other shapes.

A collection of shapes displaying fibonacci sequence ratios
Other shapes with their Golden ratios (Let’s Talk Science using an image by primo-piano via iStockphoto).
Image - Text Version

Shown is a diagram of intersecting triangles, pentagons, squares and circles with blue lines drawn over them.

The largest triangle is acute, and contains seven other, smaller triangles. Its left outside edge is covered by a blue line, labelled 1.618. The blue line turns the corner and continues on its bottom edge, labelled 1. This triangle is divided into an isosceles triangle and a scalene triangle. The scalene triangle is further divided into another acute and another isosceles triangle. The blue line continues around the long, then the short edge of the acute triangle. This triangle is divided again, into another acute and another isosceles. The blue line continues along the base of the acute triangle, which is further divided into another acute and another isosceles. The blue line continues along this base, and the triangle is divided again. The blue line finishes as it turns a tight corner along the base of the smallest acute triangle. In total, the blue line forms a sort of spiral, with series of acute angles and straight edges that become shorter and closer towards a point in the bottom right of the largest triangle.

The largest pentagon is labelled 8. Its bottom edge is covered with a blue line. Inside that, a smaller pentagon is labelled 5 and the blue line continues along one of its edges. Inside this are smaller pentagons labelled 3 and 2, where the line continues along an edge of each. Two even smaller pentagons are unlabelled, but they follow the same pattern. All the segments of the blue line forms a spiral with obtuse angles and straight edges that become shorter and closer towards a point in the lower right of the largest pentagon.

The largest square is labelled 8. A smaller one, butted up against it on the right, is labelled 5. A blue line is drawn from the top left corner of the large square, to the top left corner of the smaller square, forming a slope down to the right. Smaller and smaller squares, labelled 3, 2 and 1, follow the same pattern, and the blue line continues in a straight slope down to the smallest square, on the far right. 

The largest circle is labelled d=8. A blue line covers the upper left part of the circumference. Inside it, smaller circles are labelled 5, 3, 2 and 1. The blue line continues along part of the circumferences of each circle, forming a spiral. This curls to its smallest point in the top right of the largest circle.

So, whether you think the Golden ratio makes things more beautiful is up to you. But perhaps we can agree that the Fibonacci Sequence and the Golden Ratio are mathematically interesting. They can also give us a new way of looking at nature and art.

Learn More

How To: Draw Golden Rectangle Fibonacci Sequence (2016)
This video (4:08 min.) from sein Selbst sein provides a step-by-step tutorial on how to draw Golden rectangles and spirals.

What is the Fibonacci Sequence & the Golden Ratio? Simple Explanation and Examples in Everyday Life (2021)
This video (5:09 min.) from Science ABC offers a simplified explanation of the Fibonacci Sequence and the Golden Ratio, along with examples for use in the classroom.

Facial Analysis and the Marquardt Beauty Mask (2014)
This article from Goldennumber.net explores the relationship between the Golden ratio and beauty.

What is the Golden Ratio? (2020)
This video (4:39 min.) from Interesting Engineering discusses the history of the Golden Ratio and its appearance in works of engineering and art across the world.

Is the Nautilus Shell Spiral a Golden Spiral? (2014)
This article, written by Gary Meisner and featured on Goldennumber.net, examines the controversy over the nautilus shell being an example of the Golden Ratio, and makes the argument that there is more than one way to form a Golden Ratio.

The Golden Ratio (why it is so irrational) - Numberphile (2018)
This video (15:12 min.), from Numberphile, uses flower seed distribution and fractional turns to show what happens numerically when Golden Ratio spirals form.

References

Be Smart (2021). The Golden Ratio: Is It Myth or Math? YouTube.

Huffman, C. J. (n.d.). Mathematical Treasure: Fibonacci's Liber Abaci. Mathematical Association of America.

Mann, A. (Nov 25, 2019). Phi: The Golden RatioLiveScience.

Phyllotaxis (n.d.). Fibonacci Numbers - Golden Angle

Roos, D. (Jun 16, 2021). Why Do We Get So Much Pleasure From Symmetry? HowStuffWorks.

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