(Note: The Fibonacci Forecourt was removed due to recent construction.)
Image © GIS Bureau of the City of Norfolk
The Fibonacci Forecourt, located in the upper right corner of this aerial image of Nauticus, is based on a mathematical concept known as the Golden Spiral which in turn is based on Fibonacci Numbers.
The Fibonacci sequence starts with 0 and 1, and the next number in the sequence is produced by adding the two previous Fibonacci numbers. For example:
the sequence begins with 0 and 1
now add 0 and 1
and the sequence is now 0 1 1
now add 1 and 1
and now the sequence is 0 1 1 2
add the last two numbers
and now the sequence is 0 1 1 2 3
add the last two numbers
and now the sequence is 0 1 1 2 3 5
Continue on, and the series is 0 1 1 2 3 5 8 13 21 34 55 89…
The Fibonacci sequence is found nearly everywhere in nature. We will explore the Golden Ratio (1.618003398875…), a number derived by dividing two consecutive Fibonacci numbers and explore its role in nature.
Most of us probably haven't examined the arrangement of petals on a flower or bothered to count them. If we did, we would discover that the number of petals on a flower is very often a Fibonacci number. Let's look at some examples.
Ever wonder why a four-leaf clover is so hard to find? That's because 4 is not a Fibonacci number.
The presence of Fibonacci numbers in plants is not exclusive to the number of petals on a flower. Let’s look at a drawing of the sneezewort plant. New branches generally grow out at the axil, the space between a leaf or branch and the stem to which it is attached.
If we were to draw a horizontal line through each axil, we would observe obvious stages of growth in the sneezewort plant. The main stem produces new branches at the beginning of each growth stage. These new branches rest for two stages of growth, then they begin to produce new shoots at the beginning of each growth stage.
It should come as no surprise that the number of new shoots at any given stage of development will be a Fibonacci number.
The drawing of the sneezewort plant has been presented to you in one dimension. This explains how the development of branches results in a sequence of Fibonacci numbers, but it does not demonstrate the fact that leaves and branches generally grow in a spiral pattern around the main stem.
The scales of a pinecone are really modified leaves that are close together and connected to a short stem in the pinecone's center. While we do not find an arrangement of leaves with a pinecone as we would on the shoot of a plant with true leaves, we can observe two distinct arrangements of spirals which grow outward from the point where the scales are attached to the stem.
|In this image, eight clockwise spirals are clearly seen
ascending up the pinecone...
|whereas in this image, thirteen counterclockwise spirals are observed ascending more steeply.|
The pinecone examples introduces us to the Golden Spiral which has a ratio of 1.618003398875… and is derived by taking any two consecutive Fibonacci numbers and dividing them.
We can construct a Golden Spiral by creating a Golden Rectangle. A golden rectangle is one that uses the golden ratio. The ratio of the side lengths is 1:1.618. That is, if one side was 1 unit, the other would be 1.618 units.
We can use the Fibonacci Sequence to construct our Golden Rectangle which we can use to create a Golden Spiral. Start with one square, 1 x 1 wide. 1 is the first number in the Fibonacci Sequence.
The next number in the Fibonacci Sequence is 1. Go to the square to the right of the 1. Outline that square to represent the next number in the pattern, another 1. You'll notice we have made a rectangle. This is the pattern we will follow, add a square equal to the next Fibonacci number to create a rectangle.
The next number in the Fibonacci Sequence is 2. Use the line above the two 1 squares to outline a square that is 2 squares long and 2 squares high to create another rectangle.
3 is the next number in the Fibonacci Sequence. Now move to the right of the 1 and 2 squares. Use the right side of the 2 square and the right side of the second 1 square to draw a square that is 3 squares high and 3 squares long.
5 is the next number in the Fibonacci Sequence. Use the bottom of both 1 squares and the bottom of the 3 square to make the next number in the pattern, a square that is 5 squares long and five squares high.
8 is the next number in the Fibonacci Sequence. Move to the left of the 2 square, the 1 square, and the 5 square. Use their left edges to make the 8 x 8 square.
13 is the next number in the Fibonacci Sequence. Use the top of the 8 square, the top of the 2 square, and the top of the 3 square to make a 13 x 13 square. Notice in all of these drawings we always end up with a rectangle after we add a square.
Now you are ready to draw the Golden Spiral. All you have to do is connect one corner of each square with the opposite corner of that square with a quarter circle. This concept is best shown in an illustration. Your Golden Spiral should resemble the image below. This image goes beyond number 13 in the Fibonacci Sequence to demonstrate that the spiral will grow and grow. The image below contains numbers 21 (8+13) and 34 (21+13).
Does the spiral look familiar? It is the shape of the Fibonacci Forecourt at Nauticus.
Another example of the Golden Spiral can be found in the growth patterns of the Nautilus shell.
Golden Spirals are also found in the shapes of galaxies.
Golden Spirals can be found in the patterns of hurricanes.
The golden ratio is found throughout our world, whether it is by coincidence or by design. It is beautiful and fascinating at the same time. It can be found almost everywhere but is rarely noticed.