Fibonacci spiral
The fibonacci spiral is very easy to make. Start with a golden rectangle. Create a golden section along the bottom of the rectangle. Draw a perpendicular line through it. Then create a section of the line you just drew. Keep doing this until it gets too small to draw. Next, draw arcs centered at the golden sections with the radius having the length of the line through the section. You could, if you wanted to, continue drawing the spiral into infinity. It never ends. Look at the picture and you will understand.
The golden spiral also resembles something in nature. Can you guess what? A seashell of course! The Golden ratio is everywhere.
A Fibonacci Spiral
A Fibonacci Spiral is almost the same as a Golden Spiral, except for one difference. A Fibonacci Spiral uses Fibonacci numbers instead of the Golden Mean. Fibonacci numbers are numbers of the pattern 1,2,3,5,8,13,21,34... . If you divide consecutive Fibonaccis, you will get an approximation of Phi.
First, draw two squares with side length of 1 next to each other. Draw a circle centered at A and radius 1. Draw a square with lenth 2, and at point B, draw a circle with radius 2. Then draw a square with side 3,5,8,13 and so on. See the pattern? They're just a bunch of Fibonacci numbers. As you continue the pattern, your rectangle will get closer and closer to the real golden rectangle.
Geometrical consideration for generation of the Fibonacci spiral in three dimensions specify a constant angular turning d(theta) equal to 1/tau between successive radii and therefore a constant crossing angle, also equal to1/tau.The Fibonacci equiangular spiral is then given by the relation
The Fibonacci spiral is traced with mathematical precision in nature in the dynamical growth processes of plants as seen in the geometrical placement on the shoot, of primordia, which later develop into the various plant parts. In a majority (92%) of plants studied world wide, successive primordia always subtend angle equal to the golden angle at the apical center (Jean, 1994References). Primordia placement in space and time may therefore be resolved into the precise geometrical pattern of the quasiperiodic Penrose tiling pattern.
Here is a spiral drawn in the squares, a quarter of a circle in each square. The spiral is not a true mathematical spiral (since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller) but it is a good approximation to a kind of spiral that does appear often in nature. Such spirals are seen in the shape of shells of snails and sea shells and, as we see later, in the arrangment of seeds on flowering plants too. The spiral-in-the-squares makes a line from the centre of the spiral increase by a factor of the golden number in each square. So points on the spiral are 1.618 times as far from the centre after a quarter-turn. In a whole turn the points on a radius out from the center are 1.6184 = 6.854 times further out than when the curve last crossed the same radial line.
Cundy and Rollett (Mathematical Models, second edition 1961, page 70) say that this spiral occurs in snail-shells and flower-heads referring to D'Arcy Thompson's On Growth and Form probably meaning chapter 6 "The Equiangular Spiral". Here Thompson is talking about a class of spiral with a constant expansion factor along a central line and not just shells with a Phi expansion factor.
Below are images of cross-sections of a Nautilus sea shell. They show the spiral curve of the shell and the internal chambers that the animal using it adds on as it grows. The chambers provide boyancy in the water. Click on the picture to enlarge it in a new window. Draw a line from the center out in any direction and find two places where the shell crosses it so that the shell spiral has gone round just once between them. The outer crossing point will be about 1.6 times as far from the centre as the next inner point on the line where the shell crosses it. This shows that the shell has grown by a factor of the golden ratio in one turn.
On the poster shown here, this factor varies from 1.6 to 1.9 and may be due to the shell not being cut exactly along a central plane to produce the cross-section.
You can see that the orange "petals" seem to form spirals curving both to the left and to the right. At the edge of the picture, if you count those spiralling to the right as you go outwards, there are 55 spirals. A little further towards the centre and you can count 34 spirals. How many spirals go the other way at these places? You will see that the pair of numbers (counting spirals in curing left and curving right) are neighbours in the Fibonacci series. Click on the picture on the right to see it in more detail in a separate window.
The spirals are patterns that the eye sees, "curvier" spirals appearing near the centre, flatter spirals (and more of them) appearing the farther out we go.
So the number of spirals we see, in either direction, is different for larger flower heads than for small. On a large flower head, we see more spirals further out than we do near the centre. The numbers of spirals in each direction are (almost always) neighbouring Fibonacci numbers! Click on these links for some more diagrams of 500, 1000 and 5000 seeds.
Click on the image on the right for a Quicktime animation of 120 seeds appearing from a single central growing point. Each new seed is just phi (0�618) of a turn from the last one (or, equivalently, there are Phi (1�618) seeds per turn). The animation shows that, no matter how big the seed head gets, the seeds are always equally spaced. At all stages the Fibonacci Spirals can be seen.
One thing about this Fibonacci sequence is that it creates a curve. The sequence is blended into this curve or spiral. Some Fibonacci spirals are really noticeable. For example, the horn of a ram curls around with the same ratio and so do nautilus shells. Sometimes you'll see spirals going in both directions at once. You can see this on a pine cone or a sunflower. If you count the number of spirals going one way relative to the number of spirals going in the opposite direction, they are going to create the Fibonacci series. An example would be 34 spirals going one way and 55 spirals going the other way. In sunflowers, you will find the ratios 5-8, 8-13...all the way up to 144-233 counter rotating spirals.