Which flowers have 8 petals

Technical work: The Fibonacci numbers

In mathematics, structures and relationships are examined - when looking at the spiral structure of a sunflower, the distribution of stars, clouds of gas and dust in a star galaxy, or the intricate symmetry of old pottery.
Mathematicians instinctively look for geometric or numerical patterns and symmetry.
Even the ancient Greeks achieved seemingly impossible measurements with the help of geometry - starting with the determination of the circumference of the circle to the surprisingly exact drilling of straight tunnels.
Perhaps no other area has captivated mankind for centuries as has geometry. We observe symmetry and patterns in nature and try to understand the order and beauty behind it.

5.1. symmetry

The spirals of a sunflower, for example, or the spiral-shaped growth of leaves around a stem reveal one aspect of symmetry - regularity. Spirals are regular, but by no means uniform; they seem to want to spin, or at least want to start the next moment. Even symmetry is more likely to be found in things that are made by humans (in pottery and wallpaper or in architecture).
In mathematics, the various types of evenness are described with the help of the geometric concept of movement - or isometry - which means something like "of equal measure". A movement is a transformation of the plane (or space) in which the original object and its image are congruent or mirror images of one another. There are four different types of movements in the plane that have this congruence:
  • Reflection on a straight line, so that the mirror image of the object is created. One speaks of bilateral symmetry because a figure created in this way is divided by the mirror line and the two halves behave like mirror images.
  • Twist or rotation. A figure has rotational symmetry if it has a center around which the figure can be rotated through a certain angle without changing its shape.
  • Displacement or translation, which moves the object in a certain direction by a certain amount. Only figures that are infinitely extended can be moved without changing their appearance.
  • Sliding reflection, consists of a translation by a certain distance along a straight line; followed by a reflection on this straight line.
Phi also appears surprisingly often in geometry. For example, there is the ratio of the sides of a pentagon to its diagonals. If you draw all the diagonals in it, they intersect with each other in the same way in the golden ratio. The resulting pentagram shows a star that is found in many flags around the world.

5.2. Spirals in nature

Nature is abundant in patterns. Botanists have known for a long time that special numbers appear in natural spirals, be it the spiraling leaf structure around a plant stem, the spirals on the surface of pine cones or pineapple fruits and the arrangement of the individual flowers on the flower base of a daisy family . These are the Fibonacci numbers.
The seeds of the sunflower and the successive chambers in the housing of a nautilus lie, as it were, on a long spiral of growth.

In the sunflower you can see two kinds of spirals running clockwise and counterclockwise, with the result, if you counted them, two neighboring Fibonacci numbers. The same thing happens in real seed heads in nature. The reason for this seems to be that the seeds form an optimal spacing of the seeds so that regardless of the size of the seed head, they are evenly divided. All seeds are the same size, they are not too densely packed in the middle and not too sparsely distributed around the edges. Fibonacci numbers are not always found in the number of petals or in the spiral of the seed heads, etc.; although they often approach them.
This type of pattern is called a Fibonacci spiral because the mathematics used for this is associated with a Fibonacci series.

5.2.1. Fixed ratio in spirals

In each concentric layer of a Fibonacci spiral, the ratio of an individual component to one of the next inner layer is a constant that varies according to the number of radially symmetrical parts of the spiral. This makes the increasing series of layers a Fibonacci series: each is a certain linear combination of the previous one. It seems that every growing layer of an organism has to show itself in a certain extended version one layer further outside. Hence the fibonacci pattern is guaranteed.

5.2.2. Fibonacci numbers in branched plants

Above all, one plant shows the Fibonacci row in the number of its "branching points". It is believed that a plant will produce a new shoot that will have to grow for two months before it is strong enough to support branching.
A plant that grows similar to this pattern would be the Achillea Ptarmica.

5.2.3. Petals and flowers

In many plants, the number of their petals is a fibonacci number:
  • Buttercups have 5 petals,
  • Lilies and irises have 3 petals,
  • Delphinium has 8 petals,
  • Marigolds have 13 petals,
  • some asters have 21 petals,
  • Daisies can be found with 34, 55, or 89 petals.
Pine cones show the fibonacci spirals very clearly. Here is a picture of a pine cone seen from its base and one with the spirals exposed:

5.3. Leaf arrangements

Many plants show the Fibonacci numbers in the arrangement of the leaves around their stems. If you look at a plant from above, you will notice that the leaves are often arranged in such a way that the upper leaves do not hide the ones below. This means that each leaf receives a fair amount of sunlight, and that it will intercept most of the rain to direct it through the leaves and stems to the roots.
The Fibonacci numbers show up once you count the number of times you go around a stalk, from leaf to leaf, and so do the leaves in between, until you come across a leaf that is directly above the one you started with Has.
If you start counting in the other direction, you get a different number of spirals for the same number of leaves. The number of spirals in each direction and the number of leaves encountered are three consecutive Fibonacci numbers.

The spines of the cacti also have the same spirals as you could already see on the pine cones, the petals and the leaf arrangements, but they are more visible.

Why do the golden ratio and fibonacci numbers appear in nature?
The reason for this seems to be the same as for the arrangements of the seeds and petals. All are arranged in 0.618034 ... objects (leaves, petals, seeds) per revolution in one direction or in 1- 0.618034 ... objects per revolution in the other direction. If there are Phi (1.618 ...) seeds (leaves or petals) per revolution (or correspondingly phi = 0.618 ... revolutions per seed, leaf or petal etc.), then they have the best possible arrangement to make optimal use of the sunlight. This also enables them to take in the optimal amount of rain and then feed it directly, along the leaf and through the stem, to the roots.
The whole plant seems to be built up from these fibonacci patterns.

Next: The Fibonacci Numbers in Astronomy

Written in 1998 by Susanne Berendt
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The technical work has been restored as much as possible from an old Word document. Unfortunately, some parts could no longer be reconstructed, these were replaced accordingly. The original graphics have been omitted due to possible copyright issues.