The file was prepared as a part of the Comenius project Why Maths? by the polish studetns from Anna Vasa school in Golub-Dobrzyń (Poland)

1. Gimnazjum
im. Anny Wazówny
Golub-Dobrzyń
Poland

2. Probably most of us have never taken the time to
examine very carefully the number or arrangement
of petals on a flower. If we were to do so, we would
find that the number of petals on a flower, that still
has all of its petals intact and has not lost any, for
many flowers is a Fibonacci number:

3. •3 petals: lily, iris
•5 petals: buttercup, wild rose, larkspur, columbine
(aquilegia)
•8 petals: delphiniums
•13 petals: ragwort, corn marigold, cineraria,
•21 petals: aster, black-eyed susan, chicory
•34 petals: plantain, pyrethrum
•55, 89 petals: michaelmas daisies, the asteraceae family

7. There is also a link between Fibbonacci sequence and and a
special number that ancient civilizations called “the golden
ratio”.

8. “Logarithmic Spiral” of a
common shell.
The Fibonacci numbers increase at
a ratio that is revealed in objects and
spirals. The Chambered Nautilus
(which was so special to my husband
and I) if cut in half reveals a series of
chambers. Each chamber increases
in size as the mollusk grows. They
also grow in a spiral shape.

11. This same spiral and ratio is present in a great
many products of nature; the pinecone, the
pineapple
Look at the bottom of a pinecone. It has those same kinds of
spirals. They don’t go around and around in a circle – they go out
likefireworks. Look at the pictures above to see what that looks
like.

12. Golden Spiral also appears
in hurricanes,
ram's horns,

13. sea-horse tails
growing fern leaves

14. seed patterns of sunflowers
All the sunflowers in the world show a number of spirals
that are within the Fibonacci Sequence.

15. Look at the following images of a sunflower:
By observing closely the seeds configuration you will see how appears a kind of spiral
patterns. In the top left picture we have highlighted three of the spirals typologies
that could be found on almost any sunflower.
Well, if you look at one of the typologies, for example the one in green, and you go
to the illustration above right you can check that there is a certain number of spirals
like this, specifically 55 spirals.

16. We have more examples in the two upper panels, cyan
and orange, they are also arranged following values that
are within the sequence: 34 and 21 spirals.

17. A lot of people love honey
made by tiny bees. These
insects use so much
mathematical strategy
throughout their daily lives.
Just their hives use angles,
shape, tessellation and
addition. Wasps and bees exhibit
polygons in their nests.
Hexagons create nests
that require less material
and work to build. It is an
efficient way of
partitioning that also
saves energy.

18. Hexagonal cell requires
minimum amount of wax
for construction while it
stores maximum amount of
honey.
Why hexagons? Not squares or triangles?
Hexagons fit most closely together without any
gaps, so they are an ideal shape to maximise the
available space.

19. Fractals aren't just something we learn about in math
class. They are also a gorgeous part of the natural world.
Here are some of the most stunning examples of these
repeating patterns.
Romanesco broccoli is a particularly symmetrical
fractal.

20. The fern is one of many flora that are fractal; it’s an
especially good example.
Each part is the roughly the
same as the whole. When
we break a leaf off of the
original and it looks like the
original – break a leaf off of
that leaf and that looks like
the original also.

21. The delicate Queen Anne’s Lace, which is really just wild
carrot, is a beautiful example of a floral fractal. Each
blossom produces smaller iterative blooms. This particular
image was shot from underneath to demonstrate the
fractal nature of the plant.

22. The Giant's Causeway, located in Ireland, is an fascinating
formation found in nature. It is a collection of hexagons
tesselating the ground - even in 3D at some points.
In nature we can see samples of tessellations. This
phenomenon is really beautiful and incredible. Here you can
see some examples :

23. Rock formation in "White Pocket", Vermillion Cliffs
National Monument,

24. Veins in a leaf
Dragonfly

25. Cracked dried mud

26. Links:
http://www.scienzainrete.it/en/content/article/nature-numbers
http://nutters.edublogs.org/maths-in-nature/
http://static.panoramio.com/photos/large/2776550.jpg
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