The document discusses how mathematics is present in nature. It provides examples of symmetry, shapes, geometry, pi, fractals, Fibonacci sequences, golden ratios, and geometric sequences that can be observed in nature. These include symmetrical faces and flower petals, hexagonal bee hives, pi relationships in circles, fractal patterns in trees, Fibonacci sequences in rabbit populations and nautilus shells, and geometric growth of bacteria populations. The conclusion states that nature is inherently mathematical and science depends on mathematics to accurately describe nature.

1. Submitted to:Jaseena teacher Submitted by:Fathima Shahnaz

2. INTRODUCTION
"The laws of nature are but the mathematical thoughts of
God"
- Euclid
Mathematics is everywhere in this universe. Mathematics
is all around us. As we discover more and more about our
environment and our surroundings we see that nature can
be described mathematically. The beauty of a flower, the
majesty of a tree, even the rocks upon which we walk can
exhibit natures sense of symmetry ,we seldom note it. We
enjoy nature and are not interested in going deep about
what mathematical idea is in it. Here are a very few
properties of mathematics that are depicted in nature.

3. Symmetry
Many mathematical principles are based on ideals, and apply to an abstract,
perfect world. This perfect world of mathematics is reflected in the imperfect
physicalworld, such as in the approximate symmetry of a face divided by an axis
along the nose. More symmetricalfaces are generally regarded as more
aesthetically pleasing.
Five axes of symmetry are traced on the petals of this flower,fromeach dark
purple line on the petal to an imaginary line bisecting the angle between the
opposIng purplelines. The lines also trace the shapeof a star.
Shapes - Perfect
Earth is the perfect shape for minimising the pull of gravity on its
outer edges - a sphere (although centrifugal force from its spin
actually makes it an oblate spheroid, flattened at top and
bottom). Geometry is the branch of maths that describes such
shapes

4. Shapes - Polyhedra
For a beehive, close packing is important to maximise the use of space.
Hexagons fit most closely together without any gaps; so hexagonal wax
cells are what bees create to storetheir eggs and larvae. Hexagons are six-
sided polygons, closed, 2-dimensional, many-sided figures with straight
edges.
Geometry - Human induced
People impose their own geometry on the land, dividing a random
environmentinto squares, rectangles and bisected rhomboids, and
impinging on the natural diversity of the environment.

5. Pi
Any circle, even the disc of the Sun as viewed from Cappadoccia, central
Turkey during the 2006 total eclipse, holds that perfect relationship where
the circumferencedivided by the diameter equals pi. Firstdevised
(inaccurately) by the Egyptians and Babylonians, the infinite decimal places
of pi (approximately 3.1415926...) havebeen calculated to billions of
decimal places.
Fractals
Many natural objects, such as froston the branches of a tree, show the
relationship wheresimilarity holds at smaller and smaller scales. This fractal
nature mimics mathematical fractalshapes where formis repeated at every
scale. Fractals, such as the famous Mandelbrot set, cannotbe represented
by. classical geometry
Fibonacci sequence
Rabbits, rabbits, rabbits. Leonardo Fibonacci was a well-travelled Italian
who introduced the concept of zero and the Hindu-Arabic numeralsystem
to Europe in 1200AD. Healso described the Fibonacci sequenceof numbers
using an idealised breeding population of rabbits. Each rabbit pair produces
another pair every month, taking one month firstto mature, and giving the

6. sequence 0,1,1,2,3,5,8,13,... Each number in the sequence is the sumof the
previous two.
Fibonacci spiral
If you constructa series of squares with lengths equal to the Fibonacci
numbers (1,1,2,3,5, etc) and trace a line through the diagonals of each
square, it forms a Fibonacci spiral. Many examples of the Fibonaccispiral
can be seen in nature, including in the chambers of a nautilus shell.
Golden ratio (phi)
The ratio of consecutive numbers in the Fibonacci sequence approaches a
number known as the golden ratio, or phi (=1.618033989...). The
aesthetically appealing ratio is found in much human architecture and plant
life. A Golden Spiral formed in a manner similar to the Fibonacci spiralcan
be found by tracing the seeds of a sunflower fromthe centre outwards.

7. Geometric sequence
Bacteria such as Shewanella oneidensis multiply by doubling their
population in size after as little as 40 minutes. A geometric sequencesuch
as this, whereeach number is double the previous number [or f(n+1)
=2f(n)] produces a rapid increasein the population in a very shorttime.

8. CONCLUSION
Nature is innately mathematical, and she speaks to us in
mathematics. We only have to listen. When we learn more
about nature, it becomes increasingly apparent that an
accurate statement about nature is necessarily mathematical.
Anything else is an approximation. So, mathematics is not
only science but is also an exact science.
Because nature is mathematical, any science that intends to
describe nature is completely dependent on mathematics. It
is impossible to overemphasize this point, and it is why Carl
Friedrich Gauss called mathematics "the queen of the
sciences."

9. REFERENCE
 www.math.duke/edu
 www.project2061.org/publications/sfaa/online/chap2.html
 www.cimt.plymouth.ac.uk/projects/mepres/alevel/pure_ch
1.pdf