Maths in nature

Maths in Nature
Patterns in nature are visible regularities of form found
in the natural world. These patterns recur in different
contexts and are modelled mathematically. Natural
patterns include symmetries, trees, spirals, meanders,
waves, foams, arrays, cracks and stripes. Early Greek
philosophers studied these patterns, with Plato,
Pythagoras and Empedocles attempting to explain
order in nature. The modern understanding of visible
patterns developed gradually over time.
"The laws of nature are but the mathematical thoughts
of God" - Euclid rdthakur78@gmail.com

Rabbits, rabbits, rabbits. Leonardo Fibonacci was a
well-travelled Italian who introduced the concept of
zero and the Hindu-Arabic numeral system to Europe
in 1200AD. He also described the Fibonacci sequence of
numbers using an idealised breeding population of
rabbits. Each rabbit pair produces another pair every
month, taking one month first to mature, and giving
the sequence 0,1,1,2,3,5,8,13,... Each number in the
sequence is the sum of the previous two.
Fibonacci sequence
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As if this were not enough, Leonardo of Pisa gave us
another interesting, if less known gift of mathematics.
If you have never heard of the Fibonacci sequence,
don't worry. To be honest, the sequence sees little
publicity these days outside of a Dan Brown novel and
the occasionally nerdy conversation which may or may
not involve warp core propulsion mechanics. However,
the Fibonacci sequence is an amazing bit of numbers
that ties nature and mathematics together in surprising
ways. From deep sea creatures to flowers to the make-
up of your own body, Fibonacci is everywhere.
Nautilus Shell

If you construct a 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 Fibonacci spiral
can be seen in nature, including in the
chambers of a nautilus shell.
Fibonacci spiral

Sunflower
Evolutionarily speaking, the best way to ensure
success is to have as many offspring as possible (ergo
the Baldwin brothers). The sunflower naturally
evolved a method to pack as many seeds on its
flower as space could allow. Amazingly, the
sunflower seeds grow adjacently at an angle of
137.5 degrees from each other, which corresponds
exactly to the golden ratio. Additionally, the number
of lines in the spirals on a Sunflower is almost
always a number of the Fibonacci sequence.

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
spiral can be found by tracing the seeds of a
sunflower from the centre outwards.rdthakur78@gmail.com

Fibonacci spiral

Fibonacci spiral
Bighorn sheep Ovis canadensis

Spirals: phyllotaxis of spiral aloe,
Aloe polyphylla rdthakur78@gmail.com

Multiple Fibonacci spirals : seed head of
Sunflower, Helianthus annuus.rdthakur78@gmail.com

Multiple
Fibonacci
spirals : red
cabbage in
cross
section

Fibonacci
patterns
occur widely
in plant
structures
including
this cone of
Queen sago,
Cycas
circinalis

Fibonacci Spiral Aloe
Leonardo of Pisa was born around 1170 AD in (of course)
Pisa, Italy. While not quite as famous as some other
Italian or Ninja Turtle Leonardos, we do have a lot to
thank him for. His most notable contribution to your life is
probably found on the top row of your keyboard. While
traveling through North Africa, Leo discovered that the
local number system of 0-9 was far superior than the
obscure combination of X's, V's and I's the Romans had
invented a millennium earlier to confuse later generations
of elementary school students. Leonardo brought this
number system to Europe and eventually we invented
Sudoku with it. rdthakur78@gmail.com

Pine Cone
Like the sunflower, the pine cone evolved
the best way to stuff as many seeds as
possible around its core. Also, in what was
surely an accident, it evolved into perhaps
the best substitute for toilet paper when in a
pinch. The golden ratio is the key yet again.
As with the sunflower, the number of
spirals almost always is a Fibonacci number.

Human body
The golden ratio is found throughout your body,
all the way to your DNA.
Here's one you can see for yourself, dear reader, if
you're still with us. If you use your fingernail
length as a unit of measure, the bone in the tip of
your finger should be about 2 fingernails,
followed by the mid portion at 3 fingernails,
followed by the base at about 5 fingernails. The
final bone goes all the way to about the

middle of your palm, which is a length of about 8 fingernails.
Again, it's Fibonacci at work and the ratio of each bone to the
next comes very close to the golden ratio.
Continuing with the length of your hand to your arm is, again,
the golden ratio.
Fibonacci applies even down to what makes you, you. A DNA
strand is exactly 34 by 21 angstroms.
The Fibonacci sequence is truly a wonder. The examples are
vast, and go way beyond the scale of this article. The patterns
in which a tree grows branches, the way water falls in
spiderwebs, even the way your own capillaries are formed can
all be linked to Fibonacci. Science is just beginning to
understand the implications of this simple sequence and some
of the most amazing discoveries may be yet to come.rdthakur78@gmail.com

Spiral Galaxies
If we take the above spiral and rotate it around the the central axis, we
get an almost perfect approximation of a spiral galaxy.
The Golden Ratio
Most of the interesting things we find that relate to the Fibonacci
sequence are actually more closely related to a number that is derived
from Fibonacci, called the golden ratio. If we take each number of the
Fibonacci sequence and divide it by the previous number in the sequence
(i.e. 2/1, 3/2, 5/3, 8/5), a pattern quickly emerges. As the numbers
increase, the quotient approaches the golden ratio, which is
approximately 1.6180339887. Approximately. The golden ratio actually
predates Fibonacci and has been breaking the brains of western
intellectuals for around 2400 years. Applications for the golden ratio
have been found in architecture, economics, music, aesthetics, and, of
course, nature.

Geometric sequence

Zero - Placeholder and Number

Fractal spirals: Romanesco broccoli
showing self-similar form

The growth patterns of certain trees resemble
these Lindenmayer system fractals

Geometry - Human induced

Parallel lines

Shapes - Cones

Shapes - Polyhedra

Shapes - Perfect

Praise the Creator !!
He is Great !!!

Compiled By
Rupesh Dinkar Thakur,
A. V. S. Vidyamandir, Virar, India

Maths in nature

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