Math in NatureMATH 1 - THREEM TEAM

❏ Symmetry❏ Shapes and Patterns❏ Fractals❏ Fibonacci Sequence❏ Golden Ratio❏ Geometric Sequence

- Galileo Galilei

“Mathematics is the alphabet with which God has written the universe.”

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 physical world, such as in the approximate symmetry of a face divided by an axis along the nose.

five-fold symmetry

Types of Symmetry

Reflection (or Line or Mirror) Symmetry

Rotational Symmetry

Translational Symmetry

Five axes of symmetry are traced on the petals of this flower, from

each dark purple line on the petal to an imaginary

line bisecting the angle between

the opposing purple lines. The

lines also trace the shape of a

star.

Shapes• PERFECT • POLYHEDRA • CONES

Earth (perfect shape

for minimizing the pull of gravity on its outer

edges - a sphere)

Beehive (close

packing is important to maximise the use of space)

Volcano (form cones, the

steepness and height of which depends on the runniness (viscosity) of the lava)

Parallel Lines

parallel dunes in the Australian desert

Geometry (Human Induced)

People impose their own geometry on the land

Pi 3.1415926..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 circumference divided by the diameter equals piπ

Fractalsare fragmented

geometric shapes wherein you can see the reduced ‘orignal copy ’ or image of the fractal within the smaller parts of the said object

Fractals in Nature1. BranchingBlood vessels, Neurons, Lightning, Trees, Rivers, Lungs

2. SpiralsBiological spirals (Plant and Animal Kingdoms), Flowers, Fossilized

Ammonites, Non-living Spirals, Turbulent swirling of fluids, Star formations of the galaxies

This property is called

“Self-Similarity”

Fractals are extremely complex, sometimes infinitely complex - meaning

you can zoom in and find the same shapes forever.

Fibonacci Sequence

The rotation of the spiral arrangement of flowers also follow the pattern with fractions made with two successive Fibonacci numbers. For example, a half rotation is ½, 1 and 2 are fibonacci numbers. 3 over 5 ( ) is also fraction with ⅗fibonacci numbers. The spiral growth prevents the leaves from blocking other leaves from sunlight.

Spiral Leaf Growth

Fibonacci Spiral

GoldenRatio (phi)

Best value for each “turn” whenever a new cell is formed in plants, such as in a sunflower because it creates a pattern with no gaps from start to end.

So, if you were a plant, how much of a turn would you have in

between new cells?

If you don't turn at all, you would have a straight line.

But that is a very poor design ... you want something round that will hold together

with no gaps.

That is because the Golden Ratio (1.61803...) is the best solution to

this problem, and the Sunflower has found this solution in its own natural

way.

Because if you choose any number that is a simple fraction (example: 0.75 is 3/4, and 0.95 is 19/20, etc), then you will eventually get a pattern of lines stacking up, and

hence lots of gaps.

But the Golden Ratio is an expert at not being any fraction.

It is an Irrational Number (meaning you cannot write it as a simple fraction), but more than that ... it is as far as you can get from

being near any fraction.

Pi (3.141592654...) is also irrational. Unfortunately it has a decimal very close to 1/7 (= 0.142857...), so it ends up with 7 arms.

There is a special relationship between the Golden Ratio and Fibonacci Numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, ... etc)

If you take any two successive (one after the other) Fibonacci Numbers,their ratio is very close to the Golden Ratio:

Geometric Sequence

Bacteria such as Shewanella oneidensis multiply by doubling their population in size after as little as 40 minutes

Uniqueness, Proofs

Proofs are the tools used to find the rules that define maths. One such proof is by counter example - find

one duplicated snowflake, like Nancy Knight of the US National Center for Atmospheric Research did while studying

cloud climatology, and the theory of snowflake uniqueness disappears into the clouds.

InfinityIs one infinity bigger than another infinity?The size of all natural numbers, 1,2,3..., etc., is infinite. The set of all numbers between one and zero is also infinite.

The deep questions of maths can leave you feeling very small in a vast universe.

“Is space infinite? Even if the universe goes on forever, it may not be infinitely large.”

Earth’s Most Stunning

Natural Fractal Patterns

from wired.com

Romanesco Broccoli: “The Ultimate Fractal Vegetable”

& San Francisco Bay salt flats

Ammonite Sutures

Mountains

Waterfalls

The mathematical formula that describes ferns, named after Michael Barnsley, was one of the first to show that chaos is inherently unpredictable yet

generally follows deterministic rules based on nonlinear iterative equations

Peacocks attract mates with the repeating patterns in their plumage.

Sources

➢ http://munmathinnature.blogspot.com/

➢ http://www.maa.org/frank-morgans-math-chat-hales-proves-hexagonal-honeycomb-conjecture

➢ http://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html

➢ http://www.abc.net.au/science/photos/mathsinnature/

➢ http://www.wired.com/wiredscience/2010/09/fractal-patterns-in-nature/#slideid-407811

Math 1 - Threem Team