HSAN DORAMACI FOUNDATION BLKENT ERZURUM LABORATORY HIGH SCHOOL

İHSAN DOĞRAMACI FOUNDATION BİLKENT ERZURUM LABORATORY HIGH SCHOOL MATHEMATICS PROJECT NAME & SURNAME: GÖKÇEN DOĞAR LESSON: MATHEMATICS TEACHER: ÖZGE ÖZSOY SUBJECT: IT IS ONLY NATURAL

� � � You might be surprised to find that mathematics is in everything in nature. Maybe anyone think mathematics and nature together or Is there a relationship between mathematics and nature but real answer, exactly. There is a relationship between Mathematics and Nature. I and some people or scientist guess , relationship of mathematics and nature isn’t coincidence. Also mathematics and nature complete to each other. We can see this from some properties. For example : There are spirals and Fibonacci numbers , symmetry , pattern , crescent shaped sand dunes, reflections, pentagon, circles, spiral, angles, concentric polygons , spheres, cubic crystals, lines or arcs, generally circular, hexagons… etc Also there is a relationship mathematics and our life. IN BRIEF THERE IS MATHEMATICS IN NATURE AND THERE IS NATURE IN MATHEMATICS ALSO MATHEMATICS IS IN EVERYTHING

PINECONES _SPIRALS AND FIBONACCI NUMBERS If we miniaturize or make bigger , we can see the same their length. But we must miniaturize or make bigger the same ratio their length. And we are going to see the same. They are like fractals.

SUNFLOWERS_SPIRALS AND FIBONACCI NUMBERS � Pits of sunflowers rang according to Fibonacci numbers. And mean of Fibonacci numbers is to continue according to the rule. The rule is , we start from zero and then we write one and then two…In brief 0, 1, 1, 2, 3, 5, 8, 13, 21, 34…Each remaining number is the sum of previous two And there is golden ratio in the sunflower.

PINEAPPLE_SPIRAL FIBONACCI NUMBERS � Pineapple’s some part are the same and symmetric DAISY HEADS_SPIRALS While daisies grow , each branch of daisies rise according to Fibonacci numbers

MILKY WAY GALAXY_SPIRAL Circles are regularly and they move from small to big. CHAMBERED NAUTILUS (SNAIL)_SPIRAL If we take a line of snail on the shell , we can see “each line is the same to the other”.

RIPPLES ON WATER_CIRCLES The water’s wave spread from small to big. DIATOMS_CIRCLES These all circles are the same and there is pattern in the diatoms.

GROWTH RINGS(TREES)_CONCENTRIC CIRCLES These circles continue regularly or in regular. HONEYCOMBS_HEXAGONS Those honeycombs are hexagons and bears always do to their combs with hexagons shape. This shape has got a ratio and pattern.

SNOWFLAKKES_HEXAGONS This equilateral triangle is similar to snowflakes. There are golden ratio and symmetry. There are short and long branches in the snowflakes and we say golden ratio to between these of snowflakes branches

COMMON STARFISH_PENTAGONS It’s each arms are the same and symmetric STAR SAPPHIRE_PENTAGON Some geometric shapes are the similar to natural things or shapes

MANY ANIMALS (INCLUDING HUMANS)_SYMMETRY �BUTTERFULY_SYMMETERY There is a ratio and pattern the animals BUTTERFULY_SYMMETERY There is a symmetry because butterfly’s wings are the same

HUMAN _GOLDEN RATIO � If we divide the sum of two front teeth’ s length to the sum of two front teeth’s height , it gives golden ratio. Elbows divide to arms to two piece_ top and bottom. If we divide arm’s bottom piece to arm’s top piece , again it gives golden ratio. People’s height and belly ’ s height ’s ratio is golden ratio. People’ s overarm length equal to people’s height. People’s foot length equal to between length people’s overarm and carpus.

QUEEN ANNE’S LACE _SYMMETRY There is a symmetry because leafs are the same and there is a regularly ratio their between. FRUITS(CROSS-SECTIONS)_SYMMETERY They are the same shapes. If we combine to this , we can see a whole them

SALT_CUBIC CRYSTALS Salt’s molecules is the same and so they are like symmetry and they are cubic crystals. SPIDER WEBS_CONCENTRIC POLYGONS _SPIRAL We can say spiral to this shape _ spider webs and this spiral is Arsimed Spiral. Spiders start from center and it always do webs in ratio also it always do webs on a line And also there are some symmetry between of mutual webs

ELEPHANT AND Pİ NUMBER An elephant ’s foot is circle. If we measure to elephant’s foot ‘s diameter and if we multiple to this answer by 2, we can estimate of elephant ’ s high PLANETS, MOONS, STARS_SPHERES ORBITS OF PLANETS_GENERALLY CIRCULAR ORBIT OF PLANETS _GENERALLY CIRCULAR (WITH SLIGHT VARIATIONS) There is a specific orbit in planets. Each planets move the elliptic orbit around of Sun. Average radiuses of planets ’s of orbits namely measurement of each planets and turning periods of in the orbits are the same. In brief there is a pattern their between

PONDS AND LAKES _REFLECTIONS This is like a symmetry. The water effect to cloud exactly or completely. BARCHAN DUNES_CRESCENT SHAPED SAND DUNES Each wave are the same and it has got a pattern in the desert.

CONSTELLATIONS_ANGLES There is a ratio between of angles and this ratio is specific and regular ratio.

MATHEMATICS AND NATURE 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. More symmetrical faces are generally regarded as more aesthetically pleasing

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 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.

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 store their eggs and larvae. Hexagons are six-sided polygons, closed, 2 -dimensional, many-sided figures with straight edges. Shapes - Cones Volcanoes form cones, the steepness and height of which depends on the runniness (viscosity) of the lava. Fast, runny lava forms flatter cones; thick, viscous lava forms steep-sided cones. Cones are 3 -dimensional solids whose volume can be calculated by 1/3 x area of base x height.

Parallel lines In mathematics, parallel lines stretch to infinity, neither converging nor diverging. These parallel dunes in the Australian desert aren't perfect - the physical world rarely is. Geometry - Human induced People impose their own geometry on the land, dividing a random environment into squares, rectangles and bisected rhomboids, and impinging on the natural diversity of the environment.

Pi Any circle, even the disc of the Sun as viewed from Cappadoccia, central Turkey during the 2006 t otal eclipse, holds that perfect relationship where the circumference divided by the diameter equals pi. First devised (inaccurately) by the Egyptians and Babylonians, the infinite decimal places of pi (approximately 3. 1415926. . . ) have been calculated to billions of decim Fractals Many natural objects, such as frost on the branches of a tree, show the relationship where similarity holds at smaller and smaller scales. This fractal nature mimics mathematical fractal shapes where form is repeated at every scale. Fractals, such as the famous Mandelbrot set, cannot be represented by classical geometry.

Zero - Placeholder and Number Zero is one of the most important mathematical concepts. The idea of zero as a placeholder, eg to distinguish 303 from 33, developed in both Indian and Babylonian cultures. Three Indian mathematicians, Brahmagupta (about 628 AD), Mahavira (about 850 AD) and Bháskara (1114 - about 1185 AD), are credited with defining zero as a number, and defining the rules for subtracting, adding, multiplying and dividing by zero. Fibonacci spiral 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, I t forms a Fibonacci spiral. Many examples of the Fibonacci spiral 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 spiral can be found by tracing the seeds of a sunflower from the centre outwards. Geometric sequence Bacteria such as Shewanella oneidensis multiply by doubling their population in size after as little as 40 minutes. A geometric sequence such as this, where each number is double the previous number [or f(n+1) = 2 f(n)] produces a rapid increase in the population in a very short time.

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 theory of snowflake uniqueness disappears into the clouds. The theory may have originated from Wilson Bentley's extraordinary feat photographing over 5000 snowflakes in the 1930 s. He found no two alike. Infinity Is 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 - is one infinite set larger than the other? The deep questions of maths can leave you feeling very small in a vast universe.

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