A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale

this shell has a fractal because we can see the shape of each section is the same

(fractal geometry is an actual separate branch of geometry)

Snowflakes demonstrate a theorem known as Metatron's cube

Metatron's Cube is a name for a geometric figure composed of 13 equal circles with lines from the center of each circle extending out to the centers of the other 12 circles.

Also called "the fruit of life", derived from the flower of life

Sunflowers

Honey Combs

Snowflakes

Leaves

Birds

Pyrite Crystals

Right angles are very rarely seen in nature, but when they are seen, they are seen in mostly crystals, and rock formations

right angles are angles that are exactly 90 degrees

it is notated with a box, instead of a curve

Peacock Feathers

**GEOMETRY IN NATURE**

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When bees make their hives, they make the shape of their honey combs into regular hexagons

we can calculate the number of diagonals (d) of a polygon if we have the number of sides (n)

in this case the number of sides is 6 since we are using a hexagon

An Obtuse Angle is more than 90° but less than 180°

The angle of the bird's wings is an obtuse angle, because it can be added with other angles to form a straight line (180 degrees), but less than 90 degrees.

You can see acute angles in nature in the form of leaves

An acute angle is an angle that is less than 90° and greater than 0°

The veins of these particular leaves on the right side all consist of acute angles

Peacock feathers are a great example of reflectional symmetry

This means that one half of the feather is an exact reflection of the other

The line of symmetry can be in any direction, but in this case, it goes from the bottom of the feather to the top

Circumference can be calculated from a sunflower's circular center

Circumference-the linear distance around a circular object

C = 2π(r)

Starfish

If any internal angle is greater than 180° then the polygon is concave

a starfish is a concave polygon because, the area where the arms meet form an internal angle greater than 180 degrees

Ripples

Ripples are another example of natural circles

ripples are formed when a drop of water's momentum drives a wave of water outward (radial force)

radius is the distance from the center to the edge

diameter starts at one side of the circle, goes through the center and ends on the other side

Shells

Turtles

Turtles' shells are examples of tessellations in nature

Tessellation-the tiling of a plane using one or more geometric shapes with no overlaps

Trees

Tree trunks are examples of parallel lines

Parallel lines-two lines in a plane that do not intersect or touch at any point

the bottom right picture has a fallen trunk that crosses over two parallel trunks

Transversal: A line that cuts across two or more (usually parallel) lines.

Cauliflower

2

Romanesco Cauliflower

This is an example of Sierpinski's triangle in nature, a fractal named after Waclaw Sierpinski.

overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles

d=6(6-3)

2

d=6(3)

2

d=n(n-3)

d=9

BIBLIOGRAPHY

"Math Is Fun - Maths Resources." Math Is Fun - Maths Resources. N.p., n.d. Web. 12 Dec. 2014.

"25 Examples Of Perfect Geometry Found In Nature." List25. N.p., 27 Aug. 2013. Web. 13 Dec. 2014.

"Congruent - Math Word Definition - Math Open Reference." Congruent - Math Word Definition - Math Open Reference. N.p., n.d. Web. 18 Dec. 2014.

"Wolfram MathWorld: The Web's Most Extensive Mathematics Resource." Wolfram MathWorld: The Web's Most Extensive Mathematics Resource. N.p., n.d. Web. 11 Dec. 2014.

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then we get the number of diagonals which is 9

Butterfly Wings

A butterfly's wings are an example of congruency.

Congruency: equal size and shape

The butterflies to the right have two congruent wings, which help them with lift.

Flower of Life

The Flower of Life is the name that gives to a geometrical figure composed of multiple evenly-spaced, overlapping circles.

A "Flower of Life" figure consists of seven or more overlapping circles

involves sacred geometry

Sacred geometry refers to geometric forms that were once used in the design of holy sites, including western churches and cathedrals

Spiderwebs

In spider webs you can see many different polygons and geometrical shapes

In the spiderweb below, you can see squares and triangles

In some cases, you can also see dodecagons (12 sides), and rectangles.

The sum of the interior angles of a triangle is 180°.