Fractals and Symmetry

By: Group 3 ABENOJAR, GARCIA, RAVELO

Symmetry

Markus Reugels

• A photographer who showed that beauty can exist in places we don’t expect it to be.

• Most of his photographs are close-ups of water droplets and the water crown which features a special geometric figure called the crown is formed from splashing water.

Etymology

• Symmetry came from the Greek word symmetría which means “measure together”

Symmetry conveys two meanings…

The First

• Is an imprecise sense of harmony and beauty or balance and proportion.

The Second

• Is a well-defined concept of balance or patterned self-similarity that can be proved by geometry or through physics.

Symmetry

Geometry

Mathematics

Science

Reflection

Rotation

Helical

Scale/Fractals

Odd and Even Functions Inverse Functions

Music

Passage through time

Spatial relationships

Architecture

Social Interactions

Arts/Aesthetics

Religious Symbols

Knowledge

Translation

Logic

Rotoreflection Glide Reflection

Symmetry in Geometry

Symmetry in Geometry

• “The exact correspondence of form and constituent configuration on opposite sides of a dividing line or plane or about a center or an axis” (American Heritage® Dictionary of the English Language 4th ed., 2009)

• In simpler terms, if you draw a specific point, line or plane on an object, the first side would have the same correspondence to its respective other side.

Reflection Symmetry

• Symmetry with respect to an axis or a line.

• A line can be drawn of the object such that when one side is flipped on the line, the object formed is congruent to the original object, vice versa.

The location of the line matters

True Reflection Symmetry False Reflection Symmetry

Rotational Symmetry

• Symmetry with respect to the figure’s center

• An axis can be put on the object such that if the figure is rotated on it, the original figure will appear more than once

• The number of times the figure appears in one complete rotation is called its order.

Figures and their order

Order 2 Order 4 Order 6 Order 5

Order 8 Order 3 Order 7

Other types of Symmetry

• Translational symmetry – looks the same after a particular translation

• Glide reflection symmetry – reflection in a line or plane combined with a translation along the line / in the plane,

results in the same object

• Rotoreflection symmetry – rotation about an axis (3D)

• Helical symmetry – rotational symmetry along with translation along the axis of rotation called the screw

axis

• Scale symmetry – the new object has the same properties as the original if an object is expanded or

reduced in size

– present in most fractals

Symmetry in Math

• Symmetry is present in even functions – they are symmetrical along the y-axis

• Symmetry is present in odd functions as well – they are symmetrical with respect to the origin. They have order 2 rotational symmetry.

cos(θ) = cos(- θ) sin(-θ) = -sin( θ)

Symmetry in Math

• Functions and their inverses exhibit reflection wrt the line with the equation x = y

• f(f-1(x)) = f-1(f(x)) = x

ln(𝑒 x) = xln(𝑒) = x(1) = x

Passage of time

Time is symmetric in the sense that if it is reversed the exact same events are happening in reverse order thus making it symmetric. Time can be reversed but it is not possible in this universe because it would violate the second law of thermodynamics.

Perception of time is different from any given object. The closer the objects travels to the speed of light, the slower the time in its system gets or he faster its perception of time would be. This means it could only be possible to have a reverse perception of time on a specific system but not a reverse perception on the entire system.

THIS WON’T APPEAR IN THE QUIZ

Spatial relationship

Knowledge

Religious Symbols

Music

Fractals

Etymology

• Fractal came from the Latin word fractus which means “interrupted”, or “irregular”

• Fractals are generally self-similar patterns and a detailed example of scale symmetry.

Julian Fractal

History

• Mathematics behind fractals started in the early 17th cenury when Gottfried Leibniz, a mathematician and philosopher, pondered recursive self-similarity.

• His thinking was wrong since he only considered a straight line to be self-similar.

History

• In 1872, Karl Weiestrass presented the first definition of a function with a graph that can be considered a fractal.

• Helge von Koch, in 1904, developed an accurate geometric definition by repeatedly trisecting a straight line. This was later known as the Koch curve.

History

• In 1915, Waclaw Sierpinski costructed the Sierpinski Triangle.

• By 1918, Pierre Fatou ad Gaston Julia, described fractal behaviour associated with mapping complex numbers. This also lead to ideas about attractors and repellors an eventually to the development of the Julia Set.

Benoît Mandelbrot

• A mathematician who created the Mandelbrot set from studying the behavior of the Julia Set.

• Coined the term “fractal”

Mandelbrot Set

What is a fractal?

• A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension. And may fall between integers.

Fibonacci word by Samuel Monnier

Iteration

• Iteration is the repetition of an algorithm to achieve a target result. Some basic fractals follow simple iterations to achieve the correct figure.

First four iterations of the Koch Snowflake

Whut?

• Let’s look at the line on the right, when it is divided by 2, the number of self-similar pieces becomes 2. When divided by 3, the number of self-similar pieces becomes 3.

A formula is given to calculate the dimension of a given object:

log(𝑁)

log(𝜖)

where N = number of self-similar pieces

𝜖 = scaling factor

We can now substitute: log 2

log 2= 1

Whut?

• For the plane:

log 4

log 2=

log 22

log 2=

2 log 2

log 2= 2

• For the space: log 27

log 3=

log 33

log 3=

3 log 3

log 3= 3

Sierpinski Triangle

• Clue: Iteration 1 has an 𝜖 of 1, Iteration 2 has an 𝜖 of 2, Iteration 3 has an 𝜖 of 4 and so on.

• Answer: log 3

log 2= 1.584962500~1.58

That means that the Sierpinski triangle has a fractal dimension of about 1.58. How could that be? Mathematically, that is its dimension but our eyes see an infinitely complex figure.

Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5

Types of Self-Similarity

Exact Self-similarity

• Identical at all scales

• Example: Koch snowflake

Quasi Self-similarity

• Approximates the same pattern at different scales although the copy might be distorted or in degenerate form.

• Example: Mandelbrot’s Set

Types of Self-Similarity

Statistical Self-Similarity

• Repeats a pattern stochastically so numerical or statistical measures are preserved across scales.

• Example: Koch Snowflake

Closely Related Fractals

Mandelbrot Set Julia Set

Mandelbrot Set

Mandelbrot Iteration Towards Infinity

Self-repetition in the Mandelbrot Set

Zooming into Mandelbrot Set

Zoom into Mandelbrot Set Julia Set Plot

Newton Fractal

p(z) = z5 − 3iz3 − (5 + 2i) ƒ:z→z3−1

Applications of Fractals

Video Game Mapping

Meteorology

Art

Seismology

Geography

Coastline Complexity

Sources

• http://en.wikipedia.org/wiki/Symmetry

• http://ethemes.missouri.edu/themes/226

• http://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/symmetryrev2.shtml

• http://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/symmetryrev3.shtml