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”
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.
Odd and Even Functions Inverse Functions Rotoreflection Glide Reflection Religious Symbols Mathematics Rotation Scale/Fractals LogicReflection Geometry Helical Translation Social Interactions Symmetry Arts/AestheticsPassage through time Science Music Architecture Spatial relationships Knowledge
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 mattersTrue 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 orderOrder 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 • Symmetry is present in odd functions – they are functions as well – they are symmetrical along the y-axis 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( x) = xln() = x(1) = x
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.THIS WON’T APPEAR IN THE QUIZ Passage of timePerception of time is different from anygiven object. The closer the objectstravels to the speed of light, the slowerthe time in its system gets or he faster itsperception of time would be. This meansit could only be possible to have a reverseperception of time on a specific systembut not a reverse perception on the entiresystem.
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 thedimension of a given object: log ) ( log ) (where N = number of self-similar pieces = scaling factorWe can now substitute: log 2 =1 log 2
Whut?• For the plane: log 4 log 22 2 log 2 = = =2 log 2 log 2 log 2• For the space: log 27 log 33 3 log 3 = = =3 log 3 log 3 log 3
Sierpinski Triangle Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5• Clue: Iteration 1 has an of 1, Iteration 2 has an of 2, Iteration 3 has an of 4 and so on.• Answer: log 3 = 1.584962500~1.58 log 2That means that the Sierpinski triangle has a fractal dimension of about1.58. How could that be? Mathematically, that is its dimension but oureyes see an infinitely complex figure.
Types of Self-Similarity Exact Self-similarity Quasi Self-similarity• Identical at all scales • Approximates the same• Example: Koch snowflake 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 FractalsMandelbrot Set Julia Set
Mandelbrot SetMandelbrot Iteration Towards Self-repetition in the Mandelbrot Infinity Set