Math 6 Presentation: Fractals and Symmetry

1. Fractals and Symmetry
By: Group 3
ABENOJAR, GARCIA, RAVELO

2. Symmetry

3. 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.

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

5. Symmetry conveys two meanings…

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

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

8. Odd and Even Functions
Inverse Functions
Rotoreflection Glide Reflection Religious Symbols Mathematics
Rotation Scale/Fractals Logic
Reflection Geometry Helical
Translation
Social Interactions
Symmetry
Arts/Aesthetics
Passage through time
Science Music
Architecture
Spatial relationships
Knowledge

9. Symmetry in Geometry

10. 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.

11. 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.

12. The location of the line matters
True Reflection Symmetry False Reflection Symmetry

13. 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.

14. Figures and their order
Order 2 Order 4 Order 6 Order 5
Order 8 Order 3 Order 7

15. 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

16. 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)

17. 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

18. 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 time
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.

19. Spatial relationship

20. Knowledge

21. Religious Symbols

22. Music

23. Fractals

24. 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

25. 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.

26. 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.

27. 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.

28. Benoît Mandelbrot
• A mathematician who created
the Mandelbrot set from
studying the behavior of the
Julia Set.
• Coined the term “fractal”
Mandelbrot Set

29. 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

30. 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

31. 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⁡ ������)
(
log⁡ ������)
(
where N = number of self-similar pieces
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡������ = scaling factor
We can now substitute:
log 2
=1
log 2

32. 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

33. 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 2
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.

34. 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

35. Types of Self-Similarity
Statistical Self-Similarity
• Repeats a pattern
stochastically so numerical
or statistical measures are
preserved across scales.
• Example: Koch Snowflake

36. Closely Related Fractals
Mandelbrot Set Julia Set

37. Mandelbrot Set
Mandelbrot Iteration Towards Self-repetition in the Mandelbrot
Infinity Set

38. Zooming into Mandelbrot Set

39. Zoom into Mandelbrot Set Julia Set Plot

40. Newton Fractal
p(z) = z5 − 3iz3 − (5 + 2i) ƒ:z→z3−1

41. Applications of Fractals

42. Video Game Mapping

43. Meteorology

44. Art

45. Seismology

46. Geography

47. Coastline Complexity

48. 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