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MELBOURNE

MELBOURNE. Where great minds collide. PRESENTERS: Dr Marcy Robertson and Ms Emmy Amalia Emmy Amalia. Australia. Australia. No. 1 in Australia ∗ No. 33 in the World *Times Higher Education World University Rankings 2015–2016. Excellence across disciplines. Business. Humanities and Law.

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MELBOURNE

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  1. MELBOURNE Where great minds collide PRESENTERS: Dr Marcy Robertson and Ms Emmy Amalia Emmy Amalia

  2. Australia

  3. Australia

  4. No. 1in Australia∗ No. 33 in the World *Times Higher Education World University Rankings 2015–2016

  5. Excellence across disciplines Business Humanities and Law * Times Higher Education World University Rankings 2015-2016 + QS World University Subject Rankings 2015-2016

  6. Excellence across disciplines Science, Engineering and IT * Times Higher Education World University Rankings 2015-2016 + QS World University Subject Rankings 2015-16

  7. Your Overseas Adventure You could study business in NYC! 180 exchange partners in 40 countries

  8. The Melbourne Curriculum Flexible degrees, a world-standard curriculum and 100 majors to choose from

  9. The Melbourne Curriculum Graduate coursework degree Qualification for professional practice Degrees include: Architecture, Dental Surgery, Engineering, Law, Medicine, Physiotherapy, Veterinary Science • Undergraduate degree • 3 years • Agriculture • Arts • Biomedicine • Commerce • Design • Fine Arts • Music • Oral Health • Science Honours 1 year Specialise in your major area of study Graduate research degree Original, supervised research through a Masters or PhD YOUR CAREER

  10. Entry Requirements IB/A Levels 2017 Guaranteed Scores

  11. How to apply • BEFORE YOU APPLY • Meet subject prerequisites • Meet English requirements • Discuss your application with your school counsellor or overseas representative • THINGS TO CONSIDER • Student visa • Accommodation • Tuition fees • Cost of living www.futurestudents.unimelb.edu.au

  12. Fractal Geometry

  13. What is a Fractal?   • A fractal is a type of mathematical pattern that repeats itself at different scales.   • This property of repetition is called “self-similarity.” A self-similar pattern is a pattern whose of complexity increasing with magnification. If you divide a fractal pattern into parts you get a nearly identical reduced-size copy of the whole.  • Today we will create some fractals as well as learn about their properties, but first let's look at some examples.  

  14. Romanesco Broccoli: This vegetable is a perfect example of a fractal. Each large piece is made up of similarly-shaped smaller pieces, which are themselves made up of similarly-shaped smaller pieces.  

  15. Ammonites are marine cephalopods that have been extinct for 65 million years. Their spiral shells were built adding pieces that grow and twist at a constant rate. This leads to a self-similar pattern, that is at every scale, you see the same patter repeating itself.  

  16. Fractal antennas use fractal, self-similar design to minimize the size of the antenna while receiving radio signals across a range of frequencies.  These antennas are used in cell phones and other devices where size is important.  A great reference for physical applications of fractals can be found in the article “Overcoming Resistance with Fractals” in The Physics Teacher http://dx.doi.org/10.1119/1.2344109

  17. The Sierpinski Triangle • Step 1: Draw the largest equilateral triangle you can on your worksheet (each side should be about 12 units long). • On the side of your paper, start a chart where you write down the area of this triangle A_1=½(Base)(Height)

  18. The Sierpinski Triangle • Step 2: Take T_1 and connect its midpoints, splitting the triangle into four smaller triangles of equal size. Color in the triangle in the middle. You think about the shaded triangle as being removed. • Let A_2 be the area of all the uncolored triangles, record the data of A_2 in your chart.  

  19. The Sierpinski Triangle Think of T_1 we as sitting in the xy-plane

  20. The triangle T_2 is then the set of of points obtained from T_1 by : • first scaling three copies of T_1 each by a factor of r =1/2 • then translating two of the smaller triangles to form the desired arrangement. The scaled triangles would have to be translated so that the lower left vertices are at (1/2, 0) and(1/4, √3/4) respectively.

  21. The Sierpinski Triangle • Let us take T_2 and connect the midpoints of the three triangles we haven't colored in. That splits each of the three triangles into four smaller triangles of equal size. Let us color in the middle ones and call the resulting figure T_3.  • Write down A_3, which is the total area of all your uncolored triangles.  • Can you write down a function system that has input T_2 and output T_3?

  22. The Sierpinski Triangle • We can repeat this process on T_3 and connect the midpoints of the nine uncolored triangles. That splits each of the triangles into four smaller triangles of equal size. Let us color in the middle ones and call the resulting figure T_4.  • What is A_4? • Challenge Question: Write a function system that takes as input T_n and output T_{n+1}.

  23. The Sierpinski Triangle • Imagine that we take Step 5, Step 6, and so on to infinity. The resulting figure is called the Sierpinski triangle, or T.  • The technical definition is T=lim_{n→∞}T_n

  24. The Sierpinski Triangle Questions: Assume that we start with a triangle T which has a base of 10 units. For what value of n would the the Sierpinski triangle T_n would the area A_n become smaller than :  •  0.001?  • 0.000001?  • 0.000000000055?  • A tiny positive number ε? Students should notice that A_n = (3/4)^n for all n. For advanced students you can have them prove this by induction.

  25. The Sierpinski Triangle  You can modify this and ask that the students compute perimeter • Definition: If for any small number ε >0 we can always find a number N, that depends on ε, such that for any n ≥ N all the elements of the sequence of positive numbers A_n are always less than ε, then we say that the limit of the A_n as n goes to infinity is 0.  • This is a triangle with Area 0! • Challenge question: Generalize your solution of the problems above to any base length a > 0 to prove that the area of any Sierpinski triangle is always zero. 

  26. The Sierpinski Triangle  • What do you think is the dimension of the Sierpinski triangle? How would you define the dimension of a geometric figure in general? 

  27. Fractal Dimension  • "Dimension" is a term that we are used to talking about in our daily lives.  We say that we live in a three-dimensional world.  A box, for example, is three-dimensional (length, width, and height).  Whereas, an object such as a rectangle or a square is two-dimensional (length and width).  A line is considered to be one-dimensional (length).   • We say that a fractal has dimension based on how “self-similar” it is.

  28. Dimension  • What is self-similar dimension? Let us try some examples. Take a square and increase its linear size twice. How many original squares does one need to tile the enlarged square? 

  29. Dimension  • I'm sure you all have seen the answer, you look at the picture and count ... four! 

  30. Dimension  • Let’s try again. Let us take a square and increase its linear size three times. How many original squares does one need to tile the enlarged square? 

  31. Dimension  • We again see from the picture, the answer is 9. 

  32. Dimension  Question: One takes a square and enlarges it k times where k is a positive integer. How many original square does one need to tile the enlarged square?  • Answer: k^2 where k is the number of times we are magnifying the square. What is the 2? 

  33. Dimension  • This time, let us take a cube and increase its linear size twice. How many original cubes does one need to tile the enlarged cube? 

  34. Dimension  Question: One takes a cube and enlarges it k times where k is a positive integer. How many original cubes are needed to tile the enlarged cube?  • Answer: k^3 where k is the number of times we are magnifying the square. What is the 3? 

  35. Dimension  So, if we use what we have found out so far, we can develop a formula for finding dimension D.  • (magnification factor)^D  =  # of pieces needed to tile the original

  36. Dimension  In order to have a more useful equation, we can solve for D by using logarithms. • log (magnification factor^D )  =  log (# of pieces) • D log(magnification factor)  =  log (# of pieces) • D= log(# of pieces) / log(magnification factor)  

  37. Dimension  What would the dimension be for the Sierpinski Triangle ? We need to find the magnification factor.  We started with a single triangle, which was covered in four smaller triangles. After we removed the middle triangle we are left with 3 smaller triangles, or 3 "pieces."

  38. Dimension  Each of the pieces has a base length which is 1/2 the length of the original triangle.  In other words, if we wanted to look at this problem in the same way we thought about the cubes, we are increasing the little triangle by a factor of two and then covering it (minus the middle piece). Hence the magnification is 2. 

  39. D= log(number of pieces)/log (magnification) = log(3)/log(2) D = 1.58  Dimension 

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