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Surprising Connections in Math: From the Golden Ratio to Fractals

Surprising Connections in Math: From the Golden Ratio to Fractals. StFX Math Camp, May 2017. The Golden Rectangle. The Golden Ratio. A. B. Ratio of A to B is the golden ratio A = 1.618 B. Where can we find the Golden Ratio?. The Parthenon.

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Surprising Connections in Math: From the Golden Ratio to Fractals

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  1. Surprising Connections in Math:From the Golden Ratio to Fractals StFX Math Camp, May 2017

  2. The Golden Rectangle

  3. The Golden Ratio A B Ratio of A to B is the golden ratio A = 1.618 B

  4. Wherecan we find the Golden Ratio?

  5. The Parthenon

  6. The Great Pyramid of Giza2560 BC -Side lengths approximately 230m -Base covers 53 000 m^2 -Sides angled at 51.5 degrees. 1^2 + (√φ)^2 = φ^2 1+ 1.618 = 2.618 √φ φ 1 2

  7. CN Tower Base to observation deck 342 m Base to spire 553.33 m 553.33/342 = 1.618 = φ

  8. Moving on… 1,1,2,3,5,8,13,… What is the pattern?

  9. Fibonacci Numbers • Each number is the sum of the two before Fn= Fn-1+Fn-2 • Fibonacci Numbers in Nature • Youtube video on Fibonacci

  10. F2/F1=1/1=1 F3/F2=2/1=2 F4/F3=3/2=1.5 F5/F4=5/3=1.67 F6/F5=8/5=1.6 F7/F6=13/8=1.625 F8/F7=21/13=1.6154 F9/F8=34/21=1.619 F10/F9=55/34=1.6176 F11/F10=89/55=1.6182 Ratios of Fibonacci Numbers

  11. Connection • So the Fibonacci numbers and the golden ratio are connected • More about the Fibonacci Sequence and The Golden Ratio

  12. Pascal’s Triangle

  13. More on Pascal’s Triangle • All You Ever Wanted to Know About Pascal's Triangle and more

  14. Connection • So the Fibonacci numbers and Pascal’s triangle are also connected!

  15. Fractals No strict mathematical definition for fractals, but there are some common properties: • Detail at arbitrary scale • Repeated patterns • Geometric complexity • Fractal dimension

  16. Fractals in Nature

  17. Self-similarity • Patterns repeat at arbitrary scales • Object is made up of smaller versions of itself • Example: The Sierpinski Triangle

  18. Dimension • Line • Square • Cube • Pattern?

  19. Sierpinski Triangle

  20. Fractal Dimension • Doubling similarity: 2d=N • Sierpinski Gasket: N= 3, so 2d=3 • Take logs of both sides: d = log 3/log 2 ≈ 1.585 • Does this number make sense?

  21. Connection What happens if you colour all the odd numbers of Pascal’s triangle black and the even numbers white? • Sierpinski Pascal

  22. So… • Math is about much more than just numbers • Math is about finding patterns and symmetry • There can be beauty and wonder in math! Thanks! Tara Taylor, Department of Mathematics, Statistics and Computer Science St. Francis Xavier University ttaylor@stfx.ca

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