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 = B

4. Where can we find the Golden Ratio?

5. The Parthenon

6. The Great Pyramid of Giza 2560 BC
-Side lengths approximately 230m
-Base covers m^2
-Sides angled at 51.5 degrees.
1^2 + (√φ)^2 = φ^2
= 2.618 √φ φ 1 2

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

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

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

10. 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

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

19. Dimension
Line
Square
Cube
Pattern?

20. Sierpinski Triangle

21. 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?

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

23. 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