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

The Great Pyramid of Giza 2560 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

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

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

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

Ratios of Fibonacci Numbers

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

Pascal’s Triangle

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

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

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

Fractals in Nature

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

Dimension Line Square Cube Pattern?

Sierpinski Triangle

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?

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

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