Surprising Connections in Math From the Golden Ratio

The Golden Ratio A B Ratio of A to B is the golden ratio

The Great Pyramid of Giza 2560 BC -Side lengths approximately 230 m -Base covers

CN Tower Base to observation deck 342 m Base to spire 553. 33 m

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

Fibonacci Numbers • Each number is the sum of the two before Fn= Fn-1+Fn-2

Ratios of Fibonacci Numbers • • • F 2/F 1=1/1=1 F 3/F 2=2/1=2 F

Connection • So the Fibonacci numbers and the golden ratio are connected • More

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

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

$Fractals No strict mathematical definition for fractals, but there are some common properties: •$

Self-similarity • Patterns repeat at arbitrary scales • Object is made up of smaller

Fractal Dimension • Doubling similarity: 2 d=N • Sierpinski Gasket: N= 3, so 2

Connection What happens if you colour all the odd numbers of Pascal’s triangle black

So… • Math is about much more than just numbers • Math is about

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Surprising Connections in Math: From the Golden Ratio to Fractals St. FX 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 230 m -Base covers 53 000 m^2 -Sides angled at 51. 5 degrees. 1^2 + (√φ)^2 = φ^2 √φ φ 1 2 1+ 1. 618 = 2. 618

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?

Fibonacci Numbers • 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 • • • F 2/F 1=1/1=1 F 3/F 2=2/1=2 F 4/F 3=3/2=1. 5 F 5/F 4=5/3=1. 67 F 6/F 5=8/5=1. 6 • • • F 7/F 6=13/8=1. 625 F 8/F 7=21/13=1. 6154 F 9/F 8=34/21=1. 619 F 10/F 9=55/34=1. 6176 F 11/F 10=89/55=1. 6182

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!

$Fractals No strict mathematical definition for fractals, but there are some common properties: •$

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: 2 d=N • Sierpinski Gasket: N= 3, so 2 d=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