The Golden CurveThe Golden Curve

Maths, art and natureMaths, art and nature

It’s surprising who uses It’s surprising who uses mathsmaths

Many of the writers for The Simpsons have maths degrees.

Maths helps us…Maths helps us…

DesignDesign

Maths helps us…Maths helps us…

DesignDesign

PlanPlan

Maths helps us…Maths helps us…

DesignDesign

PlanPlan

UnderstandUnderstand

Where’s the maths?Where’s the maths?

Where’s the maths? (1)Where’s the maths? (1)

Where’s the maths? (2)Where’s the maths? (2)

Where’s the maths? (3)Where’s the maths? (3)

Where’s the maths? (4)Where’s the maths? (4)

The “Golden Curve”The “Golden Curve”

The Fibonacci sequenceThe Fibonacci sequence

• 1, 1, 2, 3, 5, 8, 13, 21, 34, ….1, 1, 2, 3, 5, 8, 13, 21, 34, ….

• What comes next?What comes next?

21 + 34 = 5521 + 34 = 55 34 + 55 = 8934 + 55 = 89

55 + 89 = 14455 + 89 = 144 89 + 144 = 233 89 + 144 = 233

The The ratiosratios of the sequence of the sequence

• 1 1 ÷ ÷ 1 = 11 = 1

• 2 2 ÷÷ 1 = 2 1 = 2

• 3 3 ÷÷ 2 = 1.5 2 = 1.5

• 5 ÷5 ÷ 3 = 1.6666666666… 3 = 1.6666666666…

• 8 8 ÷÷ 5 = 1.6 5 = 1.6

• 13 13 ÷÷ 8 = 1.625 8 = 1.625

• 21 21 ÷÷ 13 = 1.615384615384615384615… 13 = 1.615384615384615384615…

The The ratiosratios of the of the sequence…sequence…

• 8 8 ÷÷ 5 = 1.6 5 = 1.6

• 13 13 ÷÷ 8 = 1.625 8 = 1.625

• 21 21 ÷÷ 13 = 1.615384615384615384615… 13 = 1.615384615384615384615…

• 34 34 ÷÷ 21 = 1.619047619047619047619… 21 = 1.619047619047619047619…

• 55 55 ÷÷ 34 = 1.6176470588235294117647… 34 = 1.6176470588235294117647…

• 89 89 ÷÷ 55 = 1.6181818181818181818… 55 = 1.6181818181818181818…

The ratios tend to a The ratios tend to a number!number!

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

0 2 4 6 8 10 12

The golden ratioThe golden ratio

• The number that these ratios tend The number that these ratios tend towards is called the towards is called the golden ratiogolden ratio::

1.618033988749…1.618033988749…

Why golden?Why golden?

• It has lots of special propertiesIt has lots of special properties

• People have studied it for over 2000 yearsPeople have studied it for over 2000 years

• It occurs in art, architecture and natureIt occurs in art, architecture and nature

Golden rectanglesGolden rectangles

• A golden rectangle has A golden rectangle has one side which is one side which is 1.618033… times the 1.618033… times the length of the otherlength of the other

• The United Nations The United Nations building in New York building in New York has this shape has this shape

People like golden People like golden rectanglesrectangles

• Too long? Too short? Just right!Too long? Too short? Just right!

Nature and the golden ratioNature and the golden ratio

• Sunflowers use the Fibonacci sequenceSunflowers use the Fibonacci sequence

Shells and the golden curveShells and the golden curve

• The shell of a nautilus (sea creature)The shell of a nautilus (sea creature)