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The Divine Proportion

MA 341 – Topics in GeometryLecture 20

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What is the Divine Proportion?In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.

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What is the Divine Proportion?Other names frequently used for the golden ratio are the

golden section golden meangolden cut golden numbergolden proportion extreme ratiomean ratio medial sectiondivine proportion divine sectionmean of Phidias

Denoted by phi = φ

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Value?Does this ratio have a number associated with it, like π =3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310...

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Value?

If a/b = φ, then

φ2 = φ + 1φ2 - φ – 1 = 0

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Value?

Which is it? Is it + or -?

What do we know about φ?

a > b so φ > 1 and

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Phi

1.61803398874989484820458683436563811772030917980576286213544862270526046281890244970720720418939113748475408807538689175212663386222353693179318006076672635443338908659593958290563832266131992829026788067520876689250171169620703222104321626954862629631361443814975870122034080588795445474924618569536486444924104432077134494704956584678850987433944221254487706647809158846074998871240076521705751797883416…

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Properties of φFrom earlier we have that

Therefore,

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Properties of φ

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Φ and φSometimes authors use:

Φ = 1.6180… and

φ = 0. 6180… = 1/Φ

HistoryEuclid's Elements provides first known written definition of golden mean: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.“Euclid gives a construction for cutting a line "in extreme and mean ratio", i.e. the golden ratio.Several propositions and their proofs employ the golden ratio.Some of these propositions show that the golden ratio is an irrational number.

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Golden Rectangle

• A rectangle is called a golden rectangle if its sides are in the ratio of the golden mean:

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b

a

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Golden Rectangle

Remove a square from the golden rectangle:

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b

a

b a-b

b

Golden Rectangle

The remaining rectangle is a golden rectangle!Do it again!

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b

a

b a-b

b

This actually gives us a construction for a golden rectangle using a compass and straightedge – thus GeoGebra or SketchpadStart with a square

Golden Rectangle

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a

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Golden Rectangle

Find midpoint of the base and split square in two.

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a

Golden Rectangle

Construct diagonal MB.MC = a/2 and BC = a

MB = ?

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a

A B

CD M

Golden RectangleConstruct circle with radius MB centered at M.

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a

A B

CD M

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Golden Rectangle

Mark point of intersection E.

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a

A B

CD M E

Golden Rectangle

Construct perpendicular at E.

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a

A B

CD M E

Golden Rectangle

Extend AB to meet this perpendicular.

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a

A B

CD

F

E

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Golden Rectangle

AFED is a golden rectangle.

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a

A

D

F

E

Golden Spiral

Construct a golden rectangle ABCD.

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A

D

F

E

Golden Spiral

Construct a square inside XBCY.

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A

D

B

C

X

Y

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Golden SpiralConstruct another square inside the smaller golden rectangle..

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Golden SpiralAgain

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Golden SpiralAgain

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Golden SpiralIn each square construct a quarter circle:

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Golden SpiralIn each square construct a quarter circle:

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Golden SpiralIn each square construct a quarter circle:

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Golden SpiralIn each square construct a quarter circle:

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Golden SpiralIn each square construct a quarter circle:

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Golden SpiralIn each square construct a quarter circle:

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Golden SpiralThis does give a logarithmic spiral:

Θ = a ln(b·r), in polar coordinates

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Golden Spirals?

14-Oct-2011 MA 341 35Chambered nautilus shell

Golden Spirals?

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Spiral galaxies

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Golden Spirals?

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Cyclones

Golden Spirals?

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Mandelbrot Set

Golden Spirals?

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Flower heads

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Golden Spirals?

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Phyllotaxis(http://www.math.smith.edu/phyllo//)

How?

Consider the following list of numbers:1,1,2,3,5,8,13,21,34,55,89,144,…

This is the Fibonacci sequence {Fn}. We are interested in the quotients

Fn+1/Fn

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Fibonacci ConnectionF0=1F1=1F2=2F3=3F4=5F5=8F6=13F7=21F8=34

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F9=55F10=89F11=144F12=233F13=377F1/F0 = 1F2/F1 = 2F3/F2 =1.5F4/F3 = 1.6667

F5/F4 = 1.6F6/F5 = 1.625F7/F6 = 1.6153.F8/F7 = 1.6190F9/F8 = 1.6176F10/F9 = 1.618181F11/F10 = 1.617977F12/F11 = 1.618055F13/F12 = 1.618025

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Fibonacci ConnectionIs it true that:

Note: Since φ2 = φ + 1, multiplying by φn-1

gives a Fibonacci type relationship:φn+1 = φn + φn-1

(Fn+1 = Fn + Fn-1)And it so happens that

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Other representations?Consider the following sequence:

What is lim an ?First, we need to know that {an} has a limit. This can be shown with calculus.Let L = lim an

Then,

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Other representations?

Hey!!! L = φ, so

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Another representationsConsider the following sequence:

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Other representations?What is lim bn ?First, we need to know that {bn} has a limit. This can be shown with calculus.Let L = lim bn

Then,

Again, L = φ.

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Golden Triangle

A golden triangle is an isosceles triangle where the ratio of the longer side to the base is φ.

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φ φ

1

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Golden TriangleWhat are the angles?cos(α)= ½/φ α = 72º, making summit angle 36º

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φ φ

1α

Golden Triangle

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φ φ

1

• 1 + φ = φ2, so the larger triangle is similar to the smaller with similarity φ.

φ

3636

36 7272

Golden Triangle

Where do we find a golden triangle?

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In fact:Red/green

= green/blue= blue/pink = φ

Golden Triangle

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If ratio of arcsa/b = φ, then angle subtended by smaller arc is called golden angle.It measures approximately 137.51°, or about 2.399963 radians.

Golden Angle

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It is exactly

Golden Angle

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Golden Ratio and Art

• http://www.goldennumber.net/• http://www.keplersdiscovery.com/Divine

Proportion.html• http://spiralzoom.com/

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The Parthenon

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The Parthenon

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The Acropolis, Porch of the Maidens

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Chartes Cathedral & UN Building

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Taj Mahal

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Fra Luca Pacioli

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Pacioli’s De divina proportioneWritten in Milan in 1496–98, Published in Venice in 1509The subject - mathematical and artistic proportion, especially mathematics of golden ratio and application in architecture.Leonardo da Vinci drew illustrations of regular solids in De divina proportione while living with and taking mathematics lessons from Pacioli. Discusses use of perspective by painters such as Piero della Francesca, Melozzo da Forlì, and Marco Palmezzano

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Leonardo da Vinci

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Leonardo da Vinci

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Facial Study

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Mona Lisa

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Mona Lisa

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The Last Supper

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The Annunciation

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Madonna and Child with St. Anne and St.

John

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Michaelangelo & Raphael

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http://web.me.com/paulscott.info/place/pm10/pm10.html

The Holy Family - Michelangelo

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The Crucifixion - Raphael

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Self Portrait - Rembrandt

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Red line divides base

into golden mean.

The Bathers - Seurat

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The Perfect Face

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Dr. Stephen Marquardt (http://www.beautyanalysis.com/index2_mba.htm)Claims that this gives the most beautiful shape of human faceUsed decagons and pentagons and embodies φin all their dimensions.

The Perfect Face

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This mask of the human face is based on the Golden Ratio. The proportions of the length of the nose, the position of the eyes and the length of the chin, all conform to some aspect of the Golden Ratio.

The Perfect Face?

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The Perfect Face?

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The Perfect Smile

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•Front two teeth form a golden rectangle •Also a golden ratio in height to width of center two teeth. •Ratio of the width of the 2 center teeth to those next to them is φ. •Ratio of width of smile to 3rd tooth from center is φ

See Donald Duck in Mathemagic Land by Disney Studios

http://www.youtube.com/watch?v=oT_Bxgah9zc

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Without mathematics, there is no art.- Fra. Luca Pacioli

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