The Divine Proportion. What is the Divine Proportion? What is the Divine Proportion? 10/15/2011. MA 341 Topics in Geometry Lecture 20

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1 The Divine Proportion MA 341 Topics in Geometry Lecture 20 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. 14-Oct-2011 MA What is the Divine Proportion? Other names frequently used for the golden ratio are the golden section golden cut golden proportion mean ratio divine proportion mean of Phidias golden mean golden number extreme ratio medial section divine section Denoted by phi = φ 14-Oct-2011 MA

2 Value? Does this ratio have a number associated with it, like π = Oct-2011 MA If a/b = φ, then Value? φ 2 = φ + 1 φ 2 - φ 1 = 0 14-Oct-2011 MA Value? Which is it? Is it + or -? What do we know about φ? a > b so φ > 1 and 14-Oct-2011 MA

3 Phi Oct-2011 MA Properties of φ From earlier we have that Therefore, 14-Oct-2011 MA Properties of φ 14-Oct-2011 MA

4 Φ and φ Sometimes authors use: Φ = and φ = = 1/Φ 14-Oct-2011 MA History Euclid'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. 14-Oct-2011 MA Golden Rectangle A rectangle is called a golden rectangle if its sides are in the ratio of the golden mean: a b 14-Oct-2011 MA

5 Golden Rectangle Remove a square from the golden rectangle: a b b b a-b 14-Oct-2011 MA Golden Rectangle The remaining rectangle is a golden rectangle! Do it again! a b b b a-b 14-Oct-2011 MA Golden Rectangle This actually gives us a construction for a golden rectangle using a compass and straightedge thus GeoGebra or Sketchpad Start with a square a 14-Oct-2011 MA

6 Golden Rectangle Find midpoint of the base and split square in two. a 14-Oct-2011 MA Golden Rectangle Construct diagonal MB. MC = a/2 and BC = a MB =? A B a D M C 14-Oct-2011 MA Golden Rectangle Construct circle with radius MB centered at M. A B a D M C 14-Oct-2011 MA

7 Golden Rectangle Mark point of intersection E. A B a C D M E 14-Oct-2011 MA Golden Rectangle Construct perpendicular at E. A B a C D M E 14-Oct-2011 MA Golden Rectangle Extend AB to meet this perpendicular. A B F a D C E 14-Oct-2011 MA

8 Golden Rectangle AFED is a golden rectangle. A F a D E 14-Oct-2011 MA Golden Spiral Construct a golden rectangle ABCD. A F D E 14-Oct-2011 MA Golden Spiral Construct a square inside XBCY. A X B D Y C 14-Oct-2011 MA

9 Golden Spiral Construct another square inside the smaller golden rectangle.. 14-Oct-2011 MA Again Golden Spiral 14-Oct-2011 MA Again Golden Spiral 14-Oct-2011 MA

10 Golden Spiral In each square construct a quarter circle: 14-Oct-2011 MA Golden Spiral In each square construct a quarter circle: 14-Oct-2011 MA Golden Spiral In each square construct a quarter circle: 14-Oct-2011 MA

11 Golden Spiral In each square construct a quarter circle: 14-Oct-2011 MA Golden Spiral In each square construct a quarter circle: 14-Oct-2011 MA Golden Spiral In each square construct a quarter circle: 14-Oct-2011 MA

12 Golden Spiral This does give a logarithmic spiral: Θ = a ln(b r), in polar coordinates 14-Oct-2011 MA Golden Spirals? 14-Oct-2011 MA Chambered nautilus shell Golden Spirals? Spiral galaxies 14-Oct-2011 MA

13 Golden Spirals? Cyclones 14-Oct-2011 MA Golden Spirals? Mandelbrot Set 14-Oct-2011 MA Golden Spirals? Flower heads 14-Oct-2011 MA

14 Golden Spirals? Phyllotaxis ( 14-Oct-2011 MA How? Consider the following list of numbers: 1,1,2,3,5,8,13,21,34,55,89,144, This is the Fibonacci sequence {F n }. We are interested in the quotients F n+1 /F n 14-Oct-2011 MA F 0 =1 F 1 =1 F 2 =2 F 3 =3 F 4 =5 F 5 =8 F 6 =13 F 7 =21 F 8 =34 Fibonacci Connection F 9 =55 F 5 /F 4 = 1.6 F 10 =89 F 6 /F 5 = F 11 =144 F 7 /F 6 = F 12 =233 F 8 /F 7 = F 13 =377 F 9 /F 8 = F 1 /F 0 = 1 F 10 /F 9 = F 2 /F 1 = 2 F 11 /F 10 = F 3 /F 2 =1.5 F 12 /F 11 = F 4 /F 3 = F 13 /F 12 = Oct-2011 MA

15 Fibonacci Connection Is it true that: Note: Since φ 2 = φ + 1, multiplying by φ n-1 gives a Fibonacci type relationship: φ n+1 = φ n + φ n-1 (F n+1 = F n + F n-1 ) And it so happens that 14-Oct-2011 MA Other representations? Consider the following sequence: What is lim a n? First, we need to know that {a n } has a limit. This can be shown with calculus. Let L = lim a n Then, 14-Oct-2011 MA Other representations? Hey!!! L = φ, so 14-Oct-2011 MA

16 Another representations Consider the following sequence: 14-Oct-2011 MA Other representations? What is lim b n? First, we need to know that {b n } has a limit. This can be shown with calculus. Let L = lim b n Then, Again, L = φ. 14-Oct-2011 MA Golden Triangle A golden triangle is an isosceles triangle where the ratio of the longer side to the base is φ. 1 φ φ 14-Oct-2011 MA

17 Golden Triangle What are the angles? cos(α)= ½/φ α = 72º, making summit angle 36º 1 α φ φ 14-Oct-2011 MA Golden Triangle 1 + φ = φ 2, so the larger triangle is similar to the smaller with similarity φ. φ φ φ Oct-2011 MA Golden Triangle Where do we find a golden triangle? 14-Oct-2011 MA

18 In fact: Red/green = green/blue = blue/pink = φ Golden Triangle 14-Oct-2011 MA Golden Angle If ratio of arcs a/b = φ, then angle subtended by smaller arc is called golden angle. It measures approximately , or about radians. 14-Oct-2011 MA It is exactly Golden Angle 14-Oct-2011 MA

19 Golden Ratio and Art Proportion.html 14-Oct-2011 MA The Parthenon 14-Oct-2011 MA The Parthenon 14-Oct-2011 MA

20 The Acropolis, Porch of the Maidens 14-Oct-2011 MA Chartes Cathedral & UN Building 14-Oct-2011 MA Taj Mahal 14-Oct-2011 MA

21 Fra Luca Pacioli 14-Oct-2011 MA Pacioli s De divina proportione Written in Milan in , Published in Venice in 1509 The 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 14-Oct-2011 MA Leonardo da Vinci 14-Oct-2011 MA

22 Leonardo da Vinci 14-Oct-2011 MA Facial Study 14-Oct-2011 MA Mona Lisa 14-Oct-2011 MA

23 Mona Lisa 14-Oct-2011 MA The Last Supper 14-Oct-2011 MA The Annunciation 14-Oct-2011 MA

24 Madonna and Child with St. Anne and St. John 14-Oct-2011 MA Michaelangelo & Raphael 14-Oct-2011 MA The Holy Family - Michelangelo 14-Oct-2011 MA

25 The Crucifixion - Raphael 14-Oct-2011 MA Self Portrait - Rembrandt Red line divides base into golden mean. 14-Oct-2011 MA The Bathers - Seurat 14-Oct-2011 MA

26 The Perfect Face Dr. Stephen Marquardt ( tm) Claims that this gives the most beautiful shape of human face Used decagons and pentagons and embodies φ in all their dimensions. 14-Oct-2011 MA The Perfect Face 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. 14-Oct-2011 MA The Perfect Face? 14-Oct-2011 MA

27 The Perfect Face? 14-Oct-2011 MA The Perfect Smile 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 3 rd tooth from center is φ 14-Oct-2011 MA See Donald Duck in Mathemagic Land by Disney Studios gah9zc 14-Oct-2011 MA

28 Without mathematics, there is no art. - Fra. Luca Pacioli 14-Oct-2011 MA