Mathema'cal Beauty in Rome Lecture 10: The Golden Ra'o. Prof. Joseph Pasquale University of California, San Diego

Transcription

1 Mathema'cal Beauty in Rome Lecture 10: The Golden Ra'o Prof. Joseph Pasquale University of California, San Diego 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

2 Geometry has two great treasures: the theorem of Pythagoras, and the division of a line into extreme and mean ra'o Kepler 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

3 Euclid, 300 BC Greek mathema'cian Taught geometry in Alexandria Wrote Elements Geometry, number theory, axioms/proposi'ons During the last 2000 years, the two most widely read books have undoubtedly been the Bible and the Elements - Coxeter There is no royal road to geometry - Euclid 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

4 Extreme and Mean Ra'o A straight line is said to have been cut in extreme and mean ra'o when, as the whole line is to the greater segment, so is the greater to the less. Euclid s Elements, IV, 3 Given line of length a+b, a+b : a :: a : b 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

5 Luca Pacioli De Divina Propor'one, ~1497 On the divine propor'ons On mathema'cal and ar's'c propor'ons golden ra'o math applica'ons in architecture Illustrator: Leonardo da Vinci! 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

6 Golden Ra'o Subdivide line segment AB at H so that AB : AH :: AH : HB Alterna'vely, based on areas Rectangle AB BH = square AH (AB)(BH) = (AH)(AH) AB/AH = AH/BH A B 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

7 Algebraic Deriva'on Divide line into 2 parts of length φ and 1, φ > 1 φ+1 : φ :: φ : 1 φ is mean propor'onal (or geometric mean) between smaller part (1) and whole (1+φ) Note, only two quan''es in φ+1 : φ :: φ : 1 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

8 Solving for φ Given φ+1 : φ :: φ : 1 Mul'ply each side by φ to get Set up quadra'c equa'on and solve for φ > 0 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

9 Solving for φ Given φ+1 : φ :: φ : 1 (φ+1)/φ = φ Mul'ply each side by φ to get φ + 1 = φ 2 Set up quadra'c equa'on and solve for φ > 0 φ 2 φ 1 = 0 φ = (1 + 5)/2 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

10 Proper'es of φ φ Recall (φ+1)/φ = φ, or 1 + 1/φ = φ and φ + 1 = φ 2 Reciprocal of golden ra'o 1/φ = φ Square of golden ra'o φ 2 = 1 + φ /14/14 Mathema'cal Beauty in Rome, J. Pasquale,

11 Some Remarkable Results φ = / (1 + 1 / (1 + 1 / (1 + 1 / (1 + )))) φ = (1 + (1 + (1 + (1 + (1 + (1 + )))))) 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

12 Some Remarkable Results φ = / (1 + 1 / (1 + 1 / (1 + 1 / (1 + )))) φ = (1 + (1 + (1 + (1 + (1 + (1 + )))))) 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

13 Incommensurability Two lines are commensurable if they have a common measure A and B are commensurable if there exists a C A = mc, B = nc, m and n are integers Otherwise, A and B are incommensurable 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

14 Golden Ra'o is Incommensurable Given AB such that AB : AH :: AH : HB AH and HB are incommensurable So are AB and AH, and AB and HB Algebraically φ a/b for any integers a and b Other examples of incommensurability Diagonal and side of square: 2 Circumference and diameter of a circle: π 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

15 Proof that φ is Irra'onal Assume φ = a/b for some integers a > 0, b > 0 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

16 Proof that φ is Irra'onal Assume φ = a/b for some integers a > 0, b > 0 a = φ b = (1 + 1/φ) b = (1 + b/a) b = b + b 2 /a a 2 = ab + b 2 a 2 ba b 2 = 0 a = b ± (b 2 + 4b 2 )= b ± (5b 2 )= b ± b 5 = b (1 + 5) Since a is not an integer, contradic'on! 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

17 Hierarchy of Numbers Ra'onal Irra'onal Transcendental What type of number is 2 π φ 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

18 Geometric Construc'on Start with square ABCD Bisect it ver'cally with line EF Swing arc of radius EC to extension of AED, AG AD divides AG by the golden ra'o ABHG is a golden rectangle Ra'o of length AG to width AB is golden ra'o 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

19 Geometric Construc'on Start with square ABCD B F C H Bisect it ver'cally with line EF Swing arc of radius EC to extension of AED, AG A E D G AD divides AG by the golden ra'o ABHG is a golden rectangle Ra'o of length AG to width AB is golden ra'o 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

20 Golden Sequence Geometric progression Start with 1 Common ra'o of φ 1, φ, φ 2, φ 3, φ 4, φ 5, 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

21 Algebraic Construc'on Given φ 2 = φ + 1, mul'plying each side by φ φ 3 = φ 2 + φ φ 4 = φ 3 + φ 2 φ n = φ n- 1 + φ n- 2, n 2 Each term equals sum of two preceding terms Golden sequence is both an addi've and geometric progression! 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

22 Geometric Construc'on Lay off lengths 1 and φ With radius 1 + φ, swing arc from A to B Length of AB is φ 2 (recall φ 2 = φ + 1) With radius φ + φ 2, swing arc from B to C Length of BC is φ 3 (recall φ 3 = φ 2 + φ) 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

23 Hero s Construc'on Construct right triangle ABC with BC half of AB From C, swing an arc from B, loca'ng D From A, swing an arc from D, loca'ng E Then AE divides AB by the golden ra'o C D A E B 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

24 Golden Triangle An isosceles triangle in which ra'o between legs and base is the golden ra'o Acute: leg to base Obtuse: base to leg 1 O φ D 1/φ 1 A 1 B 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

25 Geometric Construc'on On line segment AO, locate point D that subdivides AO by golden ra'o; s is larger part Swing arcs from A and D with radius s; intersec'on is B Draw AB, OB, and DB AOB is acute golden triangle OBD is obtuse golden triangle AO/s = AO/AB = φ 1/φ A D 1 O 1 1 φ B 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

26 Magnitude of Angles Given AOB = θ ODB is isosceles OBD equals θ Exterior angle of ODB is 2θ O Therefore sum of angles is 5θ θ = 1/10 revolu'on (36 ) 1 θ φ D 3θ 1/φ 2θ 2θ 1 θ θ A 1 B 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

27 Gnomon Figure added to another to get a larger figure of the same shape Obtuse golden triangle is a gnomon 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

28 Take Three Golden Triangles θ φ φ φ φ 2θ 1 2θ 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

29 Combine: Regular Pentagon θ φ φ 2θ 1 2θ 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

30 Conversely, Take Pentagon 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

31 Add Lines to Get Golden Triangles 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

32 Pentagon to Pentagram All triangles are golden! 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

33 Kepler (Egyp'an) Triangle If the sides of a right triangle are in geometric ra'o, then the sides must be 1 : φ : φ Makes use of both Pythagorean Theorem and Golden Ra'o 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

34 Kepler (Egyp'an) Triangle If the sides of a right triangle are in geometric ra'o, then the sides must be 1 : φ : φ Makes use of both Pythagorean Theorem and Golden Ra'o Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ra'o Kepler 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

35 Construc'ng Kepler Triangle Construct a golden rectangle Construct a simple square Draw a line from midpoint of one side of square to opposite corner Use that line as radius to draw an arc that defines height of rectangle Use longer side of golden rectangle to draw arc that intersects opposite side of rectangle for hypotenuse of Kepler triangle 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

36 Construc'ng Kepler Triangle Construct a golden rectangle Construct a simple square Draw a line from midpoint of one side of square to opposite corner Use that line as radius to draw an arc that defines height of rectangle Use longer side of golden rectangle to draw arc that intersects opposite side of rectangle for hypotenuse of Kepler triangle 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

37 Squaring the Circle? Take a Kepler Triangle (1 : φ : φ) Draw a circle that circumscribes it Draw a square whose side is φ 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

38 Squaring the Circle? Take a Kepler Triangle (1 : φ : φ) Draw a circle that circumscribes it Draw a square whose side is φ Perimeter of circle perimeter of square! Error < 0.1%! It can t be exactly the same: why? 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

39 Fibonacci Leonardo Pisano (of Pisa) Fibonacci: filius Bonacci Liber abaci, 1202 Book of Calcula'on Actually, how to calculate without using an abacus Promoted Hindu- Arabic numerals over La'n Brought Fibonacci wide recogni'on Today, we know him for a par'cular problem 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

40 Liber Abaci, Chapter 12 A certain man had one pair of rabbits in a certain enclosed space, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

41 Liber Abaci, Chapter 12 Start with a pair of infant rabbits In one month, they become adults Each month: adult pair gives birth to pair of infants 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

42 Monthly Progression Jan I (infant pair) 1 Feb A (adult pair) 1 Mar A I 2 Apr A I A 3 May A I A A I 5 Jun A I A A I A I A 8 Jul A I A A I A I A A I A A I 13 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

43 Fibonacci Sequence Recurrence rela'on F n = F n- 1 + F n- 2 and F 0 = 1, F 1 = 1 This generates the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, Many interes'ng and surprising proper'es Journal: Fibonacci Quarterly 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

44 Consider Ra'o of F n / F n- 1 Recall φ F n F n / F n- 1 F n F n / F n /14/14 Mathema'cal Beauty in Rome, J. Pasquale,

45 Consider Ra'o of F n / F n- 1 Recall φ F n F n / F n- 1 F n F n / F n /14/14 Mathema'cal Beauty in Rome, J. Pasquale,

46 Fibonacci Sequence Converges to φ lim n F n / F n- 1 = φ Was proven by Johannes Kepler 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

47 Binet s Formula F n = 1/ 5 [ ((1 + 5)/2) n - ((1-5)/2) n ] or F n = 1/ 5 [φ n - (- 1/φ) n ] 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

48 Squaring Rectangles (1 1) + (1 2) + (2 3) = 3 2 (1 1) + (1 2) + (2 3) + (3 5) + (5 8) = 8 2 1x1 1x1 1x2 1x1 1x2 2x3 2x3 3x5 5x8 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

49 Penrose Tilings Only two kinds of 'les: kites and darts Can 'le en're plane without repea'ng paƒern Non- periodic: no matching with transla'on Note: 72 is angle of regular pentagon 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

50 Seven Ways to Arrange 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

51 Example of Penrose Tiling 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

52 Penrose Tiling and Golden Ra'o Kites: two acute golden triangles Darts: two obtuse golden triangles Ra'o of kites to darts in a Penrose 'ling is approximately the golden ra'o! 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

53 Specula'on Was golden ra'o used in architecture? Great pyramid? Parthenon? Bramante s Tempieƒo? Palazzo Farnese? Some find golden ra'o in various measurements: width to height, 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

54 Great Pyramid Great Pyramid b = ½ m h = m a = m Egyp'an triangle (1 : φ : φ) b = ½ m h = m a = m 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

55 Parthenon 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

56 Presence of Golden Ra'o Doub ul Measurement points seem arbitrary Also, compare φ ( ) to 8/5 = 1.6 (φ differs by 0.6%) 5/3 = 1.67 (φ differs by 2.9%) More likely these simple ra'os were used 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

57 Bramante s Tempieƒo Donato Bramante, 1502 Martyrium San Pietro in Montorio High Renaissance Harmonious propor'ons Simplicity of volumes 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

58 Eleva'on and Plan (Palladio) 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

59 Propor'ons Por'co H : W :: 3 : 5 Drum upper H : W :: 3 : 5 Total W : H :: 3 : 4 Drum W : H :: 2 : 3 Drum+dome W : H :: 2 : 4 Col vs. drum W : H :: 1 : 1 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

60 Another Analysis Height of por'co = height of upper drum and dome Dome and upper drum like Pantheon Diameter of por'co is 3/2 diameter of dome 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

61 Golden Ra'o in Tempieƒo? from John Pile, A History of Interior Design, /14/14 Mathema'cal Beauty in Rome, J. Pasquale,

62 Le Modulor Architectural system of propor'on explicitly based on golden ra'o By Le Corbusier ( ) So elas'c golden ra'o may be hard to detect 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

63 Romans and the Golden Ra'o No firm evidence of use in Roman architecture No men'on by Vitruvius Known by ancients as mean and extreme ra'o golden designa'on appeared in Renaissance Golden ra'o present in pentagons but this figure is rare in Roman architecture 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,

64 Reading and Homework Reading Chapter 2, pp Homework Chapter 2, p. 43: 5 References hƒp://en.wikipedia.org/wiki/golden_ra'o hƒp://en.wikipedia.org/wiki/fibonacci_number 7/14/14 Mathema'cal Beauty in Rome, J. Pasquale,