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Mathematics and Art: Making Beautiful Music Together

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Title: Mathematics and Art: Making Beautiful Music Together


1
Mathematics and ArtMaking Beautiful Music
Together
  • D.N. Seppala-Holtzman
  • St. Josephs College
  • faculty.sjcny.edu/holtzman

2
Math Art the Connection
  • Many people think that mathematics and art are
    poles apart, the first cold and precise, the
    second emotional and imprecisely defined. In
    fact, the two come together more as a
    collaboration than as a collision.

3
Math Art Common Themes
  • Proportions
  • Patterns
  • Perspective
  • Projections
  • Impossible Objects
  • Infinity and Limits

4
The Divine Proportion
  • The Divine Proportion, better known as the Golden
    Ratio, is usually denoted by the Greek letter
    Phi F.
  • F is defined to be the ratio obtained by dividing
    a line segment into two unequal pieces such that
    the entire segment is to the longer piece as the
    longer piece is to the shorter.

5
A Line Segment in Golden Ratio
6
F The Quadratic Equation
  • The definition of F leads to the following
    equation, if the line is divided into segments of
    lengths a and b

7
The Golden Quadratic II
  • Cross multiplication yields

8
The Golden Quadratic III
  • Setting F equal to the quotient a/b and
    manipulating this equation shows that F satisfies
    the quadratic equation

9
The Golden Quadratic IV
  • Applying the quadratic formula to this simple
    equation and taking F to be the positive solution
    yields

10
Properties of F
  • F is irrational
  • Its reciprocal, 1/ F F - 1
  • Its square, F2 F 1

11
F Is an Infinite Square Root
12
F is an Infinite Continued Fraction
13
F and the Fibonacci Numbers
  • The Fibonacci numbers fn are the following
    sequence
  • 1 1 2 3 5 8 13 21 .
  • After the first two 1s, each number is the sum
    of the preceding two numbers.
  • Thus fn2 fn1 fn

14
F and the Fibonacci Numbers
  • Amazingly, the ratio of sequential Fibonacci
    numbers gets closer and closer to F as n gets
    larger and larger.
  • That is Limit (fn1 / fn ) F

15
Constructing F
  • Begin with a 2 by 2 square. Connect the midpoint
    of one side of the square to a corner. Rotate
    this line segment until it provides an extension
    of the side of the square which was bisected.
    The result is called a Golden Rectangle. The
    ratio of its width to its height is F.

16
Constructing F
B
ABAC
C
A
17
Properties of a Golden Rectangle
  • If one chops off the largest possible square from
    a Golden Rectangle, one gets a smaller Golden
    Rectangle.
  • If one constructs a square on the longer side of
    a Golden Rectangle, one gets a larger Golden
    Rectangle.
  • Both constructions can go on forever.

18
The Golden Spiral
  • In this infinite process of chopping off squares
    to get smaller and smaller Golden Rectangles, if
    one were to connect alternate, non-adjacent
    vertices of the squares, one gets a Golden Spiral.

19
The Golden Spiral
20
The Golden Spiral II
21
The Golden Triangle
  • An isosceles triangle with two base angles of 72
    degrees and an apex angle of 36 degrees is called
    a Golden Triangle.
  • The ratio of the legs to the base is F.
  • The regular pentagon with its diagonals is simply
    filled with golden ratios and triangles.

22
The Golden Triangle
23
A Close RelativeRatio of Sides to Base is 1 to F
24
Golden Spirals From Triangles
  • As with the Golden Rectangle, Golden Triangles
    can be cut to produce an infinite, nested set of
    Golden Triangles.
  • One does this by repeatedly bisecting one of the
    base angles.
  • Also, as in the case of the Golden Rectangle, a
    Golden Spiral results.

25
Chopping Golden Triangles
26
Spirals from Triangles
27
F In Nature
  • There are physical reasons that F and all things
    golden (including the Fibonacci numbers)
    frequently appear in nature.
  • Golden Spirals are common in many plants and a
    few animals, as well.

28
Sunflowers
29
Pinecones
30
Pineapples
31
The Chambered Nautilus
32
F in biological populations
  • The ratio of female honey bees to males is F.
  • This is a result of the fact that male bees are
    drones with only one parent while females have
    two parents.
  • This all goes back to the relationship between F
    and the Fibonacci numbers.

33
Angel Fish
34
Tiger
35
Human Face I
36
Human Face II
37
Le Corbusiers Man
38
A Golden Solar System?
39
F In Art Architecture
  • For centuries, people seem to have found F to
    have a natural, nearly universal, aesthetic
    appeal.
  • Indeed, it has had near religious significance to
    some.
  • Occurrences of F abound in art and architecture
    throughout the ages.

40
The Pyramids of Giza
41
The Pyramids and F
42
The Pyramids Were Laid Out in a Golden Spiral
43
The Parthenon
44
The Parthenon II
45
The Parthenon III
46
Cathedral of Chartres
47
Cathedral of Notre Dame
48
Michelangelos David
49
Michelangelos Holy Family
50
Rafaels The Crucifixion
51
Da Vincis Mona Lisa
52
Mona Lisa II
53
Da Vincis Study of Facial Proportions
54
Da Vincis St. Jerome
55
Da Vincis The Annunciation
56
Da Vincis Study of Human Proportions The
Vitruvian Man
57
Rembrandts Self Portrait
58
Seurats Parade
59
Seurats Bathers
60
Turners Norham Castle at Sunrise
61
Mondriaans Broadway Boogie-Woogie
62
Hoppers Early Sunday Morning
63
Dalis The Sacrament of the Last Supper
64
Literally an (Almost) Golden Rectangle
65
Patterns
  • Another subject common to art and mathematics is
    patterns.
  • These usually take the form of a tiling or
    tessellation of the plane.
  • Many artists have been fascinated by tilings,
    perhaps none more than M.C. Escher.

66
Patterns Other Mathematical Objects
  • In addition to tilings, other mathematical
    connections with art include fractals, infinity
    and impossible objects.
  • Real fractals are infinitely self-similar objects
    with a fractional dimension.
  • Quasi-fractals approximate real ones.

67
Fractals
  • Some art is actually created by mathematics.
  • Fractals and related objects are infinitely
    complex pictures created by mathematical
    formulae.

68
The Koch Snowflake (real fractal)
69
The Mandelbrot Set (Quasi)
70
Blow-up 1
71
Blow-up 2
72
Blow-up 3
73
Blow-up 4
74
Blow-up 5
75
Blow-up 6
76
Blow-up 7
77
Fractals Occur in Nature (the coastline)
78
Another Quasi-Fractal
79
Yet Another Quasi-Fractal
80
And Another Quasi-Fractal
81
Tessellations
  • There are many ways to tile the plane.
  • One can use identical tiles, each being a regular
    polygon triangles, squares and hexagons.
  • Regular tilings beget new ones by making
    identical substitutions on corresponding edges.

82
Regular Tilings
83
New Tiling From Old
84
Maurits Cornelis Escher (1898-1972)
  • Escher is nearly every mathematicians favorite
    artist.
  • Although, he himself, knew very little formal
    mathematics, he seemed fascinated by many of the
    same things which traditionally interest
    mathematicians tilings, geometry,impossible
    objects and infinity.
  • Indeed, several famous mathematicians have sought
    him out.

85
M.C. Escher
  • A visit to the Alhambra in Granada (Spain) in
    1922 made a major impression on the young Escher.
  • He found the tilings fascinating.

86
The Alhambra
87
An Escher Tiling
88
Eschers Butterflies
89
Eschers Lizards
90
Eschers Sky Water
91
M.C. Escher
  • Escher produced many, many different types of
    tilings.
  • He was also fascinated by impossible objects,
    self reference and infinity.

92
Eschers Hands
93
Eschers Circle Limit
94
Eschers Waterfall
95
Eschers Ascending Descending
96
Eschers Belvedere
97
Eschers Impossible Box
98
Penroses Impossible Triangle
99
Roger Penrose
  • Roger Penrose is a mathematical physicist at
    Oxford University.
  • His interests are many and they include cosmology
    (he is an expert on black holes), mathematics and
    the nature of comprehension.
  • He is the author of The Emperors New Mind.

100
Penrose Tiles
  • In 1974, Penrose solved a difficult outstanding
    problem in mathematics that had to do with
    producing tilings of the plane that had 5-fold
    symmetry and were non-periodic.
  • There are two roughly equivalent forms the kite
    and dart model and the dual rhombus model.

101
Dual Rhombus Model
102
Kite and Dart Model
103
Kites Darts II
104
Kites Darts III
105
Kite Dart Tilings
106
Rhombus Tiling
107
Rhombus Tiling II
108
Rhombus Tiling III
109
Penrose Tilings
  • There are infinitely many ways to tile the plane
    with kites and darts.
  • None of these are periodic.
  • Every finite region in any kite-dart tiling sits
    somewhere inside every other infinite tiling.
  • In every kite-dart tiling of the plane, the ratio
    of kites to darts is F.

110
Luca Pacioli (1445-1514)
  • Pacioli was a Franciscan monk and a
    mathematician.
  • He published De Divina Proportione in which he
    called F the Divine Proportion.
  • Pacioli Without mathematics, there is no art.

111
Jacopo de Barbaris Pacioli
112
In Conclusion
  • Although one might argue that Pacioli somewhat
    overstated his case when he said that without
    mathematics, there is no art, it should,
    nevertheless, be quite clear that art and
    mathematics are intimately intertwined.
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