Title: Mathematics and Art: Making Beautiful Music Together
1Mathematics and ArtMaking Beautiful Music
Together
- D.N. Seppala-Holtzman
- St. Josephs College
2Math 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.
3Math Art Common Themes
- Proportions
- Patterns
- Perspective
- Projections
- Impossible Objects
- Infinity and Limits
4The Divine Proportion
- The Divine Proportion, better known as the Golden
Ratio, is usually denoted by the Greek letter
Phi ?. - ? 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.
5A Line Segment in Golden Ratio
6? The Quadratic Equation
- The definition of ? leads to the following
equation, if the line is divided into segments of
lengths a and b
7The Golden Quadratic II
- Cross multiplication yields
-
8The Golden Quadratic III
- Setting ? equal to the quotient a/b and
manipulating this equation shows that ? satisfies
the quadratic equation
9The Golden Quadratic IV
- Applying the quadratic formula to this simple
equation and taking ? to be the positive solution
yields
10Properties of ?
- ? is irrational
- Its reciprocal, 1/ ?, is one less than ?
- Its square, ?2, is one more than ?
11? Is an Infinite Square Root
12F is an Infinite Continued Fraction
13Constructing ?
- 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 ?.
14Constructing ?
B
ABAC
C
A
15Properties 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.
16The 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.
17The Golden Spiral
18The Golden Spiral II
19The 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 ?.
- The regular pentagon with its diagonals is simply
filled with golden ratios and triangles.
20The Golden Triangle
21A Close RelativeRatio of Sides to Base is 1 to F
22Golden 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.
23Chopping Golden Triangles
24Spirals from Triangles
25? In Nature
- There are physical reasons that ? and all things
golden frequently appear in nature. - Golden Spirals are common in many plants and a
few animals, as well.
26Sunflowers
27Pinecones
28Pineapples
29The Chambered Nautilus
30Angel Fish
31Tiger
32Human Face I
33Human Face II
34Le Corbusiers Man
35A Golden Solar System?
36? In Art Architecture
- For centuries, people seem to have found ? to
have a natural, nearly universal, aesthetic
appeal. - Indeed, it has had near religious significance to
some. - Occurrences of ? abound in art and architecture
throughout the ages.
37The Pyramids of Giza
38The Pyramids and ?
39The Pyramids were laid out in a Golden Spiral
40The Parthenon
41The Parthenon II
42The Parthenon III
43Cathedral of Chartres
44Cathedral of Notre Dame
45Michelangelos David
46Michelangelos Holy Family
47Rafaels The Crucifixion
48Da Vincis Mona Lisa
49Mona Lisa II
50Da Vincis Study of Facial Proportions
51Da Vincis St. Jerome
52Da Vincis The Annunciation
53Da Vincis Study of Human Proportions
54Rembrandts Self Portrait
55Seurats Parade
56Seurats Bathers
57Turners Norham Castle at Sunrise
58Mondriaans Broadway Boogie-Woogie
59Hoppers Early Sunday Morning
60Dalis The Sacrament of the Last Supper
61Literally an (Almost) Golden Rectangle
62Patterns
- 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.
63Patterns 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.
64Fractals
- Some art is actually created by mathematics.
- Fractals and related objects are infinitely
complex pictures created by mathematical
formulae.
65The Koch Snowflake (real fractal)
66The Mandelbrot Set (Quasi)
67Blow-up 1
68Blow-up 2
69Blow-up 3
70Blow-up 4
71Blow-up 5
72Blow-up 6
73Blow-up 7
74Fractals Occur in Nature (the coastline)
75Another Quasi-Fractal
76Yet Another Quasi-Fractal
77And Another Quasi-Fractal
78Tessellations
- 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.
79Regular Tilings
80New Tiling From Old
81Maurits 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.
82M.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.
83The Alhambra
84An Escher Tiling
85Eschers Butterflies
86Eschers Lizards
87Eschers Sky Water
88M.C. Escher
- Escher produced many, many different types of
tilings. - He was also fascinated by impossible objects,
self reference and infinity.
89Eschers Hands
90Eschers Circle Limit
91Eschers Waterfall
92Eschers Ascending Descending
93Eschers Belvedere
94Eschers Impossible Box
95Penroses Impossible Triangle
96Roger 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.
97Penrose 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.
98Dual Rhombus Model
99Kite and Dart Model
100Kites Darts II
101Kites Darts III
102Kite Dart Tilings
103Rhombus Tiling
104Rhombus Tiling II
105Rhombus Tiling III
106Penrose 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 ?.
107Luca 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.
108Jacopo de Barbaris Pacioli
109In 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.