Mathematics and Art: Making Beautiful Music Together

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Mathematics and ArtMaking Beautiful Music
Together

• D.N. Seppala-Holtzman
• St. Josephs College
• faculty.sjcny.edu/holtzman

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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.

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Math Art Common Themes

• Proportions
• Patterns
• Perspective
• Projections
• Impossible Objects
• Infinity and Limits

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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.

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A Line Segment in Golden Ratio
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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

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The Golden Quadratic II

• Cross multiplication yields

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The Golden Quadratic III

• Setting F equal to the quotient a/b and
  manipulating this equation shows that F satisfies
  the quadratic equation

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The Golden Quadratic IV

• Applying the quadratic formula to this simple
  equation and taking F to be the positive solution
  yields

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Properties of F

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

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F Is an Infinite Square Root
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F is an Infinite Continued Fraction
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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

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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

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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.

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Constructing F
B
ABAC
C
A
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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.

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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.

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The Golden Spiral
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The Golden Spiral II
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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.

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The Golden Triangle
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A Close RelativeRatio of Sides to Base is 1 to F
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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.

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Chopping Golden Triangles
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Spirals from Triangles
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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.

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Sunflowers
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Pinecones
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Pineapples
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The Chambered Nautilus
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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.

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Angel Fish
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Tiger
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Human Face I
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Human Face II
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Le Corbusiers Man
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A Golden Solar System?
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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.

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The Pyramids of Giza
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The Pyramids and F
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The Pyramids Were Laid Out in a Golden Spiral
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The Parthenon
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The Parthenon II
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The Parthenon III
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Cathedral of Chartres
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Cathedral of Notre Dame
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Michelangelos David
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Michelangelos Holy Family
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Rafaels The Crucifixion
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Da Vincis Mona Lisa
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Mona Lisa II
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Da Vincis Study of Facial Proportions
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Da Vincis St. Jerome
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Da Vincis The Annunciation
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Da Vincis Study of Human Proportions The
Vitruvian Man
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Rembrandts Self Portrait
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Seurats Parade
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Seurats Bathers
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Turners Norham Castle at Sunrise
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Mondriaans Broadway Boogie-Woogie
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Hoppers Early Sunday Morning
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Dalis The Sacrament of the Last Supper
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Literally an (Almost) Golden Rectangle
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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.

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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.

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Fractals

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

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The Koch Snowflake (real fractal)
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The Mandelbrot Set (Quasi)
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Blow-up 1
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Blow-up 2
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Blow-up 3
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Blow-up 4
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Blow-up 5
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Blow-up 6
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Blow-up 7
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Fractals Occur in Nature (the coastline)
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Another Quasi-Fractal
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Yet Another Quasi-Fractal
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And Another Quasi-Fractal
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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.

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Regular Tilings
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New Tiling From Old
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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.

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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.

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The Alhambra
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An Escher Tiling
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Eschers Butterflies
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Eschers Lizards
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Eschers Sky Water
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M.C. Escher

• Escher produced many, many different types of
  tilings.
• He was also fascinated by impossible objects,
  self reference and infinity.

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Eschers Hands
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Eschers Circle Limit
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Eschers Waterfall
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Eschers Ascending Descending
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Eschers Belvedere
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Eschers Impossible Box
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Penroses Impossible Triangle
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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.

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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.

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Dual Rhombus Model
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Kite and Dart Model
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Kites Darts II
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Kites Darts III
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Kite Dart Tilings
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Rhombus Tiling
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Rhombus Tiling II
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Rhombus Tiling III
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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.

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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.

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Jacopo de Barbaris Pacioli
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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.