mathematics and art: making beautiful music together based a presentation by d.n. seppala-holtzman...
TRANSCRIPT
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Mathematics and Art:Making Beautiful Music Together
Based a presentation byD.N. Seppala-Holtzman
St. Joseph’s College
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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: .
• 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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The Golden Quadratic III
is equal to the quotient a/b and it can be shown that is equal to:
(1+√5)/2
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Properties of is irrationalo Its reciprocal, 1/ , is one less than
o Its square, 2, is one more than o There’s even more, but we won’t
get into that.o just think of these as strange but
true facts
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Constructing 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 .
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Constructing
A
B
C
AB=AC
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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 Triangleo An isosceles triangle with two base
angles of 72 degrees and an apex angle of 36 degrees is called a Golden Triangle.
o The ratio of the legs to the base is .
o 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 Relative:Ratio of Sides to Base is 1 to Φ
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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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In Natureo There are physical reasons that
and all things golden frequently appear in nature.
o 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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A Golden Solar System?
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In Art & Architecture o For centuries, people seem to have
found to have a natural, nearly universal, aesthetic appeal.
o Indeed, it has had near religious significance to some.
o Occurrences of abound in art and architecture throughout the ages.
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The Pyramids of Giza
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The Pyramids and
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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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Cathedral of Chartres
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Cathedral of Notre Dame
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Michelangelo’s David
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Michelangelo’s Holy Family
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Rafael’s The Crucifixion
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Da Vinci’s Mona Lisa
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Mona Lisa II
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Da Vinci’s Study of Facial Proportions
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Da Vinci’s St. Jerome
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Da Vinci’s Study of Human Proportions
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Rembrandt’s Self Portrait
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Seurat’s Bathers
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Turner’s Norham Castle at Sunrise
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Mondriaan’s Broadway Boogie-Woogie
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Dali’s 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 mathematician’s
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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Escher’s Butterflies
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Escher’s Lizards
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Escher’s 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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Escher’s Hands
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Escher’s Circle Limit
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Escher’s Waterfall
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Escher’s Ascending & Descending
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Escher’s Belvedere
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Escher’s Impossible Box
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Penrose’s 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 Emperor’s 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
Kite Dart
72 72
72
144
36 36
72
216
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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 .
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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 Φ the Divine Proportion.
Pacioli: “Without mathematics, there is no art.”
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Jacopo de Barbari’s 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.