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Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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Page 1: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Mathematics and Art:Making Beautiful Music Together

Based a presentation byD.N. Seppala-Holtzman

St. Joseph’s College

Page 2: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 3: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Math & Art: Common Themes Proportions Patterns Perspective Projections Impossible Objects Infinity and Limits

Page 4: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 5: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

A Line Segment in Golden Ratio

Page 6: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Golden Quadratic III

is equal to the quotient a/b and it can be shown that is equal to:

(1+√5)/2

Page 7: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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

Page 8: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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 .

Page 9: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Constructing

A

B

C

AB=AC

Page 10: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 11: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 12: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Golden Spiral

Page 13: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Golden Spiral II

Page 14: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 15: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Golden Triangle

Page 16: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

A Close Relative:Ratio of Sides to Base is 1 to Φ

Page 17: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 18: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Chopping Golden Triangles

Page 19: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Spirals from Triangles

Page 20: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 21: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Sunflowers

Page 22: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Pinecones

Page 23: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Pineapples

Page 24: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Chambered Nautilus

Page 25: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

A Golden Solar System?

Page 26: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 27: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Pyramids of Giza

Page 28: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Pyramids and

Page 29: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Pyramids were laid out in a Golden Spiral

Page 30: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Parthenon

Page 31: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Parthenon II

Page 32: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Cathedral of Chartres

Page 33: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Cathedral of Notre Dame

Page 34: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Michelangelo’s David

Page 35: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Michelangelo’s Holy Family

Page 36: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Rafael’s The Crucifixion

Page 37: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Da Vinci’s Mona Lisa

Page 38: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Mona Lisa II

Page 39: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Da Vinci’s Study of Facial Proportions

Page 40: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Da Vinci’s St. Jerome

Page 41: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Da Vinci’s Study of Human Proportions

Page 42: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Rembrandt’s Self Portrait

Page 43: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Seurat’s Bathers

Page 44: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Turner’s Norham Castle at Sunrise

Page 45: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Mondriaan’s Broadway Boogie-Woogie

Page 46: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Dali’s The Sacrament of the Last Supper

Page 47: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Literally an (Almost) Golden Rectangle

Page 48: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 49: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 50: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Fractals Some art is actually created by

mathematics. Fractals and related objects are

infinitely complex pictures created by mathematical formulae.

Page 51: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Koch Snowflake (real fractal)

Page 52: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Mandelbrot Set (Quasi)

Page 53: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Blow-up 1

Page 54: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Blow-up 2

Page 55: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Blow-up 3

Page 56: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Blow-up 4

Page 57: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Blow-up 5

Page 58: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Blow-up 6

Page 59: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Blow-up 7

Page 60: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Fractals Occur in Nature (the coastline)

Page 61: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Another Quasi-Fractal

Page 62: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Yet Another Quasi-Fractal

Page 63: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

And Another Quasi-Fractal

Page 64: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 65: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Regular Tilings

Page 66: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

New Tiling From Old

Page 67: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 68: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 69: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

The Alhambra

Page 70: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

An Escher Tiling

Page 71: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Escher’s Butterflies

Page 72: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Escher’s Lizards

Page 73: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Escher’s Sky & Water

Page 74: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

M.C. Escher Escher produced many, many

different types of tilings. He was also fascinated by

impossible objects, self reference and infinity.

Page 75: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Escher’s Hands

Page 76: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Escher’s Circle Limit

Page 77: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Escher’s Waterfall

Page 78: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Escher’s Ascending & Descending

Page 79: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Escher’s Belvedere

Page 80: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Escher’s Impossible Box

Page 81: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Penrose’s Impossible Triangle

Page 82: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 83: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.

Page 84: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Dual Rhombus Model

Page 85: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Kite and Dart Model

Page 86: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Kites & Darts II

Page 87: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Kites & Darts III

Kite Dart

72 72

72

144

36 36

72

216

Page 88: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Kite & Dart Tilings

Page 89: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Rhombus Tiling

Page 90: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Rhombus Tiling II

Page 91: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Rhombus Tiling III

Page 92: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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 .

Page 93: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.”

Page 94: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

Jacopo de Barbari’s Pacioli

Page 95: Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

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.