Mathematics and Art: Making Beautiful Music Together 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.

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

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.

A Line Segment in Golden Ratio

 : The Quadratic Equation The definition of  leads to the following equation, if the line is divided into segments of lengths a and b:

The Golden Quadratic II Cross multiplication yields:

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

The Golden Quadratic IV Applying the quadratic formula to this simple equation and taking  to be the positive solution yields:

Properties of   is irrational Its reciprocal, 1/ , is one less than  Its square,  2, is one more than 

 Is an Infinite Square Root

Φ is an Infinite Continued Fraction

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 .

Constructing  A B C AB=AC

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.

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.

The Golden Spiral

The Golden Spiral II

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 . The regular pentagon with its diagonals is simply filled with golden ratios and triangles.

The Golden Triangle

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

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.

Chopping Golden Triangles

Spirals from Triangles

 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.

Sunflowers

Pinecones

Pineapples

The Chambered Nautilus

Angel Fish

Tiger

Human Face I

Human Face II

Le Corbusier’s Man

A Golden Solar System?

 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.

The Pyramids of Giza

The Pyramids and 

The Pyramids were laid out in a Golden Spiral

The Parthenon

The Parthenon II

The Parthenon III

Cathedral of Chartres

Cathedral of Notre Dame

Michelangelo’s David

Michelangelo’s Holy Family

Rafael’s The Crucifixion

Da Vinci’s Mona Lisa

Mona Lisa II

Da Vinci’s Study of Facial Proportions

Da Vinci’s St. Jerome

Da Vinci’s The Annunciation

Da Vinci’s Study of Human Proportions

Rembrandt’s Self Portrait

Seurat’s Parade

Seurat’s Bathers

Turner’s Norham Castle at Sunrise

Mondriaan’s Broadway Boogie- Woogie

Hopper’s Early Sunday Morning

Dali’s The Sacrament of the Last Supper

Literally an (Almost) Golden Rectangle

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.

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.

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

The Koch Snowflake (real fractal)

The Mandelbrot Set (Quasi)

Blow-up 1

Blow-up 2

Blow-up 3

Blow-up 4

Blow-up 5

Blow-up 6

Blow-up 7

Fractals Occur in Nature (the coastline)

Another Quasi-Fractal

Yet Another Quasi-Fractal

And Another Quasi-Fractal

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.

Regular Tilings

New Tiling From Old

Maurits Cornelis Escher ( ) 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.

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.

The Alhambra

An Escher Tiling

Escher’s Butterflies

Escher’s Lizards

Escher’s Sky & Water

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

Escher’s Hands

Escher’s Circle Limit

Escher’s Waterfall

Escher’s Ascending & Descending

Escher’s Belvedere

Escher’s Impossible Box

Penrose’s Impossible Triangle

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.

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.

Dual Rhombus Model

Kite and Dart Model

Kites & Darts II

Kites & Darts III

Kite & Dart Tilings

Rhombus Tiling

Rhombus Tiling II

Rhombus Tiling III

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 .

Luca Pacioli ( ) 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.”

Jacopo de Barbari’s Pacioli

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.