Mathematics and Art Making Beautiful Music Together

Mathematics and Art: D.N. Seppala-Holtzman St. Joseph’s College faculty.sjcny.edu/~holtzman Making Beautiful Music Together 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: Φ. Φ 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 Φ: The Quadratic Equation  The definition of Φ leads to the following equation, if the line is divided into segments of lengths a and b: ab a  a b The Golden Quadratic II  Cross multiplication yields: a  ab  b 2 2 The Golden Quadratic III  Setting Φ equal to the quotient a/b and manipulating this equation shows that Φ satisfies the quadratic equation:    1  0 2 The Golden Quadratic IV  Applying the quadratic formula to this simple equation and taking Φ to be the positive solution yields: 1 5   1.618 2 Properties of Φ    Φ is irrational Its reciprocal, 1/ Φ = Φ - 1 Its square, Φ2 = Φ + 1 Φ Is an Infinite Square Root   1  1  1  1  ..... Φ is an Infinite Continued Fraction   1 1 1 1 1 1 1 1  ... 1 Φ and the Fibonacci Numbers     The Fibonacci numbers {fn } are the following sequence: 1 1 2 3 5 8 13 21 …. After the first two 1’s, each number is the sum of the preceding two numbers. Thus fn+2 = fn+1 + fn Φ and the Fibonacci Numbers  Amazingly, the ratio of sequential Fibonacci numbers gets closer and closer to Φ as n gets larger and larger.  That is: Limit (fn+1 / fn ) = Φ Constructing Φ  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 Φ. Constructing Φ B AB=AC A C 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 Φ. 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 Φ In Nature   There are physical reasons that Φ 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. Sunflowers Pinecones Pineapples The Chambered Nautilus Φ in biological populations    The ratio of female honey bees to males is Φ. 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 Φ and the Fibonacci numbers. Angel Fish Tiger Human Face I Human Face II Le Corbusier’s Man A Golden Solar System? Φ In Art & Architecture    For centuries, people seem to have found Φ to have a natural, nearly universal, aesthetic appeal. Indeed, it has had near religious significance to some. Occurrences of Φ abound in art and architecture throughout the ages. The Pyramids of Giza The Pyramids and Φ 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: The Vitruvian Man Rembrandt’s Self Portrait Seurat’s Parade Seurat’s Bathers Turner’s Norham Castle at Sunrise Mondriaan’s Broadway BoogieWoogie 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 (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. 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 72 144 Dart 72 72 216 36 72 36 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 Φ. 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.” 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.