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Explore the captivating connection between mathematics and art through the lens of the divine proportion, also known as the Golden Ratio. Discover common themes, properties of Phi, Fibonacci numbers, golden rectangles, and spirals in this mesmerizing fusion of disciplines.
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Mathematics and Art:Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College faculty.sjcny.edu/~holtzman
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.
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:
The Golden Quadratic II • Cross multiplication yields:
The Golden Quadratic III • Setting Φ equal to the quotient a/b and manipulating this equation shows that Φ satisfies the quadratic equation:
The Golden Quadratic IV • Applying the quadratic formula to this simple equation and taking Φ to be the positive solution yields:
Properties of Φ • Φ is irrational • Its reciprocal, 1/ Φ = Φ - 1 • Its square, Φ2 = Φ + 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 C A
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 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.
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.
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.
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.
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.
