Golden Ratio Proof

This is a physical demonstration of the recursive definition of he Golden Ratio.

The golden ratio is the ratio between the width and height of a golden rectangle, a shape that has both mathematical and aesthetic beauty. The fundamental property of a golden rectangle is that if you cut out a square from it, the result is another, smaller golden rectangle. You can repeat this process infinitely, splitting the original golden rectangle into infinitely many squares and golden rectangles. No other dimensions of rectangle have this property.

This geometrical definition of the golden ratio, represented by the Greek letter phi, leads to an algebraic definition:

phi = 1/(phi-1)

As shown in the engraving on the model, this definition can be converted into a quadratic formula, and solved to give the value:

phi = (1 + sqrt(5)) / 2 = 1.61803…..

The value of the golden ratio is an irrational number, a number that cannot be represented by the ratio of any two whole numbers. The discovery of this fact, along with the fact that pi is also irrational, was a cause of great concern to the ancient Greeks, whose entire philosophical world view was challenged by the existence of such numbers.

This model demonstrates one of the ways of representing the definition of phi:

phi (phi - 1) = 1

It does this by allowing a quantity of beautiful 0.8mm (super-small) metal beads to settle either into a square of area 1, or into a rectangle whose dimensions are phi by (phi - 1), giving it an area of phi (phi - 1).