More about the Golden Ratio

Contents

Continued fractions
Golden spirals

Continued fractions

A continued fraction is a complex fraction with further fractions in the denominator, and fractions in the denominator of the denominator, and so on. A continued fraction can be finite or infinite. One example at right is a finite continued fraction. You can easily simplify it to the form a/b. The other example is an infinite continued fraction. The main question for an infinite continued fraction is how to express it as a real number (fraction, repeating or nonrepeating decimal, radical, or other expression).

The notation [a; b,c,d] describes the continued fraction with integer part a, 1s in the numerators, and with b, c, d, b, c, d, b, c, d, . . . repeating successively in the denominators. (The repeat can be any finite list of integers.) So the infinite continued fraction above is written [1;1].

Problems

1. Find a numerical value for the continued fraction [1;1]:
   1. It's not obvious what the numerical value of the infinite continued fraction at right is, so we'll temporarily call that value x. Notice that the part outlined in the rectangle is the same as the whole fraction, since the (...) part means to continue forever.
   2. Write an equation with x on the left, and the part on the right in the box replaced by x.
   3. Write this equation in the standard form for a quadratic polynomial, a x^2 + b x + c = 0 and solve for x. You will need to use the quadratic formula.
2. Find a numerical value for [2;2].
3. Find a numerical value for [1;2,1].
4. Find a numerical value for [1;3,6,1].
5. Can you make a general conjecture about infinite repeating continued fractions? Prove it if possible.
6. Find a continued fraction (with numerators all 1) for the rational numbers 7/2, 114/47, 2000/99.
7. Can you find a general procedure for converting a rational number to a continued fraction? Will it work for any real number?

Golden spirals

Construct a golden spiral.

Construct a golden rectangle and the square inside it as before. Inscribe a quarter-circle in the square as shown.
In the remaining smaller golden rectangle, make another square. Inscribe a quarter-circle in this square so that the arc connects with that of the first quarter-circle.
Keep constructing squares and arcs in the smaller golden rectangles that appear.

Extensions

1. Consider a golden spiral inscribed in a golden rectangle 1 by g.
• Find a formula for the length of the nth arc of the golden spiral.
• Find a formula for the total length of the golden spiral.
• Find a formula for the area of the nth sector of the golden spiral.
• Find a formula for the total area of the golden spiral.
• The centers of the arcs making up the golden spiral are getting closer and closer to a point; let's call this the center of the spiral. Find a formula for the coordinates of this point if the containing golden rectangle has opposite corners at (0,0) and (g,1).