Golden ratio

problem
Gold again

Age

16 to 18

Challenge level

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.
problem
Golden thoughts

Age

14 to 16

Challenge level

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.
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Whirling Fibonacci squares

Draw whirling squares and see how Fibonacci sequences and golden rectangles are connected.
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The golden ratio, Fibonacci numbers and continued fractions

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.
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Leonardo of Pisa and the golden rectangle

Leonardo who?! Well, Leonardo is better known as Fibonacci and this article will tell you some of fascinating things about his famous sequence.
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About Pythagorean golden means

What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?