A first trail through the mysterious world of the Golden Section.
A voyage of discovery through a sequence of challenges exploring
properties of the Golden Ratio and Fibonacci numbers.
Leonardo who?! Well, Leonardo is better known as Fibonacci and this article will tell you some of fascinating things about his famous sequence.
You add 1 to the golden ratio to get its square. How do you find higher powers?
Find the link between a sequence of continued fractions and the
ratio of succesive Fibonacci numbers.
When is a Fibonacci sequence also a geometric sequence? When the
ratio of successive terms is the golden ratio!
Nick Lord says "This problem encapsulates for me the best features
of the NRICH collection."
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
Find a connection between the shape of a special ellipse and an
infinite string of nested square roots.
Draw a square and an arc of a circle and construct the Golden
rectangle. Find the value of the Golden Ratio.
A rhombus PQRS has an angle of 72 degrees. OQ = OR = OS = 1 unit. Find all the angles, show that POR is a straight line and that the side of the rhombus is equal to the Golden Ratio.
ABCDE is a regular pentagon of side length one unit. BC produced
meets ED produced at F. Show that triangle CDF is congruent to
triangle EDB. Find the length of BE.
Solve an equation involving the Golden Ratio phi where the unknown
occurs as a power of phi.
Draw whirling squares and see how Fibonacci sequences and golden rectangles are connected.
Explain how to construct a regular pentagon accurately using a
straight edge and compass.
Without using a calculator, computer or tables find the exact
values of cos36cos72 and also cos36 - cos72.
Three triangles ABC, CBD and ABD (where D is a point on AC) are all
isosceles. Find all the angles. Prove that the ratio of AB to BC is
equal to the golden ratio.
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.
The diagram shows a regular pentagon with sides of unit length.
Find all the angles in the diagram. Prove that the quadrilateral
shown in red is a rhombus.
Show that the arithmetic mean, geometric mean and harmonic mean of
a and b can be the lengths of the sides of a right-angles triangle
if and only if a = bx^3, where x is the Golden Ratio.
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?