Without using a calculator, computer or tables find the exact
values of cos36cos72 and also cos36 - cos72.
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.
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.
Published December 2005,June 2005,December 2011,February 2011.
There are some side trips to explore properties of the Fibonacci numbers which are not essential for the main voyage of discovery. You can take these side trips if you have time or maybe return to them later.
Now divide each term by the term before it and keep doing this for say 13 terms or more. What do you notice?
Set up a spreadsheet like the one illustrated. One cell in the spreadsheet is outlined and you can see that the formula '=A3+A4' has been defined for that cell. The Fibonacci sequence has been produced by copying the formula down the column. In your spreadsheet change the first two terms to any values you choose. What happens to the sequence? What happens to the ratio of successive terms?
You will find that, whatever the first two terms in the sequence, the ratio of successive terms quickly approaches a constant value. A later challenge in this trail leads to a proof that this value is the golden ratio.
Imagine drawing these rectangles on a large tiled courtyard so that you can go on making bigger and bigger rectangles. Notice that the sides of these rectangles are the Fibonacci numbers and as you draw bigger rectangles they get closer in proportions to the Golden Rectangle.
Now draw a spiral starting in the bottom left hand corner of the unit square on the left; draw a smoothly curving arc to the opposite corner of the square, move into and across the next square in a smoothly curving arc, and so on across each square. This is called a logarithmic spiral.
In a section of a Nautilus shell, in the arrangements of seeds on flower heads and in the segments of a pine cone we can see similar spirals.
Now draw a regular pentagon using a ruler and a protractor to measure the angles of 108 degrees. Draw in the five chords to form a pentagram star inside your pentagon. Measure the length of one of the chords and the length of a side of the pentagon and divide the chord length by the side length. You should get a ratio about 1.62 and later we'll prove that the exact value of this ratio is the