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AB Euclid of Alexandria (325 – 265 BC.) THE GOLDEN SECTION In Book VI of the Elements, Euclid defined the "extreme and mean ratios" on a line segment. He wished to find the point (P) on line segment AB such that, the small segment is to the large segment as the large segment is to the whole segment. P In other words how far along the line is P such that: We use a different approach to Euclid and use algebra to help us find this ratio, however the method is essentially the same. 1  Let AP be of unit length and PB = .Then we require  such that Solving this quadratic and taking the positive root. We get the irrational number shown.  is the Greek letter Phi.

Euclid of Alexandria (325 – 265 BC.) THE GOLDEN SECTION In Book VI of the Elements Euclid defined the "extreme and mean ratios" on a line segment. He wished to find the point (P) on line segment AB such that, the small segment is to the large segment as the large segment is to the whole segment. 1  If the smaller and larger segments are thought of as the sides of a rectangle then we can apply the same procedure to get the Golden Rectangle. A golden rectangle has a length 1.618… times its width. It is thought to be aesthetically pleasing as the proportions look harmonious and balanced. It has been used throughout history especially in art and architecture.  w

THE GOLDEN SECTION Can you pick out the Golden Rectangles? a b c d e f g h i j  

THE GOLDEN SECTION Some properties of  Checking this on a calculator: … + 1 = … and: … 2 = …

THE GOLDEN SECTION The ancient Egyptians may have been the first people to apply the golden ratio to their art and design. It may have been used in the construction of the Great Pyramid at Giza. 1  2 Measurements through the cross section of the pyramid have shown that the ratio of slant height to the distance to the centre point is

THE GOLDEN SECTION The Parthenon in Athens is believed to have been designed using the golden section. The Ancient Greeks of the golden age of Pericles appear to have used the golden ratio throughout the structure. The overall dimensions of it’s façade are very close to being a golden rectangle.

THE GOLDEN SECTION Leonardo da Vinci ( ) made a close study of the human figure and had shown how all its different parts were related by the golden proportion. Michelangelo ( ) applied the golden proportion to some elements of his famous sculpture of David. The Sacrament of the Last Supper by Salvador Dali ( ) is painted inside a golden rectangle and it contains other elements using this proportion.

THE GOLDEN SECTION Many common objects are good approximations to a Golden Rectangle. Try to find some more by looking at and measuring books/cards/widescreen TV pictures, picture frames, doors, windows, etc.

THE GOLDEN SECTION The pentagram was the symbol of the Pythagorean brotherhood. Inscribing the regular pentagon like so below produces a pentagram. The Pentagram consists of five congruent isosceles triangles, five congruent obtuse-angled triangles and a central regular pentagon. a a b The diagonals exhibit the golden ratio as shown. The two large triangles are congruent and we see that the pentagon sides are a + b a + b We can see from this that there are many golden triangles inside the pentagram as shown. b a a a b a a + b a b a

Constructing a Golden Rectangle. 1. Construct a square and the perpendicular bisector of a side to find its midpoint p. 3. Set compass to length PM and draw an arc as shown. 2. Extend the sides as shown. L M 4. Construct a perpendicular QR. O N Q THE GOLDEN SECTION LQRO is a Golden Rectangle. 1 P R

L M O N Q THE GOLDEN SECTION 1 P R A Golden Rectangle. PM = by Pythagoras = ½ ½ 1

THE GOLDEN SECTION Each time you cut off a square from a golden rectangle a new smaller golden rectangle is formed. A special type of logarithmic spiral called the Golden Spiral can be inscribed through the vertices of the squares. The red spiral shown consisting of ¼ circles is a very close approximation.

THE GOLDEN SECTION Johannes Kepler "Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel."