Rectangles

• On scrap paper, each sketch or draw a rectangle• Measure the sides of your rectangle• What’s the ratio of length to width for your

rectangle?• Find the average ratio for the rectangles of all

class members

In the eye of the beholder…• Beauty is in the eye of the beholder• …but it is said that humans find one particular

ratio more appealing than others• It is known as the Golden Ratio, or Phi (Φ) and

is said to appear in artwork and architecture as well as in the human face.

Finding Phi – by experimentation• Draw a rectangle• Find the ratio of length:width• From one end, cut off a square with side length equal to the

width of the original rectangle, leaving you with a new rectangle (the pale one)

• Find the length:width ratio of the new rectangle

• Repeat – cut off a square, find the ratio of the rectangle’s dimensions

• If all the ratios are the same, you have found Phi

• If not, start with a rectangle with new dimensions

Finding Phi – by drawing

• A Pentagram (regular, 5 pointed star) has the Golden Ratio within it

• Using one of the two methods shown (or one of your own) find the ratio of lengths AB:CD to find Phi

Finding Phi – by calculation

• Read the information on how to find Phi by experimentation (or better still, try it out)

• Use the diagram below to set up an equation, then solve it to find Phix

x

1

The Golden Ratio in Nature• Many plants seem to link to Phi• It is said that counting the segments on the skin

of a pineapple or the segments in a spiral on a pine cone will always give a number from the Fibonacci sequence (1 1 2 3 5 8 13 21…)

• You might like to verify this• Find the ratio of consecutive terms of the

Fibonacci sequence• Do they converge to Phi?

The Golden Ratio in Art• One of the most famous paintings

in the world ‘The Mona Lisa’ is said to contain numerous instances of the Golden Ratio, for instance her face surrounded by a rectangle, the ratio of the length of the lower part of her face (eyes down) compared to the height of her forehead, and the shape of her forehead surrounded by a rectangle.

• How could you check that these are true?

Image from http://en.wikipedia.org/wiki/Mona_Lisa

The Golden Ratio in ArchitectureThe Parthenon in Ancient Greece is also said to contain many instances of Phi. • How many can you find? • Do you think they are deliberately included?

Image from http://employees.oneonta.edu/farberas/ARTH/Images/109images/greek_archaic_classical/parthenon/parthenon_proportions.jpg

Teacher Notes: Finding Phi

• This activity is accessible to most pupils, but the approach adopted may vary according to the students’ algebraic competence

• Experimentation followed by the algebraic approach gives students a good insight into the problem, which means that they should then be able to construct the equation for themselves

• If using a drawing approach, students can draw the pentagram for themselves – perhaps having the class initially consider how to construct a pentagram in a circle with the use of a protractor. The result could then be verified using a more accurate medium such as dynamic geometry software package

Teacher notes: Finding Phi – by calculation• Use the diagram below to set up an equation

and solve it to find Phix

x

1

2

2

1

1

1

1 0

1 5

21.61803...

x x

x

x x

x x

x

x

PhiIn nature• Successive terms of the Fibonacci sequence converge to Phi.• By the 10th ratio (89/55) it is correct to 3dp

In Art

This is a debatable. The only way to be certain that the ratios are there by design would be to ask the artist, however, we can use the basic ideas of probability to explore the possibility that they are deliberate.• Aspects to consider:

– When taking measurements from the painting, where should they be taken?

– How often might the ratio appear simply by chance?– How many pairs of measurements used to calculate a ratio don’t result

in Phi? How many do?

This activity should cause students to question claims, as well as considering how they might have arisen and how claims might be tested.

The difficulty with testing this claim is deciding where rectangles ‘should’ be drawn, as slight variations in positions will affect the result.

PhiIn Architecture• This is easier to explore since there are obvious lines to the construction

and it is therefore easier to agree upon which measurements might be taken. There is also less scope for variation.

Classroom use• These activities are designed to develop problem solving skills• Group work and discussion should be encouraged • Asking groups to present ‘their case’ to convince others of their findings will

allow students to develop the basic ideas of proof • Asking questions such as “How many instances of Phi compared to ‘non-

instances’ of Phi would you be convinced by?” will help students to understand the basics of hypothesis testing

• Print out the following 2 slides to allow students to draw, debate and take measurements