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
• 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

• 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)
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 Phi
x
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 Architecture
The 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 Phi
x 1 x
1
x

x
1
x  1  x2
x
x2  x 1  0
1 5
x
2
x  1.61803...
Phi
In 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.
Phi
In 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 ‘noninstances’ 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