Rectangle construction
1) Draw a 1 by 1 rectangle on your grid paper centered
vertically and about a quarter of the way from the top.
2) Add a square to this figure whose length equals the
longest length of the current figure. Repeat this step
as much as you can, placing the squares so that they
spiral out from the first square.
3) After adding each square, determine the length of the
longest side and try to determine a pattern.
Fibonacci sequence
• What rule governs this sequence?
• History: named for Fibonacci’s problem
in the book Liber abaci published in
1202:
A certain man put a pair of rabbits in a place
surrounded on all sides by a wall. How many pairs
of rabbits can be produced from that pair in a year
if it is supposed that every month each pair begets
a new pair which from the second month on
becomes productive?
Connections with nature
• Flower petals - many varieties of flowers
have a number of petals equal to a
Fibonacci number
• Pinecone spirals
Finding ratios
• Compute F1 through F20.
• Compute the ratios Fn/Fn-1 for values of
n ranging from 2 to 20.
– what patterns do you observe?
– are the ratios converging?
• Call this number of convergence Φ and
find a formula.
Solving equations
• A reminder: to solve ax 2 bx c 0
use the quadratic formula
b b2 4ac
x
2a
• Use this to find an expression for Φ.
The Golden Mean
• Φ is called
– the golden mean
– the golden ratio
– “the divine proportion”
• If you break a bar into two pieces so
that the ratio of the long piece to the
short equals the ratio of the whole piece
to the long, both ratios are Φ.
The Golden Rectangle
• A rectangle is golden if the ratio of its
long side to its short side is Φ.
Source: Mark
Frietag’s site Phi:
That Golden
Number
Constructing a Golden Rectangle
• Draw a square and bisect the bottom side.
• Draw a line from that side to the top right
corner.
• Draw a line with that same length extending
from the bisection point parallel to the bottom
side.
• Complete the rectangle. (You can measure
the sides to check the ratio.)
Why does this work?
• Reminder: given a right triangle with
short sides a and b and hypotenuse c,
we have the Pythagorean Theorem:
a b c
2 2 2
• Use this to find the ratio.
A beautiful result
• Given a golden rectangle, if you divide it
into a square and a rectangle, the
rectangle is golden.
A example (with a
logarithmic spiral) from
Mark Frietag’s site Phi:
That Golden Number
Question for Friday
• Clearly, Φ has some important
mathematical properties.
• Does it have important aesthetic
properties?