Rectangle construction: 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: 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: Connections with nature Flower petals - many varieties of flowers have a number of petals equal to a Fibonacci number
Pinecone spirals Finding ratios: 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: Solving equations A reminder: to solve
use the quadratic formula
Use this to find an expression for Φ. The Golden Mean: 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: 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: 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?: Why does this work? Reminder: given a right triangle with short sides a and b and hypotenuse c, we have the Pythagorean Theorem:
Use this to find the ratio.
A beautiful result: 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: Question for Friday Clearly, Φ has some important mathematical properties.
Does it have important aesthetic properties?