Fibonacci and the Golden Ratio - Ms. Owen's Math Pages
Transcript of Fibonacci and the Golden Ratio - Ms. Owen's Math Pages
Golden Ratio
Fibonacci and the
Golden Ratio
Warm-Up
Find the next five terms
in the following
sequence:
1, 1, 2, 3, 5, …
Fibonacci Sequence
Background of Golden Ratio
Euclid of Alexandria (300 B.C.) defined the
golden ratio in his book, “Elements.” Since
then, artists and architects who deem this
ratio as being the most aesthetically pleasing
ratio have used it as a basis for their art and
buildings.
The golden ratio is called phi, , and is
approximately 1.61803.
Golden Ratio in Art
http://library.thinkquest.org/trio/TTQ05063/phibeauty3.htm
Mona Lisa
by
Leonardo daVinci
The Vetruvian Man
(The Man in Action)
by
Leonardo daVinci
http://www.world-mysteries.com/sci_17.htm
“We can draw many lines of the
rectangles into this figure.
Then, there are three distinct sets of
Golden Rectangles:
Each one set for the head area, the
torso, and the legs.”
The Sacrament of the Last Supper – Salvador Dalihttp://britton.disted.camosun.bc.ca/goldslide/gold38.jpg
Golden Ratio in
Architecture
Parthenon in Athens, Greecehttp://britton.disted.camosun.bc.ca/goldslide/gold08.jpg
Pyramids in Egypt
The angle of inclination is 1.61818
http://creativesagest.blogspot.com/2009/03/golden-ratio-secret-to-aesthetics.html
Tahjmahal, India
http://creativesagest.blogspot.com/2009/03/golden-ratio-secret-to-aesthetics.html
Golden Ratio in Nature
Nautilus Shell
Shells - A Fibonacci Spiral is created by
drawing arcs connecting the opposite
corners of squares, whose relative sizes
follow the Fibonacci Sequence. Many
shells follow the shape of the Fibonacci
Spiral.
Sunflower
http://hynesva.com/blogs/character_and_excellence/archive/2009/11.aspx
Pinecone
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#pinecones
Romanesque
Broccoli/Cauliflower
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#pinecones
Constructing a Golden Rectangle
Draw a squarewith sides 1 cm in the lower right portion of your paper.
Now, let's build another, congruent square right next to the first one.
Now we have a rectangle with width 1 and length 2 units.
Let's build a square on top of this rectangle.
Let’s continue to build squares. Draw the next one to the right.
What size should it be?
Draw the next one below.
What size should it be?
Draw the next one to the left.
What size should it be?
Draw the last one above.
What size should it be?
What do you notice
about the lengths of the
sides of the golden
rectangle you drew?
They are the Fibonacci
sequence: 1, 1, 2, 3, 5, 8, 13
Connect the corners with a smooth curve to form a Golden Spiral.
Making a Collage – Teacher Sample
Instructions
You are to create a collage using
images relating to the golden ratio.
You may use the pictures I have
printed or bring in your own.
Identify the golden ratio in at least
three of the images.
Pictures to Use for Collage
Retrieved 2/25/11 from: http://www.world-mysteries.com/sci_17.htm