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

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/turku.jpg