Mathematical Connections

Table of Contents The Fibonacci Series The Golden Section The Relationship Between Fibonacci and The Golden Section References

The Fibonacci Sequence “The greatest European Mathematician of all ages.” Leonardo of Pisa of Fibonacci: Born 1175 AD

The Fibonacci Sequence He gave us our 10 digit number system! He recognized a series of numbers that often occur in nature. These are now called the Fibonacci numbers. The series starts with 0 & 1. All following numbers are the sum of the 2 previous numbers! Create A Fibonacci Seq. Leonardo of Pisa or Fibonacci: Born 1175 AD Next

Create A Fibonacci Series Remember, just add the 2 previous numbers together to get the next number... Here's how it starts... 0,1,1,2,3,5..... So, click on the button that has the correct choice for the next several numbers in the Fibonacci series! 8,13 7,15 9,17 Back

The Fibonacci Spiral This computer drawing was created using Fibonacci Numbers. This is called a Fibonacci Spiral

The Fibonacci Spiral Remember , the series is 0,1,1,2,3,5,8,13,21..... Place squares with those lengths for each side as shown in the drawing ( excluding 0). The two squares where each side has a length of 1 are in the center. Find them and click on them!

The Fibonacci Spiral On top of the two squares with sides of length 1 is a square where each side has a length of 2. To the right of those three squares is a square with a side of length 3. Continue with this, then draw a quarter circle in each square as shown!

The Fibonacci Spiral In this drawing, side E will have a length of 3 + 2 = 5. See if you can figure out what the length of the sides marked F, G and H are! Side F is 8 7 9 Side G is 10 13 11 Side H is 15 18 21

The Fibonacci Spiral Example of the Fibonacci Spiral in Nature.... Click to see the relationship!

The Fibonacci Spiral Example of the Fibonacci Spiral in Nature.... Can you see the spiral in this pine cone? Many seeds and seed heads have the Fibonacci Spiral in them! Click to see the relationship!

The Fibonacci Spiral Example of the Fibonacci Spiral in Nature.... The Fibonacci Spiral can be seen in the seed head of a sunflower seed too! Can you see it? Click to see the relationship! This concludes the Fibonacci portion... care to explore the Golden Section?

The Golden Section is defined as the ratio most pleasing to the eye. B Here line AB is divided at point M. The ratio of AB to MB is the same as the ratio of AM to MB. This means the line is divided into golden sections!

The Golden Section This ratio is the same as the ratio between 1 and the number phi (1.6180339887). By constructing a rectangle where the sides have the golden ratio, we create a golden rectangle! Most people would select the middle one, the "golden rectangle“, as most pleasing to the eye.

The Golden Section Leonardo da Vinci 1452-1519. Italian painter, engineer, musician, and scientist. The most versatile genius of the Renaissance, Leonardo used the Golden Rectangle in his paintings.

* The rectangle around her face represents a Golden rectangle. The Golden Section Leonardo's famous Mona Lisa reflects the artist’s use of the Golden Section. * The rectangle around her face represents a Golden rectangle. * If you subdivide the rectangle at the eyes the vertical side of the rectangle is divided by the golden ratio. Click To See!

The Golden Section This painting by George Seurat, "La Parade" was designed by dividing the canvas into the Golden Section. Click to see the artist's planning sketch!

The Golden Section Even in the time of the ancient Greeks, the golden rectangle was considered a pleasing shape. It appears in many of the proportions of the Parthenon in Athens, Greece. Continue on to explore the relationship between the Fibonacci Numbers and the Golden Section.

The Golden Section & Fibonacci Numbers Remember the spiral we created using the Fibonacci Numbers? Look closely at the rectangle made by any adjacent squares.

The Golden Section & Fibonacci Numbers By doing the math we can see that the larger our rectangles get, the closer to the perfect golden section the proportions are!

The Golden Section & Fibonacci Numbers The sides on the 6th and 7th squares are lengths 13 & 21. 21 / 13 = 1.61538... and this result gets ever closer to the ratio between 1 and the number  as the rectangles grow!

The Golden Section & Fibonacci Numbers The End!!

References http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibonat.html http://www.mos.org/sln/Leonardo http://www.atrsnet.getty.edu http://www.criba.edu.ar/girasol http://www.rit.edu/~sxa7544/structural.htm http://share3.esdros.wednet.edu/2/lbohl/goldensppiral.html http://pass.math.org..uk/issue3/fibonacci/alt.html http://mongo.tvi.cc.nm.us/art_math/golden.html http://sunsite.unc.edu/wm/paint/auth/vinci http://familiar.sph.umich.edu/cjackson/s http://www.notam.uio.no/~oyvindha/loga.html http://www.summum.org/phi.htm