Fibonacci, Art, and Quilts
1,1,2,3,5,8,13,21,34,55,89,144,233,...
I have long been fascinated by the Fibonacci sequence, which has connections throughout the history of art, ancient Greece, nature, music, and science: in fact, just about anywhere you look. The sequence is named after an Italian mathematician who described the sequence in the thirteenth century.
Characteristics of the Fibonacci sequence:
  • each number is the sum of the previous 2 numbers.
  • the ratio of each consecutive 2 numbers (called Phi and pronounced "fee") is near .618, but like Pi is an irrational number, i.e. the digits go on forever without repeating.
  • Phi: that number .618 (or rounded off to .62 or .6 or 60%) is often called the Golden Ratio, the Golden Mean, or the Golden Proportion, and it has often been suggested that the golden ratio might be an underlying explanation for what is aesthetically pleasing: both in a natural object and in an artistic masterpiece.
  • Dynamic Symmetry: when you use the Golden or Fibonacci proportion to divide something (anything!), the ratio of the small part to the large part is the same as the ratio of the large part to the whole. [Euclid called this "extreme and mean ratio."] Thus an object using the golden ratio can be said to have "dynamic symmetry", having both variety and unity, i.e. it is interesting AND goes together as as whole.

Just a few of the places you might stumble across the Fibonacci sequence:
  • Nature:
    • pattern of seeds in a sunflower: 55 rows of seeds spiral in one direction, 89 in the other.
    • pinecones: 8 rows of bracts spiral in one direction, 13 in the other.
    • many flowers have a fibonacci number of petals: Iris 3, Buttercup 5, Cosmo 8, Daisies 13 and 21. There are sunflowers with 34, 55 and even 89 petals!
    • This all arises from the dynamics of plant growth which leads to the most efficient spacing of leaves and seeds. It turns out that this spacing is based on the angle 137.5, the 360 degrees of a circle times the fibonacci proportion. This is sometimes called the golden angle.
  • Music:
    • 5 black keys
    • 8 white keys
    • 13 keys in an octave: these 13 notes make up the chromatic scale
    • Some say the major sixth chord is "the one our ears like best": The note E vibrates at a ratio of .62 to note C.
  • The Human Body
    • The ratio of the length of the each finger bone is .62 to the length of the next finger bone.
    • To the Greeks, for a perfectly proportioned body, the ratio of the height of the navel to the total height of the body should equal the Golden Ratio (.62).
  • Architecture, Art, and History:
    • The Great Pyramid and the Parthenon can be shown to demonstrate the Fibonacci proportion.
    • Mondrian used the Fibonacci proportion in his art
    • Leonardo da Vinci called it the "divine proportion"
  • Poetry
    • Limerick: 5 lines with 13 beats, grouped in 3's and 2's.
    • Virgil's Aeneid is a far more sophisticated example
  • Geometry
    • Pentagon:
      • the ratio of the side of a regular pentagon to a diagonal is .62
      • the point where two diagonals cross divides them into segments of the fibonacci proportion to each other.

I use Fibonacci numbers, or at least the Fibonacci proportion (Golden proportion, .618, rounded to 60%) frequently: in the proportion of one color to another, in the size of stripes, in the size of borders, in the ratio of width to height in rectangular quilts, etc. But these 3 quilts were specifically made using Fibonacci numbers: Frank Lloyd Fibonacci, Molto Fibonacci, and Hidden Fibonacci. All three demonstrate many uses of fibonacci: click on the images for more details.
Molto Fibonacci
©1997 67" x 96"
Frank Lloyd Fibonacci
©1996 39" x 65"
Hidden Fibonacci
©1997 23" x 32"
 

References:

  1. Ball, P: The Self-Made Tapestry: Pattern formation in nature. Oxford 1999. ISBN 0-19-850244-3.
  2. The Fibonacci Association website.
  3. Garland, TH: Fibonacci Numbers in Nature (poster): Dale Seymour Publications, 1988. ISBN 0-86651-454-6.
  4. Glyka, M: The Geometry of Art and Life. Dover 1977. ISBN 0-486-23542-4.
    Huntley, HE: The Divine Proportion: A Study in Mathematical Beauty. Dover 1970. ISBN 0-486-22254-3.
  5. Knott, Ron: Surrey University, UK multimedia website on the Fibonacci numbers, the Golden section and the Golden string.
  6. Livio, Mario: The Golden Ratio: The Story of PHI, the World's Most Astonishing Number. Broadway 2003. ISBN: 0-76790816-3
  7. Stewart, I: Life’s Other Secret: The New Mathematics of the Living World. Wiley 1998. ISBN 0-471-15845-3.
  8. Stewart, I: What Shape is a Snowflake? Freeman 2001. ISBN 0-7161-4794-4.
  9. Willis, D: The Sand Dollar and the Slide Rule: Drawing Blueprints from Nature. Addison-Wesley 1995. ISBN 0-201-48831-0.
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