|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.
of the Fibonacci sequence:
- each number is the sum of the previous
- 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
- 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
- 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
- 5 black keys
- 8 white keys
- 13 keys in
an octave: these 13 notes make up the
- Some say the major sixth chord
is "the one our ears like best": The note
E vibrates at a ratio of .62 to note
- 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"
- Limerick: 5
lines with 13 beats, grouped in 3's
- Virgil's Aeneid is a far more
- 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.
67" x 96"
39" x 65"
23" x 32"
- Ball, P: The Self-Made Tapestry: Pattern formation in
nature. Oxford 1999. ISBN 0-19-850244-3.
- The Fibonacci
- Garland, TH: Fibonacci Numbers in Nature (poster):
Dale Seymour Publications, 1988. ISBN 0-86651-454-6.
- 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.
- Knott, Ron: Surrey University, UK
multimedia website on the Fibonacci numbers, the Golden section
and the Golden string.
- Livio, Mario: The Golden Ratio: The Story of PHI, the
World's Most Astonishing Number. Broadway 2003. ISBN:
- Stewart, I: Life’s Other Secret: The New Mathematics
of the Living World. Wiley 1998. ISBN 0-471-15845-3.
- Stewart, I: What Shape is a Snowflake? Freeman
2001. ISBN 0-7161-4794-4.
- Willis, D: The Sand Dollar and the Slide Rule: Drawing
Blueprints from Nature. Addison-Wesley 1995. ISBN 0-201-48831-0.