The Fibonacci Sequence in Art

Painting by Numbers
The Fibonacci Sequence in Art
Curtis Belmonte
& Conor Pappas
Table of Contents
What are the Fibonacci Numbers? ............Error! Bookmark not defined.
What Makes a Sequence?.......................Error! Bookmark not defined.
Recursion .........................................Error! Bookmark not defined.
The Golden Ratio................................Error! Bookmark not defined.
A Similar Sequence .............................Error! Bookmark not defined.
Famous Fibonacci Patterns & Shapes ........Error! Bookmark not defined.
The Fibonacci Spiral ............................Error! Bookmark not defined.
Fibonacci in Nature .............................Error! Bookmark not defined.
Fibonacci’s Sequence in Famous Art .........Error! Bookmark not defined.
Fibonacci in Famous Architecture ............Error! Bookmark not defined.
Stained Glass Fibonacci Patterns .............Error! Bookmark not defined.
Furniture by Fibonacci .........................Error! Bookmark not defined.
Fibonacci Around the House ...................Error! Bookmark not defined.
Glosary............................................Error! Bookmark not defined.
Image Credits ....................................Error! Bookmark not defined.
About the Authors...............................Error! Bookmark not defined.
What are the
Fibonacci Numbers?
You might notice that the name ―Fibonacci‖ sounds familiar, especially
if you’ve ever done some extra math-related research in or outside of
school. In fact, the Fibonacci numbers, also known as the Fibonacci
sequence, make up one of the most famous number patterns in the
history
of
math!
The
sequence
is
named
after
the
famous
mathematician Leonardo Fibonacci, who studied and realized the
importance of the unique properties of the numbers that made up this
sequence. But what exactly are these special numbers, and what did
Fibonacci think was so important about them? Well, to answer those
questions, we’ll first look at some rabbits…
Hello! I am the famous mathematician
Leonardo
Fibonacci.
Yes,
the
same
Fibonacci mentioned above. My friends
and I will be dropping in from time to, so
be sure to keep your eye out for us.
Did you know?
Say we start out with one pair of rabbits, a mom and
dad. Except the rabbits aren’t old enough to be
parents just yet. It takes them another whole month
before they’re mature enough to have their own baby
rabbits. So after two full months, they have a pair of
babies and there are now two pairs of rabbits in total.
The baby rabbits, like their parents, take a month to
mature before they have rabbits of their own.
Meanwhile, the first two rabbits have another pair of
babies for a total of three pairs. This pattern continues
every month until there are five pairs, eight
pairs, thirteen pairs, and so on. As this
pattern continues, the numbers of pairs
of rabbits each month are all numbers in
the Fibonacci sequence.
I know that may not seem like
a lot of us at first, but the
pattern grows quickly. After a
year, there will be as many as
233 pairs of rabbits! How
many of us will there be after
two years?
For centuries Roman
numerals were the
preferred style of
numbers to use…but
it was really difficult
to do math with
them. Fibonacci was
the one who made
the Arabic numbers
we use today popular
in his book Liber
Abaci, which means
Book of Calculation.
It was published in
1202, when Fibonacci
was 32 years old.
What Makes a
Sequence?
Sequence may sound like a fancy mathematical term, but a
sequence is nothing more than a pattern of numbers. The following
are all examples of sequences:
1, 2, 3, 4, 5
1, 3, 7, 15, 31, 63
The first sequence is just all of the whole numbers starting with 1.
The pattern in the second sequence may be a little harder to
identify, but it starts with the number 1, and then each number is
one more than twice the previous number. So to form a sequence,
all you need is some sort of pattern to follow.
We’ve been calling it the Fibonacci ―sequence,‖ so what is the
pattern for this special set of numbers? Well, the first and second
Fibonacci numbers are both 1, and every other Fibonacci number is
the sum of the previous two numbers. That’s all there is to it!
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
I’ve been working on this problem for a
while. Why don’t you give it a try? See if
you can find the pattern for this sequence:
1, 3, 6, 10, 15, 21, …
Recursion
Think about the definition of the Fibonacci sequence. We start
with two initial values, and then every other term in the sequence
is based on the previous two. Now imagine that we want the 100th
Fibonacci number. Well, we will need the 99th and 98th numbers for
that. But to find those numbers we are going to need the 97th and
96th numbers, and then the 95th and 94th. You need to go all the
way to the 1st and 2nd numbers and build your way back up to the
100th number. This is what recursion is. You begin with some
starting value, and then every value after that is based on the
previous values. Then, in order to find a particular number, you
need to find every single number in the sequence up to that point.
You can think of recursion like the tower below. If I want the
lowest ring, I’m going to need to remove every single ring above it.
The Golden Ratio
The Fibonacci sequence has a lot of neat properties, and you’ll find that
playing around with it can yield some pretty interesting results. Try this:
Take the second Fibonacci number and divide it by the first. As it turns
out, this is pretty easy. The first and second Fibonacci numbers are both
1, and 1 divided by itself is clearly 1. Now take the third Fibonacci
number and divide it by the second. The third Fibonacci number is 2, so 2
divided by 1 gives us 2. Following this pattern, you would then take the
fourth Fibonacci number, 3, and divide it by the third, so 3 divided by 2
gives you about 1.67. If you keep trying this with higher and higher
Fibonacci numbers, you’ll see that your answers get closer and closer to a
certain number. This number, approximately equal to 1.618034, is known
as the golden ratio and is a very special value, indeed. In mathematical
terms, it is defined as the number ratio between two sides, call them a
and b, such that the ratio of (a + b) to a is the same value.
The golden ratio is usually
represented by the Greek letter phi
(φ) in honor of the Greek sculptor
Phidas, who used it in his artwork.
A Similar Sequence
Recall that the first two numbers of the Fibonacci sequence are 1
and 1. Now imagine if we chose two different numbers to start
with but still added every two numbers to create the next number
in the sequence. Well, François Lucas did exactly that and started
a sequence with 1 and 3. This sequence is named the Lucas
sequence and the first few terms of the sequence are
1, 3, 4, 7, 11, 18, 29, …
The Lucas sequence is formed in the same way as the Fibonacci
sequence, so it may not come as a surprise that the ratio between
consecutive Lucas numbers also gets closer and closer to the
golden ratio.
My name is François Lucas. While
I lived about 500 years after
Leonardo Fibonacci, I worked with
a similar sequence to his. It is
used mostly for discovering and
working with prime numbers.
Famous Fibonacci
Patterns & Shapes
Lots of famous geometric shapes, many of which can be found in
works of art, are related to the Fibonacci sequence. For example:
The Fibonacci Pinecone:
A three-dimensional image, in which the ―petals‖
of the pinecone are counted in a spiral downwards
from the top of the cone. The Fibonacci numbers
can be seen highlighted on this cone.
The Fibonacci Star:
A two-dimensional regular five-pointed star is a
great example of the golden ratio in action. The
ratio of an outer side to a side of the regular
pentagon in the middle of the star is
approximately, 1.618034, the golden section.
The Golden Fractal:
Using the golden ratio, you can construct
this unique fractal image. The area of each
big square in the figure is in the golden
ratio with the next smaller square, and this
pattern is repeated indefinitely.
The Fibonacci Spiral
Of all of the geometric shapes that can be related to the Fibonacci
sequence, the Fibonacci spiral—also known as the golden spiral or a
logarithmic spiral—is by far the most famous. To construct this unique
pattern, first start with a drawing of the golden fractal design described
on the previous page. Now, using the top-right corner of the biggest
square as a center point, draw an arc from the top-left to the bottomright corner of the square. Now use the top-left corner of the second
biggest square as a new center point to draw an arc from the bottom-left
to the upper-right corner. Then use the bottom-left corner of the next
square as a center point and continue the pattern by drawing another arc.
Because the fractal has, by definition, an infinite number of smaller and
smaller squares inside it, you can keep doing this on and on to infinity,
producing a spiral that loops in on itself again and again as it approaches a
single point. This simple, elegant pattern shows up everywhere, from
nature to manmade art and everything in between, as you’ll soon see…
Fibonacci in Nature
One of the most fascinating things about the Fibonacci sequence is that
they are not just a manmade concept. When we look a little closer, we
find countless examples of the Fibonacci sequence and the golden ratio
in nature. For example, in the image on this page, the leaves of the
plant are arranged in logarithmic spiral patterns around its center. The
fact that these numbers show up in the natural world around us on a
regular basis makes it sound like they must have some sort of special
significance, doesn’t it? Experts in science, mathematics, and art
certainly think so. Because humans tend to look at nature as especially
beautiful, many believe that the close connection between the
Fibonacci sequence and nature makes designs that incorporate the
Fibonacci numbers more aesthetically pleasing. For this reason, many
artists have tried to work these numbers into their art in order to
capture the beauty of nature.
Just how often do the Fibonacci
numbers show up in nature, you ask?
Well, let’s take a look at some
examples and see for ourselves.
Conch Shells:
An obvious example of the Fibonacci sequence
in nature is the conch shell. The spiral pattern
of these shells traces out a perfect Fibonacci
spiral shape.
Florets:
Many of those little seedlike parts of
flowers that you see, known as florets,
also behave according to the golden
ratio and are arranged in a Fibonacci
spiral pattern.
Petals:
Many flowers, such as the one shown
to the right often contain a special
number of petals. Can you guess what
kind of special number? That’s right, a
Fibonacci number!
Fibonacci’s Sequence
in Famous Art
In addition to in nature, you can find many examples of
of the Fibonacci in famous works of art. This is because
the Fibonacci sequence is considered a naturally
elegant number sequence and using it is said to create
very aesthetically pleasing images, so many artists have
used it to increase the visual appeal of their artwork.
Let’s take a look of some of the more notable examples
of the Fibonacci sequence and golden ratio in artwork.
The Mona Lisa:
Leonardo Da Vinci’s famous portrait of a mysterious
woman incorporates the Fibonacci sequence into its
design in a very subtle way. As you can see
highlighted in the image to the left, the woman’s
figure traces out a Fibonacci spiral
Vitruvian Man:
Another piece of artwork by Leonardo Da
Vinci, The Vetruvian Man, uses the Fibonacci
sequence in a different way. The ratio of the
distance from the man’s feet to his stomach
to the distance from the man’s stomach to
his head is approximately the golden ratio.
Crucifixion:
In this piece of art by the famous painter
Rafael, you can see the focal points of the
image—from the figure on the cross to the
angels in the background to the spectators
down below—connect to form a Fibonacci
star, as shown to the left.
Fibonacci in Famous
Architecture
The ancient Greeks, Romans, Egyptians, and many other civilizations knew
about the golden ratio. They called it by different names, but they all saw
this particular ratio as very pleasing and designed architecture that
utilized it. The Parthenon, which the Greeks constructed, has the golden
ratio present in many different places. The ratio of the width to the
height of the building and the ratio of the height of the building to the
height of the roof are both golden. Plus, the pillars in the front are placed
so that the width of the building is split into a golden segment. The
Egyptians used the golden ratio to construct the Pyramids of Giza. The
ratio of the side length of the pyramid to half the length of the base is,
you guessed it, the golden ratio.
Stained Glass
Fibonacci Patterns
A few companies have come up with a new way to
incorporate the Fibonacci sequence and the golden
spiral
in
their
artwork.
They
are
stained
glass
manufacturers and the golden spiral is the main focus of
a few of the pieces they have made. A few examples of
these glass masterpieces are shown here.
Furniture by
Fibonacci
Using the Fibonacci sequence in artwork and other creations makes it
more pleasing to an observer. For this reason, many furniture designers
are now using the golden ratio in their work to make their furniture not
only functional, but also aesthetic. They often times use a set of calipers
similar to the ones shown below. The middle prong of the calipers splits
the length between the other two prongs into a golden proportion. A
designer can then use these to measure and plan using the golden ratio.
Fibonacci Around
the House
The Fibonacci numbers can come up anywhere if you just look hard
enough. Jason and Marcus found the sequence while enjoying an
afternoon snack of nachos, but where around the house can you find the
Fibonacci sequence? Is the number of teeth on your comb a Fibonacci
number? What about the number of paper clips you can drop into a full
glass of water before it overflows? Experiment and see what you can
find! These things may seem silly, but you never know where a pattern
is going to show up. The Fibonacci sequence is everywhere: you just
need to know where to look!
Glossary
Aesthetic - Concerned with beauty or the appreciation of beauty
Calipers - An instrument for measuring having two hinged legs
Fractal - A curve or geometric figure that reapeats itself over and over
again, usually on a smaller scale each time
François Lucas - A French mathematician known for his study of the
Fibonacci sequence and the related Lucas sequence, which was named
after him
Golden Section - The division of a line so that the ratio of the larger part
to the smaller part is the same as the ratio of the whole to the larger
part. That ratio is the golden ratio, 1/2 (√5 + 1). It is a proportion that is
considered to be particularly pleasing to the eye.
Leonardo Fibonacci - Also known as Leonardo of Pisa, he was an Italian
mathematician famous for his study with the Fibonacci numbers.
Logarithmic - Constructed so that successive points represent values that
are in an equal ratio.
Parthenon - The temple of Athena Parthenos, built on the Acropolis in
447–432 BCE to commemorate the recent Greek victory over the Persians
Prime Number - An integer that has no integral factors but itself and 1
Pyramids of Giza – A complex of ancient pyramids, also known as the
Great Pyramids, located just outside of Cairo, Egypt
Ratio - A fraction that divides two quantities. For example, 3 girls to 2
boys is a ratio of 3:2.
Regular – A polygon is regular if all of its sides are the same length
Sequence – A list of numbers that follows a specific pattern
Image Credits
http://fc07.deviantart.net/fs70/i/2010/085/4/6/Fibonacci_Spiral_Art_by_Grwobert.jpg
http://pics.livejournal.com/skywardprodigal/pic/00e9xgwy
http://math2033.uark.edu/wiki/images/3/35/Fibonacci2.jpg
http://www.morrischia.com/david/portfolio/boozy/research/fibonacci%27s_20rabbits_img_2.png
http://www.petsintouch.com/nwposter/wp-content/uploads/2010/09/rabbit-big.jpg
http://theurbandaily.com/files/2010/10/algebra-equation-on-blackboard-1.jpg
http://office.microsoft.com
http://thewoodenwagon.com/Merchant2/graphics/00000001/Stacking_towerL.jpg
http://www.scheerimages.com/files/phi_gold.png
http://4.bp.blogspot.com/_OtWAadOXEMo/TAqyjW9mAQI/AAAAAAAADNA/zOmE-29hjxM/s400/
Golden_rectangle.png
http://www.freeimagehosting.net/uploads/90711c7f41.png
http://faculty.evansville.edu/ck6/bstud/elucas.jpg
http://www.segerman.org/2ndlife/fibonacci_pinecone.jpg
http://scienceblogs.com/goodmath/upload/2006/08/pentaphi.jpg
http://farm1.static.flickr.com/16/21152695_35a8e05d1a_m.jpg
http://www.jananas.com/wp-content/uploads/2009/01/golden-rectangle.gif
http://fc06.deviantart.net/fs70/f/2010/075/5/9/Fibonacci_Spiral___Flash_Art_by_Rahzizzle.jpg
http://amazingdata.com/mediadata6/Image/amazing_fun_featured_2561778790105101600S600x
600Q85_200907231856306879.jpg
http://www.zinn-x.com/images/fibonacci-nature-nautilus4.jpg
http://en.wikipedia.org/wiki/File:FibonacciChamomile.PNG
http://library.thinkquest.org/C005449/media/petalsfib.jpg
http://www.discoveredartists.com/uploadedImages/188/717/lightbox_Mac_Tracy_Fibonacci.jpg
http://25.media.tumblr.com/tumblr_kw26ndcjWT1qzyb10o1_500.jpg
http://2.bp.blogspot.com/_Su6QxOINZ3I/TCnfgT5NRJI/AAAAAAAAARg/bUkv28QVhlo/s1600/3862.jpg
http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio/golden.html
http://capnbob.us/blog/wp-content/uploads/2008/04/caracol.jpg
http://britton.disted.camosun.bc.ca/goldslide/gold08.jpg
http://ny-image3.etsy.com/il_570xN.106851039.jpg
http://www.library.mun.ca/qeii/FirstSpace/images/AH09T.jpg
http://www.helixglass.co.uk/images/leaded/thumbnails/fibonacci.jpg
http://16.media.tumblr.com/M5i2XO58Jiq9tcqrZ92qtZSco1_500.jpg
http://bp1.blogger.com/_UWWH1YUB1Fs/Rw0rxk7g3cI/AAAAAAAAAFk/HsBN_et2W8o/s320/
DSC03634_2.jpg
http://29.media.tumblr.com/FpXBuePy3k2t7k21gl23JORco1_500.gif
About the Authors
Conor Pappas and Curtis Belmonte are both juniors at the
Massachusetts Academy of Mathematics and Science. Conor lives
in Oxford, Massachusetts and Curtis lives in Shrewsbury,
Massachusetts. They are both published in the Scientia Review
and have a passion for learning. Conor has a passion for
Mathematics and Computer Science and hopes to major in those
subjects. Curtis likewise aims to pursue a major and later a
career in the field of computer science, and enjoys other
topics, such as psychology and physics.