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Carolyn,
When I do the lesson, I try to talk to the kids about various times in human
history when artists, designers and architects held an interest in the
potential for mathematics and mathematical proportion as the source of
prescriptions for beauty. None of the formulas which were developed seem
to have been confirmed as time marched on. Like so many things in the world
of art, those formulae which seemed unquestionable in one age was upended in
the next. Even an objective system like scientific perspective, which does
have hard and fast rules, can not be employed absolutely, without exception.
There are cases when it is technically wrong (see Andrea Mantegna ‘Dead
Christ' c. 1501, Tempera on canvas, Brera Gallery, Milan) but, at the same
time, artistically right for the picture. — because content affects form
just as form affects content.
Some time during the third century before Christ, Euclid discovered a
geometric proportion which has been considered the key to formal beauty by
many theorists in art. It is a ratio between the two dimensions of a plane
figure or the two divisions of a line such that the smaller element is to
the larger as the larger is to the whole. His ‘Golden Proportion' works best
in the lesson if one has some visuals to aid the teaching and learning.
For example, show your students ‘The Golden Section' ....draw a dark line
13" and then measure 5" out on the line and mark (or cut) that point. You
will have a visual relationship of 5 to 8. Then you can mark (or cut) the
5" line at 2" and it will be divided into a relationship of 2 to 3. Then
you can point out a visual relationship between the 2" line and the 3" line
as the same proportion is represented between the 3" line and the 5" line.
Then show that the same proportion can be represented by the relationship of
the 5" line to the 8" line or the 8" line to the 13" line.
In short, the size if the compared lines change but the proportion stays the
same. And the relationship of those proportions hold a natural balance. The
‘Golden Section' results in a natural, balanced proportion.
A is to B as B is to A+B ...or...
A B
___ = _____
B A=B ....or....
If A is 1, then B is 1.618 ...or....
If B is 1, then A is 0.618
[Warning!!! this next paragraph may be a bit much for some of us
right-brained art teachers.)
If you extend Euclid's relationship to arrive at a longer sequence you will
discover Fibonacci numbers. (1..2..3..5..8..13..21..34..55..89..144..etc..
) The theory of Fibonacci numbers is just a theory. In 1202, the Italian
mathematician Fibonacci (also known as Leonardo da Pisa) found the
relationship, the number of rabbit pairs in the nth breeding season (Un)
will be equal to the number of pairs one season earlier (Un-1) {because none
have died} plus the number of rabbits two seasons earlier (Un-2) because all
of those rabbits are now mature and each pair produces a new pair. His
recurrent sequence was stated as: ‘U1, U2, U3, ............... Un'. His
recurrent relation was stated as ‘Un = Un-1 + Un-2'. And this equation is
obviously inadequate, when n < 2. Therefore we must establish certain
supplementary conditions. Because for instance, the first term of the
sequence has no terms preceding it, and the second term is preceded by only
one. This means that in addition to condition Un = Un-1 +Un-2 we must know
the first two terms of the sequence in order to define it. If we start with
one immature pair of rabbits (U1 = 1, U2 = 1), it is easy to calculate the
numbers of rabbits pairs each successive season (1, 1, 3, 5, 8, 13, 21, 34,
55, 89,.......where each number is the sum of its two predecessors). The
sequence and numbers are called the ‘Fibonacci Sequence' and ‘Fibonacci
Numbers' respectively.
Now, back to the art lesson.......
After talking about the natural beauty of Euclid's proportion, show how it
can be translated into either of the ‘Golden Triangles'..In the two types of
‘Golden Triangles' the proportion is maintained (1.) width to height or (2.)
base to sides
Another shape to include is the ‘Golden Cuboid‘ with the relations stepped
from height to width to depth (example= 3 to 5 to 8 or 8 to 5 to 3, etc. ).
The ‘Golden Cross' has the longer vertical member (say.. 13") divided by a
horizontal member (say.. 8") at a point down the vertical (say... 5"). Here
again the size can change but the proportion must remain true to the ‘Golden
Proportion'.
You can go on to show the kids a ‘Golden Rectangle', a ‘Golden Spiral'
(drawn using the "Golden Rectangle' divided in a progression), a Golden
Ellipse (with the proportion (say...5 to 8) shown in the relationship
between the width to the length of the oval), and the 'Golden Angle' (137.5
degrees, dividing a circle from its center like a pie.
After showing the kids the proportion, turn to man-made objects such as the
U.N. Building, the face of the Classical Greek Parthenaon, the Pyramid of
Giza, a Greek amphora (vase), or a Violin where the ‘Golden Rectangle', the
‘Golden Triangle' and the ‘Golden Section' have been used extensively. There
are many more examples which can be used.
Next, you can show nature as a wonderful source for the ‘Golden Proportion.'
Show the students the proportion in the growth of the chambers in a shell.
Draw a rectangle around an egg and see what results. Many trees seem to grow
in this proportion and one can check it out by measuring their height to
width. The proportion often shows up in microscopic cell structure. Show
your students the relationship of the human body from the top of the head to
the navel (belly button) and from the navel to the bottom of the foot. Then
show how the same proportion can be found in the relationship between the
distance from the top of the head to the base of the neck and from the base
of the neck to the navel. Then show the same proportion as it can be seen
between the distance from the bottom of the foot to the knee and from the
knee to the navel. The same general proportion shows up in the relationship
between the distance from the eyes to the tip of the nose and from the tip
of the nose to the chin in most people. It can also be seen in the
relationship between the width of the face to the height of the face (from
the chin to the top of the head).
At this point you may want to talk a bit about Fibonacci, the man and then
show the kids some Fibonacci numbers in nature.
Many flowers exhibit Fibonacci numbers...An Iris with three blooms from a
but, a Buttercup with 5 petals to a bloom, a Cosmo with 8 , Daisies with 13
and some with 21 petals to the blossom.
If you look at the petals as they spiral around and up an Artichoke you will
find the Fibonacci numbers of 5 rows in one direction and 8 row in the
other. A pinecone shows the relationship in 8 rows one direction to 13 rows
in the other direction. A pineapple has scales in rows of three different
directions and they will number 8,13 and 21 from any side view. The
arrangement of new buds on many plants show Fibonacci ratios in the
relationship between the number of buds and the circles of growth
Fibonacci relationships can be seen in the spiral of seeds in rows as they
are counted out from the center of a blossom. Fibonacci numbers can be
found by counting chambers and appendages of natural forms such as
vegetables, fruit, starfish, sand dollars, fingers on a hand or toes on a foot.
Please remember that this lesson must have plenty of visuals if it is to be
effective. The kids need to see the beauty of the proportion. They need to
see pictures of the buildings and the natural objects (or better yet, they
need to have the "real thing" to hold and examine) .
Using rulers with old picture magazines, they can be involved in a search
for more examples of the proportion in the world around them. When thy find
it, they can draw the ‘Golden Rectangle'( Golden Triangle, Golden Ellipse,
etc) to record it over its location on the picture. They can measure the
proportion as it can be found in their faces and bodies, then draw portraits
and overlay the ‘Golden Proportion" with the triangles, rectangles, etc. as
that proportion shows up in may places in their drawings. Others may choose
to design drawings for buildings, automobiles, tools, landscapes,
neighborhoods, or playground designs where the Fibonacci numbers are used to
create the Golden Proportion throughout the formal order of their project.
There is no question that the ratio is common in nature and it is appealing
to the human eye. Artists have attempted to paint perfectly composed
pictures by applying the ‘Golden Section' to their compositions.
Unfortunately, the fact that this ratio is a persistent feature of great
works does not mean that its presence will assure greatness. It does not
even imply that a work containing it will be good or that those that lack it
will be poor. Yet it is a fascinating phenomena which can help kids connect
math to art and nature.
Hope this helps,
Bob