The Golden Ratio
SECRETS
OF EGYPTIAN ARCHITECTURE REVEALED
Introduction
- Throughout
much of art history, artists and architects were concerned with
the proportions of the parts of their works. In fact, there
were not only particular ratios that were preferred, but sometimes
entire systems of proportions.
Rather than individual parts of a painting satisfying different
proportions, the painting would relate as a whole,
many parts were related by the same proportion. Surprisingly
enough, many of these systems of proportions were based on musical
intervals and the human body, as well as a unique ratio known
as the Golden Ratio. (http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html).
The Ancients believed the human body to be formed based on those
of the gods, so universal and divine proportional relationships
could be observed in their make-up. (Roth)
- The
Golden Ratio, also referred to as the Golden Section or the Golden
Mean, is often designated by the Greek letter phi, for Phideas
(490-430 BC). Phideas was an Athenian sculptor who supposedly
used the golden ratio in his work. Phideas was an artistic
director of the construction of the Parthenon in ancient Greece.
The Golden Ratio can be seen throughout ancient architecture
including both the Greek and Egyptian cultures.
(http://www.geom.umn.edu/~demo5337/s97b/fibonacci.html)
Exploring Phi
Phi found
in nature:
- The
Golden Ratio finds its roots in nature. The value of phi
can be found in numerous areas of nature that display
the beauty
of its unique mathematical properties. These naturally occurring
examples show the importance of order and how it acts on the
Golden Ratio. The Chambered Nautilus is one example of the Golden
Ratio in nature. These spiral seashells expand from the
center at the ratio of phi, which provides the greatest structural
strength because the partitions found in its shell never line
up. The position of plant leaves on a stalk also follow
the ratio of phi. This proportion guarantees that the same placement
of the leaves is never repeated and allow sunlight to reach the
underlying leaves.
- In certain
flowers, such as daisies
or
aster, the number of pedals is related to the diameter of the
pod by the Golden Ratio. The Golden spiral formations also
extend to such things in nature as the positioning of the seeds
on pine cones, the shape of the pods of sunflowers, and the placement
of the leaves on lettuce heads, which "correspond with radius
vectors generated by nested golden section rectangles."
(http://www.tomgilmore.com/phi.htm)
Modern examples of phi:
- The
Golden Ratio mysteriously has an intrinsic aesthetic value.
This quality of beauty makes the Golden Ratio a popular ratio
for a variety of modern converntions. Television sets,
table tops, and even 3x5 photographs hold true to the Golden
Ratio. The human body also corresponds to the Golden Ratio
in many ways. This ratio is seen in facial features
and associated with beauty. A classic example is with the
face of Leonardo di Vinci's Mona Lisa. Studies show that people
unknowingly chose this ratio as the most pleasing, "responding
to an innate sense of balance." Combining mathematics with
an aesthetic evaluation, it can be say that "symmetry considerations
in perceptions of beauty combine with golden section rectangle
intersection zones in defining attractive proportions in the
features." (http://www.tomgilmore.com/phi.htm)
- The
notion of incorporating the Golden Ratio into architecture has
also been included by modern architects. Le Corbusier is
among the most frequent users of proportional systems in his
designs. His "Modular" proportion system resembles
the Egyptians ideas about proportions in the human body and geometric
figures. (Roth)
Euclid's
definition of phi:
- The
third definition of book VI of Euclid's Elements states
that "a straight line is said to have been cut in extreme
and mean ratio (EMR) when, as the whole line is to the greater
segment, so is the greater to the less." We shall let the
smaller part equal the number 1 and the larger part equal phi,
and so phi is the Golden Ratio.
Algebraic
derivation of phi (according to Euclid):
- Now
let's look at an algebraic derivation of the Golden Ratio. Considering
the above definition, we have that: (phi)/ 1 = (1 + phi) / (phi),
so (phi)^2 = 1^2 + 1(phi), and therefore, we get the quadratic
equation, (phi)^2 - (phi) - 1 = 0. When we solve this quadratic
equation for phi, we get that phi = {[1/2 + (or -) sqrt(5 )]/
2}, which is approximately 1.618. Also stated in Theorem
II, 11, we find related to area, a construction which defines
the partitioning of a line "according to division in extreme
and mean ratio (DEMR)." In Euclid's Elements, the
method for finding DEMR is "to cut a straight line so that
the area of the rectangle contained by the whole line and one
of the segments is equal to the area of the square on the remaining
segment." (Hertz-Fischler)
A
simple construction of phi:
- There
are many other possible ways to find a geometric construction
of the Golden Ratio. One of the simplest is to subdivide a square
of side length 1 into two equal rectangles. Next, we set out
a distance equal to the diagonal of one of the rectangles, plus
half the side length of our original square. The ratio of this
new distance to the original side, with a length of 1, is the
Golden Ratio. (http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html)
Golden Polygons
The
definition of a Golden Rectangle:
- As explained
by Euclid, a Golden Rectangle is a rectangle in which the length
is divided into two pieces such that the ratio of the left part
(a) to the right part (b) is the same ratio as the whole (a+b)
to the left part (a). Therefore, we receive the formula: a/b
= (a+b)/a, which is the same formula we investigated earlier
with the segment cut into two parts 1 and phi. So, from the Golden
Rectangle, we see that the ratio a/b=1/2+sqr(5)/2, which is approximately
1.618, or phi.
An investigation
with the Golden Rectangle:
- When
we divide a Golden Rectangle into two pieces by dividing up its
length, we get a square on one side and a rectangle on the other,
which is similar to (the same shape as) our original Golden Rectangle.
Interestingly, if more squares were cut off from the remaining
rectangles, you will continue to receive smaller and smaller
Golden Rectangles! Once we cut off the last two remaining squares
in the center of the construction, we will end with two unit
squares side by side. From this point, we can construct a close
approximation of a spiral or a seashell shape if we choose to
do so.
- In order
to perform this construction of the Golden spiral, we will use
a compass and construct an arc of a circle, beginning in the
unit squares and passing through each of the successive squares,
with our compass set at a radius of the length of a side of each
corresponding square. (http://www.geom.umn.edu/~demo5337/s97b/spiral.html)
Supposedly, Pythagoras discovered
this ratio for rectangles, and the ancient Greeks incorporated
it into their architecture and works of art. Golden Rectangles
were thought to be the most pleasing to the eye, causing many
architects to incorporate them in the buildings they constructed.
The Parthenon, mentioned earlier in the introduction, is an example
of one such building. (http://www.mcn.net/~jimloy/golden.html)
The
Golden Triangle in pentagons:
- Besides
the Golden Rectangle being related to the Golden Section, the
crossing lines from the corners of a regular pentagon are also
related to this ratio. The vertices of a regular pentagon are
set on a circle at 72 degrees
apart, as
seen by the computation 5 x 72 = 360. In the center of the pentagon,
there are five intersections of the diagonals at which the Golden
Section can be seen. Here, the triangle formed by two adjacent
diagonals is determined by the angles 36 - 72 72. This
triangle is known as the Golden Triangle. If we bisect one of
the 72-degree base angles, we can see that the small triangle
formed will also be a 36 - 72 72 triangle. Therefore, the
sides of the original triangle and the small triangle are proportional.
And since the triangles are isosceles, we can then say that the
side opposite the bisected angle is divided into the Golden Ratio.
Surprisingly, there are said to be at least 200 phi ratios in
a pentagon. (Hertz-Fischler)
Fibonnacci Sequence
The Fibonacci
Sequence: 
- Also
related to the Golden Ratio, is the famous Fibonacci sequence
named after Leonardo Fibonacci, an 11th century mathematician.
The Fibonacci Sequence is expressed by the sequence 1, 1, 2,
3, 5, 8, 13, ...etc., where each term is the sum of two previous
terms, so the general form is phi^(n) = phi^(n-1) + phi^(n-2).
Therefore, in the sequence we have for instance 2 + 3 = 5, 3
+ 5 = 8, ...etc. By studying the ratio of two successive terms
in this sequence, we will see that a pattern will begin to emerge.
As we approach larger and larger numbers, the ratio will get
closer and closer to the Golden Ratio or 1.618. This can easily
be seen by setting up a spreadsheet using Fibonacci numbers and
the ratios between successive numbers.
A
construction of the Fibonacci Sequence:
- Beginning
with the Fibonacci Sequence, we can perform the operations that
we used to find the Golden Rectangle, but this time in reverse
order. Therefore, we will start with a square with its side length
equal to one unit, and add a square of the same side length to
form a new rectangle. Next, we will continue by adding another
square whose side length is the length of the longer side of
the rectangle already formed by the first two unit squares placed
side by side, therefore this length would be two units. Remarkably,
by inspecting the lengths of the longer sides of the rectangles,
we will notice that they will turn out to be successive numbers
in the Fibonacci sequence! Finally, we will see that the large
rectangle (composed of all the added squares) will eventually
look more and more like a Golden Rectangle! (http://forum.swarthmore.edu/dr.math/faq/faq.golden.ratio.html)
Two
specific cases for the Fibonacci Sequence:
- While
traveling in Egypt, Fibonacci was exposed to two formulas that
are specific cases of the general formula phi^(n) = phi^(n-1)
+ phi^(n-2). The two formulas were (1 + 1/phi = phi) and (phi
+ 1 = phi^2). The first formula, (1 + 1/phi = phi), illustrates
the case that when the length of the long segment in the Golden
Section, phi, is equal to 1 unit, the short segment in the Golden
Section would be expressed as the reciprocal of the entire line.
Therefore, the length of short segment would be equal to 1/1.618,
which equals .618. Surprisingly, if you look at the value of
phi, in the tenths, hundredths, and the thousandths place, you
will notice the same number that we just received as the reciprocal
of phi, .618! The appearance of this particular decimal in both
phi and its reciprocal illustrate the symmetry of phi. Due to
the explanation above, phi is often times known as the "natural
balance" number.
- The
second formula that Fibonacci came across during his travels
to Egypt, (phi + 1 = phi^2), is known to be a handy formula used
when performing constructions with phi as a basic unit. This
formula can basically be interpreted as stating that there is
a series of numbers, where each number is related to its previous
number by the value phi. This explanation about the ratios was
explained earlier in the section, and can be thought of as alternative
interpretation of the Fibonacci Sequence. (http://www.tomgilmore.com/phi.htm)
Egyptian Pyramids
An Introduction to the Egyptian Pyramid:
- The
pyramids have fascinated the world generation after generation.
Whether it is their gigantic size, the mystery and secrets that
the pyramids conceal, or just the breathtaking beauty of the
structures, people have been drawn to Egypt to experience one
of the Seven Wonders of the World. In Egypt, the pyramids were
built in the lifetime of a single king, and their main purpose
was to help grant the king immortality. The pyramids were constructed
in a period of time that was known as the 4th dynasty of the
Old Kingdom, which was approximately 2800 BC. Around 440 BC,
an ancient scholar named Heroditus (484?-425 BC), also known
as the Father of History, was the first to record information
about the pyramids.
- These
mysterious pyramids are known to hold numerous secrets; many
anthropologists and archeologists have devoted their lives to
learning their secrets. Some claim hat the pyramids form
part of an enormous star chart and that the shafts leading down
into the chambers of the pyramids are aligned with certain stars.
Others believe that they are models of the earth, and that they
are part of a communication system with other forms of life.
(http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html)
Phi and the Egyptians:
- Scholars
have argued for centuries about whether the Egyptians really
had some knowledge of the Golden Ratio, and if so, how that knowledge
came about. Some argue that
the
Egyptians did not have the mathematical capabilities to use phi
in their constructions of the pyramids and temples, and others
respond stating that they gained knowledge of phi from simpler
methods that were not mathematically complex. Some of these examples
included using a string in a method known as the rope-stretcher's
method (explained later on) or even using a measuring wheel to
measure specific distances along the ground.
- All
other information points to the fact that the Egyptians intentionally
used the Golden Ratio, and even believed that it was sacred,
hence called the "sacred ratio." Therefore, this ratio
was extremely important in their religion. Besides using the
Golden Ratio in their pyramids, they also used it when building
temples and places for the dead. The use of this ratio was central
to many burial procedures. Some believed that if the proportions
of their buildings were not according to the golden ratio, that
the deceased might not make it to the afterlife or that the temple
might not appease the gods. As well as the role of the Golden
Ratio in their religion, the Egyptians found the ratio to be
visually pleasurable . Therefore, they used it many other aspects
of their lives, including in their system of writing, in the
form of hieroglyphics, in statues, and in the arrangement of
their temples. (http://www.geocities.com/capecanaveral/station/8228/arch.htm)
- The
Egyptians noted that the 3-4-5 triangle displays a Golden Ratio
between its 5-unit side and 3-unit base. The Egyptians thought
this observation was of extreme importance and therefore, used
it in many ways, such as a surveying tool and in the construction
of the pyramids. The Egyptians used other sets of paired numbers
as well, and most extensively, the repeating value of 1.618181818...
from the two paired numbers 55 and 89 which were used in building
the Great Pyramid. (http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html)
- Interestingly,
phi was utilized in ways other than just mathematically for their
constructions. For example, as explained in the introduction
about the value of phi found in nature, the massive blocks used
in building the lower levels of the Great Pyramid were overlapped
by the ratio phi to optimize stability. Also, an in-depth study
of the dimensions reveals that the balance of the Great Pyramid
depends on the Golden Section being applied to its dimensions.
(http://www.tomgilmore.com/phi.htm)
- With
the Golden Ratio found in a regular pentagon as explained earlier,
in Egypt, the pentagon was a symbol of an "underground womb"
and was therefore symbolically related to the pyramid as well.
This idea can be seen in the layout of the Giza pyramids. (http://www.celt.drak.net/alliekat/pentagram.htm)
Golden Triangles and Rectangles related
to the Great Pyramid:
- Although
many ancient scholars gave accounts of the constructions of the
pyramids, a paper known as The Ahmes Papyrus of Egypt presents
an account from 4700 BC of the building of the Great Pyramid
of Giza. The Great Pyramid was constructed with the proportions
given according to a "sacred ratio." Today, modern
measurements show that the ratio of half the base to the slant
edge of the Great Pyramid is almost exactly 0.618. Therefore,
the faces of the pyramid appear to be Golden Triangles, which,
as mentioned earlier are isosceles triangles with base angles
of 72º and vertex angles of 36º. We can also take a
cross-section through the pyramid to get a special type of triangle
know as the Egyptian Triangle, also called the Triangle of
Price, and the Kepler Triangle. From studying the calculations,
we learn that the Egyptian triangle has a base of 1 and a hypotenuse
equal to phi.
- Also,
we can see that its height (which we will call h), by the
Pythagorean theorem, is given by: (h^2) = ((phi)^2) (1^2).
So, solving for the height h,
we get a value
of the square root of phi! Therefore, we learn that the sides
of the Egyptian triangle are in the ratio of 1: sqrt(phi) : phi.
At a later period in history, the astronomer Johannes Kepler
(1571-1630) became fascinated by the golden ratio. He came to
the conclusion that if the sides of a right triangle are in geometric
ratio, then the sides are in the ratio 1: sqrt(phi) : phi, which
is another reason why this special triangle is also named the
Kepler Triangle.
(http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html)
(http://www.geocities.com/capecanaveral/station/8228/earch.htm)
- It has
also been found that the apothem, which is another name for the
slant height, of any of the side triangles of the pyramid (the
pyramid's faces) relate to half the base in a proportion equivalent
to the Golden Section. The ancient Egyptians used a basic measurement
called the cubit, and here we have that half of the base is approximately
equal to 220 cubits and the height of the apothem is approximately
equal to 356 cubits. Therefore, we can see that the ratio of
356 to 220 is equal to 1.618! Oddly enough, the cubit measure
is defined according to the length of the human forearm, yet
again showing how inter-connected the human body is with the
Golden Ratio. (http://www.geocities.com/capecanaveral/station/8228/earch.htm)
The Pyramids and the Fibonacci Sprial:
- For
another interesting relationship to the Golden Ratio, the three
oldest Giza pyramids lay on a Fibonacci spiral. Also, the opening
to the Hall of Records, a place that apparently contains the
history of the earth, is geometrically marked by the lines that
bisect the length and the width of the Golden Rectangle, from
which the spiral is created. The opening to this famous Hall
is found in the right shoulder of the great Egyptian monument,
the Sphinx, and has been marked geometrically. Interestingly,
if we chose to bisect the Golden Rectangle that encompasses the
spiral around the Giza plateau, the line would cross straight
through the headdress of the Sphinx. (http://www.geocities.com/capecanaveral/station/8228/earch.htm)
- Finally,
the line that bisects the Golden Rectangle mentioned above and
a line extended from the southern face of the middle pyramid
in the set of the three oldest Giza pyramids, form a cross that
marks a significant spot according to the Egyptians, on the right
shoulder of the Sphinx. (http://www.geocities.com/capecanaveral/station/8228/earch.htm)
The Fibonacci Sequence in the Great Pyramid:
- If we
choose to study
the ancient Egyptian measure of the cubit, we will see that it
reveals directly how the Fibonacci Series was utilized. Considering
the Egyptian Triangle that we looked at earlier, where we end
up with the ratios of the sides as 1: sqrt(phi): phi, it is indicated
that the builders of the Great Pyramid used 4 x 55 = 220 for
the length of half the base and 4 x 89 = 356 for the apothem,
which incorporates the two paired numbers 55 and 89, mentioned
earlier. These two numbers are numbers in the Fibonacci Sequence,
and therefore, not only was the Golden Ratio, Rectangle, and
Triangle utilized in the Egyptian architecture, but the Fibonacci
Series was used as well!
- Interestingly
enough, we can calculate the circumference of a circle from the
diameter using the Fibonacci Series and a formula, which relates
pi and the value of (6/5). Due to this formula, building with
the dimensions in the Fibonacci Sequence, such as with the Great
Pyramid using the dimensions of 4 x 55 and 4 x 89, provides a
built-in conversion that squares the circle.
Squaring
the Circle in the Great Pyramid:
- To say
that the base of the Great Pyramid squares the circle, we would
be stating that the perimeter of the base of the Great Pyramid
equals the circumference of a circle whose radius is equal to
the height of the pyramid. On the other hand, another definition
states that the area of the same circle mentioned above, with
radius equal to the pyramid height, equals that of a rectangle
whose length is twice the pyramid height and whose width is the
width of the pyramid.
- So,
if we choose to look at the first definition, we will use the
dimensions of height and width, so that if the base of the Great
pyramid is equal to 2 units in length, then the pyramid height
is equal to the square root of phi. Therefore, the perimeter
of base is equal to 4 x 2, which equals 8 units. Then for a circle
with its radius equal to pyramid height of the square root of
phi, then the circumference of circle is equal to 2(pi)(sqrt(phi))
= 27.992. From these measurements, we find that the error of
the difference between the perimeter of the square and the circumference
of the circle is very small. Therefore, using the square of the
pyramid base and a circle with the radius of the height of the
pyramid, we find that the perimeter of the square and the circumference
of the circle have almost the exact same values.
- From
the previous observation, we can now find an approximate value
for pi in terms of phi. Because we just found out that the value
for the circumference of the circle, which was given by the formula
2(pi)(sqrt(phi)), is almost exactly equal to the perimeter of
the square around the base of the Great Pyramid, which is equal
to 8 units, we can now set the two dimensions equal to each other
and solve for phi. We get that pi is approximately equal to 3.1446,
therefore, the error compared to the true value of phi is extremely
trivial as well.
- Now,
we can see check the second definition, which states that the
area of that same circle, with radius equal to the pyramid height
equals that of a rectangle whose length is twice the pyramid
height (sqrt(phi)) and whose width is the width (2)
of the pyramid. So, we calculate that the area of
the rectangle is 2 (sqrt(phi)) ( 2 ) = 5.088, and the area of
circle of radius (sqrt(phi)) is (pi)(r^2) = (pi)(sqrt(phi)^2)
= (pi)(phi) = 5.083. The error between the values of the area
of the rectangle and the circle of radius sqrt(phi) is almost
negligible.
Explanations for the Egyptians vast knowledge
of the Golden Ratio:
- For
thousands of years, scholars were baffled about how the Egyptians
came to have such vast understanding of pi and the Golden Ratio.
Two theories, namely the Pizza Cutter Theory and the Rope-Stretcher's
Triangle aim to help uncover some possible explanations. If we
suppose that the Egyptians knew nothing explicitly about phi,
surprisingly, they could have constructed the pyramid dimensions
using only a measuring wheel, like those used today to measure
distances along the ground. In order to use this method, they
would have taken a wheel of any diameter and laid out a square
base that was equal to one revolution on a side. Then, the Egyptians
would have to make the pyramid height equal to two diameters
of the wheel. Oddly enough, this simple method will give us a
pyramid having the exact shape of the Great Pyramid, including the squaring of the
circle for both the perimeter and the area, as well as containing
the Golden Ratio!
- For
the second possible explanation, the Egyptians could have used
a tool known today as the Rope-Stretcher's Triangle. This
is a rope knotted into 12 sections, stretched out to form a 3-4-5
triangle (3 + 4 + 5 = 12). According to the converse of the Pythagorean
Theorem, if the square of one side of a triangle equals the sum
of the squares of the other two sides, then we have a right triangle.
So, we have that 52 = 32 + 42, so 25 = 9 + 16. Therefore, using
a rope with twelve knots, we can create a right triangle. This
type of triangle is also known as the Rope-Knotter's triangle,
and the Pythagorean triangle. As mentioned earlier, there is
evidence that the Egyptians used the 3-4-5 right triangle in
the construction of their pyramids. This triangle displays a
Golden Ratio between its 5-unit side and 3-unit base, and therefore,
this may help to explain how at least part of their knowledge
of the Golden Ratio came about.
- Whether
or not the Egyptians held specific knowledge about the proportions
they were using, the fact remains that the Golden Ratio and Fibonacci
sequence are recurring throughout Egyptian culture. Even
when spans of several hundred years separated construction dates
of the architecture, the proportions remain the same. (Giedion).
Teacher Resources
1) Fibonacci
Numbers, the golden section, and the golden string; This site
details construction methods for the Golden ratio and ideas for
how to involve students in construction discovery. This site also
looks at the relationship between the Golden ratio and trigonometry.
take
me there
http://www.mcs.surrey.ac.uk/personal/R.Knott/Fibonacci/phi2Dgeomtrig.html#cons1
a) constructing
phi with compass and protractor
b) making a paper knot to show the Golden section in a pentagon
c) basic ratios (the shape of a piece of paper)
d) Fibonacci Paper
e) Phi and Trig graphs (cos x, sin x, tan x)
2) Phi
and the Fibonacci Numbers: this is a site rich in information
about calculating numerically Phi and Fibonacci numbers and analyzing
the calculations (the ratio of successive Fibonacci numbers gets
closer and closer to phi). It also includes activities on how
to calculate these numbers using varying methods. take
me there
http://www.mcs.surrey.ac.uk/personal/R.Knott/Fibonacci/phi.html
a) graph
of y=phi(x)-each integer coordinate point closest to phi line
are the successive Fibonacci number's
b) Phi^2=Phi + 1, so Phi = 1 + 1/Phi (definition that number
which is 1 more than its reciprocal) (Formula for phi using a
continued fraction)
c) Same as above with using your calculator-way to discuss number
convergence.
3) Basic
Lessons. This site takes the visitor on a guided "tour"
of the Fibonacci Sequence and the Golden Ratio. There site includes
many questions with pop-up answers that target the key points
about the topic. The tour is page by page each adding new information
about the Golden Ratio. take
me there
http://www.geom.umn.edu/~demo5337/s97b/fibonacci.html
a) explanation
of Fibonacci sequence
b) Fibonacci in Nature (construction of spiral using compass,
ruler and protractor)
c) Golden Ratio: includes activitiy ideas for exploring the ratios
of common rectangles, i.e. 3x5 photo, notebook paper, etc.
4)A Golden
Greek Face: This site features a great activity to print out and
do in a classroom. It involves measuring the dimiensions of a
Greek statue and finding the golden ration from these measurements.
take
me there
http://markwahl.com/golden-ratio.htm
Links
In addition
to the links found throughout our site, here are a few other sites
we liked:
Project Sources
Even though
we would love to take credit for discovering all of the information
found on this site, an abundance of credit is due to books and
web-sites we consulted. The one thing we did discover about
the Golden Ratio is that there seems to be an infinite amount
of information on the subject. Below are the authors of
most of the ideas seen on our page and the sites from which we
borrowed the beautiful images on our page. The web-sites
listed under teacher resources and links also offer additional
information about this fascinating topic.
books:
- Giedion,
S. The Beginnings of Architecture. Princeton:
Princeton University Press, 1964.
- Herz-Fischler,
Roger. A Mathematical History of the Golden Number. Mineola,
Dover Publications, INC., 1998.
- Huntley,
H.E. The Divine Proportion, A study in mathematical beauty.
New York:Dover Publications, INC., 1970.
- Roth,
Leland M. Understanding Architecture. New
York: HarperCollins Publishers, 1993.
web-sites:
- http://forum.swarthmore.edu/dr.math/faq/faq.golden.ratio.html
- http://www.celt.drak.net/alliekat/pentagram.htm
- http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html
- http://www.geocities.com/capecanaveral/station/8228/index.htm
- http://www.geocities.com/capecanaveral/station/8228/arch.htm
- http://www.geocities.com/capecanaveral/station/8228/earch.htm
- http://www.geocities.com/capecanaveral/station/8228/etemples.htm
- http://www.geom.umn.edu/~demo5337/s97b/fibonacci.html
- http://www.geom.umn.edu/~demo5337/s97b/spiral.html
- http://www.mcn.net/~jimloy/golden.html
- http://www.pbs.org/wgbh/nova/pyramid/explore/age.html
- http://www.tomgilmore.com/phi.htm
This page
is the final project of Elizabeth Jones and Natalie Smith for
Math5200.
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