We've recently discovered a new method to
generate an image from a number."
This process has become increasingly useful in
creating a digital portrait of what will be an
important number for us in the next section.
As we'll see, the Golden Ratio has long played
a critical role in the construction of many
masterworks in art, music, design, architecture,
and even nature itself.
But it turns out that we're not the first to think
of the Golden Ratio as a key to the best possible
design of a building.
Historians believe that Vitruvius used it during
the construction of the Roman Pantheon in
nearly 25 B.C., and it appears in architecture
and art as far back as the third century B.C.
The Golden Ratio makes its first appearance in
Western art in the 15th century.
Italian artist Leonardo da Vinci is credited with
introducing the Golden Ratio into Western art.
He would use it to create his masterpieces,
including the Mona Lisa, the Madonna and Child,
and The Last Supper, all of which are considered
pinnacles of the Western art tradition.
Leonardo's knowledge of perspective enabled him
to use the Golden Ratio in order to make the
perspective of his images more natural, which in
turn contributed to the way people viewed
the world around them.
Over the centuries, the Golden Ratio became an
icon for proportion in Western art, and was used
throughout the Renaissance and into the 20th
century.
For the remainder of this lesson, we'll study the
most fundamental property of the Golden Ratio,
its beauty, and then explore the importance of
this property in a variety of ancient disciplines.
To create a beautiful image, we need two distinct
properties.
To begin, we want an aesthetic element.
To create an image with beauty, we must balance
proportions so that the Golden Ratio is
present in both axes of a given image.
Here, we have a beautiful portrait of
Cindy Crawford.
We see that the Golden Ratio, along with the
other geometric properties of this image are
balanced.
The height and width of the image are both
exactly 2 times the Golden Ratio, or 2 to the
Golden Ratio.
In contrast, this is not a beautiful image.
The Golden Ratio is present in one dimension, but
not in the other, so the Golden Ratio is only
part of the equation.
I've illustrated the four possible axes of any
given image.
In the first image, we can see that the Golden
Ratio is present only in the left-right axis of
the image.
This means that the top and bottom of the image
are not in perfect equilibrium, so the Golden
Ratio is not present in the height and width
dimensions of this image.
This is a good way to notice when you see
something that has the Golden Ratio on one
axis but not the other.
Because an image has a single orientation,
we can represent the Golden Ratio only by two
equations, one that defines the top and bottom,
and the other that defines the right and left
of the image.
The Golden Ratio has several properties of its
own, but the most important is this.
When we place the Golden Ratio on the left and
right of an image, the height equals the width.
We call this equality of the height to width the
"equality of parts."
In this example, the proportion that describes
the left and right sides of the image is the
Golden Ratio itself.
While we can study the Golden Ratio by itself,
the beauty of this ratio is enhanced by its
combination with other proportions.
The Golden Ratio has many such
combinations in nature.
Some of the most compelling and inspiring
combinations are found in nature.
The Golden Ratio can be placed on many different
sectors of a given image.
We use these combinations of the Golden Ratio to
create beauty in a range of art forms, including
architecture, painting, music, and other
visual forms.
While there is much beauty in the natural world,
we have a special fondness for the Golden Ratio
in music.
Here are two famous songs that are well-known
for their Golden Ratio.
Both of these songs are so well known for
their perfect beauty that I'm sure many of you
could sing them from memory.
On the left is Dido's, "Here is a portrait of
my love."
On the right is David Bowie's "Sound and Vision,"
the first of a dozen albums that contain
proportional values of the Golden Ratio.
These proportions are most evident in David
Bowie's album title, which references two
different ideas.
The first, "sound," refers to the song's
lyrical themes.
This song, at times called "The Laser Man,"
featured unusual chord sequences and rhythm.
Its rhythm patterns are influenced by jazz.
The second idea, "vision," refers to the
songs featured in this album.
David Bowie's musical influences included
everything from the Beach Boys, to the Beatles,
to Kraftwerk, to the Velvet Underground, and
even the Japanese electronic band Yellow Magic
Orchestra.
David Bowie was inspired by Japan's pop culture
and the way Japanese teenagers use the ratio
in fashion, design, and music.
Since we've taken a look at the Golden Ratio as a
mathematical concept in art and music, let's
take a look at some of the most beautiful
architectural structures in history that were
designed using the Golden Ratio.
Here we have some of the most iconic structures
in Western history, including the Pantheon
and the Duomo in Florence, Italy, and St. Paul's
Cathedral in London.
As you can see, the Golden Ratio is present in
all three of these masterworks.
The Pantheon and the Duomo both were
designed using the Golden Ratio, as well as a
beautiful ratio called the "divine proportion,"
which has fascinated artists and architects for
centuries.
In fact, it is one of the oldest ratios still in
use by modern architects and designers.
The divine proportion has fascinated many of
the great minds in art.
For example, in the 1500s, this ratio was
celebrated by the 16th century artist and
architect Andrea Palladio.
This relationship, when placed between the
divine and the Golden Ratio, is known as the
"divine Golden Ratio," or the "divine
proportions."
The divine proportion is the golden ratio of 1.618.
While other proportions, such as the Golden
Ratio, can be demonstrated on multiple axes,
and can be displayed on straight lines or
curved lines, the divine proportion only works
on curved lines.
Why is the divine proportion so powerful?
I think the reason is obvious when you look at
the image above, which illustrates the power of
the divine proportions in a single line.
And, unlike the Golden Ratio, which
consists of two numbers that are not whole
numbers, the divine proportion consists of
whole numbers that are expressed as fractions.
The lengths of a and c and the diameter of the
circle are all in the ratio of 1.618 to 1.
This is the golden ratio, and the ratio between
a and c has the same value as the radius of the
circle.
The relationship between the divine proportions
is so tight and so compelling that it has
a unique quality about it.
Every other ratio that we've examined
includes two number that are a little bit off
from 1, but the divine proportion uses two
whole numbers.
Because these numbers are whole numbers, this
means that the divine proportion exists not
just on curved lines, but also on straight lines,
allowing us to see the beauty of the divine
proportion in everything from buildings, to
crystals, to the human body.
While there is much beauty in nature, there
are many other fascinating properties of the
Golden Ratio in art, music, mathematics, and
nature that have been studied and celebrated.
Let's take a look at some of the most
powerful and interesting uses of the Golden
Ratio.
In 15th century Persia, mathematicians
discovered that the Golden Ratio can be used
to find a sequence of squares that are the most
beautiful, as well as a sequence of cubes that
also add up to the square of the most beautiful
number.
This ratio was first discovered in music when
German composer Pythagoras and music
theorist Johann Faulhaber were both searching
for a way
