Essay 2 | Page 1 Dr. Shanyu Ji | Math 4388, Section 18577
ART USES MATHEMATICS AS ITS ULTIMATE TOOL:
The Golden Ratio and Linear Perspective
Have you ever wondered how painters are able to paint three dimensions onto a flat
surface making it look almost like a photograph? How are they able to paint the subjects of their
piece proportional in size to one another to make it more realistic? Before the Renaissance
period, which was between 1400 and 1600, artists did not use certain mathematical tactics in
their work. Perspective was considered but applied using inaccurate methods and proportions
were not always apparent. During the Renaissance, individuals like Leonardo da Vinci and
Filippo Brunelleschi contributed to society and mathematics by rediscovering and reintroducing
ideas such as proportion and perspective. Not everyone sees art as a science, but it truly is
because many works are made inconsideration with mathematical calculations and tactics.
Before the Renaissance period, many countries in Europe were experiencing poverty and
disease. Religion played a vital role in society and culture in European countries. Due to
religious constraints, individuals during this time were not able to explore and discover new
techniques in mathematics.1 The birthplace of the Renaissance era was in Italy, where Italian
cities were able to grow, flourish, and establish trading networks, which gave citizens room to
discover and practice academics and other trades. It was a movement towards humanism, where
humans strived to become the best they can be rather than be limited to study theology and
medieval practices. This allowed individuals to rediscover the work of previous mathematicians
and scientists, which is why this era was known as the ‘rebirth’ of society.2
1
"What Was Life like before the Renaissance?" Birthplace of the Renaissance. 2013. Accessed September 29, 2016.
Shuttleworth, Martyn. "Renaissance Science - The Path to Enlightenment." Explorable. Accessed September 29,
2016.
2
Essay 2 | Page 2 Individuals reexamined and added onto intellectual ideology from works previously
written by Greeks, Egyptians, and others before the conquest of the Roman Empire. Revisiting
previous works led Leonardo da Vinci, who was an artist, mathematician, and inventor during
this era, to rediscover Euclid’s formula for the ‘Golden Ratio’ (Figure 1). Euclid was the first to
record the definition of the Golden
Ratio in his work, “Elements” around
300 B.C.3 The ‘Golden Ratio’ is
represent as the Greek uppercase letter
Φ or 𝜑 for lower case modern Greek,
Figure 1: The ‘Golden Ratio’ by Euclid
(Used by Artists Like Leonardo da Vinci)
Taken from http://www.livescience.com/37704-phi-golden-ratio.html
which is known as Phi and its unique
value is rounded off to 1.618. It is also
known as the Golden Number, Golden Proportion, Golden Mean, and many others because it has
been rediscovered numerous times.
The formula for the Golden Ratio is described as
“a + b is to a as a is to b”, which is
#$%
#
#
= =
%
1.618033988749 … = Φ .4 Prior to the Renaissance era,
the Golden Ratio has been found to have been used in
the construction of the Great Pyramids in Egypt. In
figure 2, it shows how the Golden Ratio was applied to
calculating the height and lengths of each side of the
Figure 2: Golden Ratio Triangle
Used to Construct the Great Pyramids
Taken from http://www.goldennumber.net/phi-pigreat-pyramid-egypt/golden-triangle-pythagorus/
pyramids. The Great Pyramid of Giza has these dimensions: height = 481 feet, length of each
side = 756 Feet. When inputted into the Golden Ratio equation the ratio it is about 1.5717, which
3
Meisner, Gary. "Divine Proportion/Golden Ratio in the Art of Da Vinci." The Golden Ratio Phi 1618. July 7, 2014.
Accessed September 29, 2016.
4
Hom, Elaine. "What Is The Golden Ratio." Live Science. June 24, 2013. Accessed September 30, 2016.
Essay 2 | Page 3 is relatively close to the Golden Number. A couple other examples that were linked to the use the
Golden Ratio was the ratio between the two successive Fibonacci numbers and the symbol used
to depict a pentagram (Figure 3). The ratios within the pentagram are equal to 1.618. These
examples show how the Golden Ratio has been previously used until its rediscovery during the
Renaissance era.
In 1509, Luca Pacioli who was an Italian
accountant and mathematician, worked with
Leonardo da Vinci to illustrate the Golden Ratio
using human proportions, which they defined as
‘Divine Proportion’ (Figure 4).5 Leonardo da
Figure 3: Golden Ratio in the Pentagram
Taken From: /www.cut-theknot.org/do_you_know/GoldenRatioInRegularPentagon Vinci’s illustration of the Divine Proportion is
known as the Vitruvian Man and it was intended to
show all the Golden Proportions on the human body. After analyzing this piece, it was found that
the circle and square that frames the man has a ratio that is roughly 0.609, which is 0.009 off of
the Golden Number, 0.618. Da Vinci might
have been off by 0.009, but it is impressive
that he was able to get so close to 0.618
without the use of modern tools and
technology. Although many believe that da
Vinci used the Golden Ratio, there are some
who are skeptical, who feel that its tied with
5
Ibid.
Figure 4: Vitruvian Man by Leonardo Da Vinci
Taken from http://monalisa.org/2012/09/12/leonardo-andmathematics-in-his-paintings/
Essay 2 | Page 4 geometrical proportions and fractional measurements of the body rather than phi.6 One claim is
certain though; and it is the fact that mathematics was used as a tool to create da Vinci’s
Vitruvian Man whether it be the implementation of the Golden Ratio or calculated proportions.
Another famous artwork from da Vinci
that uses this special ratio is one of his most
famous pieces, the Mona Lisa. It is said that da
Vinci used the Golden Ratio to make her look
more aesthetically pleasing (Figure 5).7 With the
use of the Golden Ratio, features such as the size
of the nose and mouth, the gap between the eyes,
and the spacing between the nose and mouth are
carefully considered and placed. The placement
Figure 5: Mona Lisa by Leonardo da Vinci
Taken from http://monalisa.org/2012/09/12/leonardoand-mathematics-in-his-paintings/
of how the ratio was used by da Vinci can be seen in figure 5, but it has been creatively placed
by experts who have tried to link the Golden Ratio with the Mona Lisa. This is why many are
also skeptical about the usage of the
ratios in the Mona Lisa because the
Golden ratio is not clearly defined.
The Golden Ratio is most
apparent in his piece, The Last Supper,
which used the ratio extensively in all
the elements within the painting from
Figure 6: The Last Supper by Leonardo da Vinci
Taken from http://www.jaydax.co.uk/lastsupper/lastsupp.jpg
canvas size to the dimension of the room
6
Meisner, Gary. "Divine Proportion/Golden Ratio in the Art of da Vinci." The Golden Ratio Phi 1618. July 7, 2014.
Accessed September 29, 2016
7
Hom, Elaine. "What Is The Golden Ratio." Live Science. June 24, 2013. Accessed September 30, 2016.
Essay 2 | Page 5 to the table size to the window placement on the wall (Figure 6). These ratios are clearly depicted
in figure 6 using various colors to help differentiate between the other subjects and objects in the
piece. Each boxed area shows a larger segment, which will be referred to as ‘a’ (larger portion of
the division) with a smaller segment referred to as ‘b’ (smaller portion of the division). Given a
and b, the ratios within the piece can be calculated using the formula:
#$%
#
#
= , which shows
%
how da Vinci carefully used mathematical calculations in The Last Supper. By using the Golden
ratio in his piece, da Vinci was able to create a work of art that is extraordinarily pleasing to the
eye.
Many artists after da Vinci used the Golden Ratio in their work to enhance its physical
proportional appeal, which include artists like Raphael, Michelangelo, and Rembrandt. Overall,
the rediscovery of the Golden Ratio during the Renaissance era allowed artists to use it as a
mathematic tool to enhance their work using a variety of media. Leonardo da Vinci might have
not contributed a great deal to mathematics, but some of his artwork do show how powerful
mathematics can be when used as a tool in art.
An individual who was able to make a
substantial contribution in both art and
mathematics was Filippo Brunelleschi.
Brunelleschi was an Italian engineer and
architect who rediscovered linear perspective and
applied mathematics to calculate perspective in
art. Before the Renaissance era, the ancient
Figure 7: Art Prior to Perspective,
Kaufmann Haggadah 14th Century
Greeks have somewhat touched on the idea of
perspective but were not able to demonstrate it in
Taken from https://math.dartmouth.edu
a flat image. An example of what perspective was depicted in the 14th century would be the piece
Essay 2 | Page 6 from the Kaufmann Haggadah, which is rather one dimensional and perspective lines are applied
incorrectly (Figure 7). Building onto the ideology of perspective laid out by ancient Greeks,
Brunelleschi was able to redefine linear perspective and apply it to make flat images look three
dimensional and more realistic.8 This laid out the groundwork for future work with mathematical
perspective theory.
To explore perspective, Brunelleschi
used a mirror with a small peephole to help
him determine and calculate how linear
perspective can be replicated through one
vanishing point (Figure 8). He painted his
subject and as he painted he checked
proportions using the method mentioned.
Figure 8: Brunelleschi’s Discovery of Perspective
Using Mirrors to Replicate Vanishing Points
Taken from https://maitaly.wordpress.com/2011/04/28/brunelleschiand-the-re-discovery-of-linear-perspective/
After painting his subject, he took the mirror
and reflected it back onto the canvas to check
how well he was able to replicate it with exact proportions and depth using linear perspective. He
was able to paint a realistic image of the Florentine streets and buildings using this method, but
his original work was lost.9 Essentially, Brunelleschi was able to take a two dimensional image
and turn it into a three dimensional image on a flat surface using very precise mathematical
calculations to define vanishing points.
Brunelleschi found that to create an image with the illusion of depth and realism, these
are the components needed: horizontal line, vanishing point, and parallel lines. The horizontal
line is typically defined as the division between the sky and the ground, which is called the
8
"Filippo Brunelleschi." Bio. Accessed September 29, 2016.
Hyman, Isabelle. "Filippo Brunelleschi." Encyclopedia Britannica Online. Accessed September 29, 2016.
https://www.britannica.com/biography/Filippo-Brunelleschi.
9
Essay 2 | Page 7 horizon, but it can also be the artist’s vantage point. The vanishing point can be placed on any
part of the horizontal line. Parallel lines are used to connect to the vanishing point with the base
shape of the original object. Then the lines can be used as a guide to create the other components
of the object to make it look like it
has depth. In figure 9, these
components are shown with an
example using a vertical rectangle.
With these three components,
artists are able to give the illusion
Figure 9: Single Point Perspective Example
(Personal Image)
that certain objects are father than
others with accurate proportions.
Brunelleschi was able to leave calculations and instructions on how to create the illusion of depth
to other artists, which probably led to our development of three dimensional film and digital
designs. Although Brunelleschi was able to use mathematics to define linear perspective, he is
more known for his extravagant architectural designs in Italian cities.
The Renaissance era was definitely a time of rebirth in the arts, mathematics, and
sciences. Evidence shows that individuals like Leonardo da Vinci and Filippo Brunelleschi were
able to rediscover forgotten ideas and transform them into lasting creations. From their work and
the work that came before and after, it shows how powerful mathematics can be as a tool.
Mathematics is the ultimate tool in art because it helps generate more aesthetically pleasing
designs using the Golden Ratio, calculate linear perspective, and much more.
Essay 2 | Page 8 REFERENCES
Blumberg, Naomi. "Linear Perspective." Encyclopedia Britannica Online. March 03, 2016.
Accessed September 29, 2016.
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September 29, 2016.
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