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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. 2 Shuttleworth, Martyn. "Renaissance Science - The Path to Enlightenment." Explorable. Accessed September 29, 2016.

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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,

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

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.

 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

 Figure 2: Golden Ratio Triangle

Used to Construct the Great Pyramids Taken from http://www.goldennumber.net/phi-pi-great-pyramid-egypt/golden-triangle-pythagorus/

 

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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

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 3: Golden Ratio in the Pentagram

Taken From: /www.cut-the-knot.org/do_you_know/GoldenRatioInRegularPentagon  

Figure 4: Vitruvian Man by Leonardo Da Vinci

Taken from http://monalisa.org/2012/09/12/leonardo-and-mathematics-in-his-paintings/

 

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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

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

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.

Figure 5: Mona Lisa by Leonardo da Vinci Taken from http://monalisa.org/2012/09/12/leonardo-

and-mathematics-in-his-paintings/  

Figure 6: The Last Supper by Leonardo da Vinci

Taken from http://www.jaydax.co.uk/lastsupper/lastsupp.jpg  

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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

Greeks have somewhat touched on the idea of

perspective but were not able to demonstrate it in

a flat image. An example of what perspective was depicted in the 14th century would be the piece

Figure 7: Art Prior to Perspective, Kaufmann Haggadah 14th Century

Taken from https://math.dartmouth.edu  

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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.

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. 9 Hyman, Isabelle. "Filippo Brunelleschi." Encyclopedia Britannica Online. Accessed September 29, 2016. https://www.britannica.com/biography/Filippo-Brunelleschi.  

Figure 8: Brunelleschi’s Discovery of Perspective

Using Mirrors to Replicate Vanishing Points Taken from https://maitaly.wordpress.com/2011/04/28/brunelleschi-

and-the-re-discovery-of-linear-perspective/  

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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

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.

Figure 9: Single Point Perspective Example (Personal Image)

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REFERENCES

Blumberg, Naomi. "Linear Perspective." Encyclopedia Britannica Online. March 03, 2016.

Accessed September 29, 2016.

"Brunelleschi and the Re-Discovery of Linear Perspective." MaItaly. April 28, 2011. Accessed

September 29, 2016.

Calter, Paul. "Brunelleschi's Peepshow & The Origins of Perspective." Geometry in Art &

Architecture. 1998. Accessed September 29, 2016.

"Filippo Brunelleschi." Bio. Accessed September 29, 2016.

Fussell, Matt. "One Point Perspective." One Point Linear. Accessed October 03, 2016.

Hom, Elaine. "What Is The Golden Ratio." Live Science. June 24, 2013. Accessed September

30, 2016.

Hyman, Isabelle. "Filippo Brunelleschi." Encyclopedia Britannica Online. Accessed September

29, 2016.

"Leonardo and Mathematics." The Mona Lisa Foundation. 2012. Accessed September 29, 2016.

"Leonardo DaVinci and the Golden Section - Math Central." Math Central. Accessed September

29, 2016.

Malkevitch, Joseph. "Mathematics and Art." American Mathematical Society. Accessed

September 29, 2016.

Meisner, Gary. "Divine Proportion/Golden Ratio in the Art of da Vinci." The Golden Ratio Phi

1618. July 7, 2014. Accessed September 29, 2016.

Shuttleworth, Martyn. "Renaissance Science - The Path to Enlightenment." Explorable. Accessed

Tyler, Christopher, and Michael Kubovy. "Perspective: The Rise of Renaissance Perspective."

Science and Art of Perspective. Accessed September 29, 2016.

"What Was Life like before the Renaissance?" Birthplace of the Renaissance. 2013. Accessed

September 29, 2016.