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The Golden Ratio: A Ubiquitous Fact or a Wild Misconception
The Parthenon, pine cones, the Great Pyramid of Khufu, the stock market, the human
face, the Aston Martin, the Notre Dame, The Birth of Venus, the violin, the Taj Mahal, the CN
Tower, The Last Supper, La Mer, and the nautilus shell each share a singular characteristic: the
alleged incorporation of the golden ratio. Evidently, the extent of the golden ratio, the proportion
of 1.618 to 1, displays itself in various forms and in various fields, often with the mathematics to
defend its existence and occasionally with intent or reason behind the crafting of a mathematical
work. However, some deny the golden ratio of any presence. These critics refute its occurrence
in almost all cases, citing these examples as superfluous flukes and misconceptions.
The debate lies in the true extent of the golden ratio’s presence. Is the golden ratio a
ubiquitous fact of life, as the “Phi enthusiasts” assert? Or is the golden ratio a nonexistent
mathematical application driven by a craze of ill-informed fanatics?
In fact, the golden ratio
exists neither as a ubiquitous presence nor a wild misconception but as a limited presence,
mathematically and intentionally, which can be represented through analysis of the Parthenon,
the Great Pyramid of Khufu, Claude Debussy’s La Mer and Images, Le Corbusier’s Le Modulor,
and Salvador Dalí’s The Sacrament of the Last Supper.
In the hopes of delineating an amalgam of misconceptions and verified existences of the
golden ratio, this document focuses on these five major works of art, architecture, and music.
While some suspect that the golden ratio plays significant roles in nature, the human body, and a
variety of other fields, for sake of specificity, this study highlights the golden ratio in mediums of
human expression.
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Background Information
The basis of the golden ratio derives from the limits of the Fibonacci sequence, relating to
the number Phi, φ or 1.618. One can recreate the sequence by starting with 0 and 1, adding the
two previous numbers to find the next (Fn=Fn-1+Fn-2). The sequence begins with 0, 1, 1, 2, 3, 5,
8, 13, 21, 34, 55, 89, 144, and 233 and continues on toward infinity (Lock 37).
With some manipulation, the Fibonacci sequence relates closely to the number Phi. Phi,
the 21st letter of the Greek alphabet, stands for the number 1.6180339887…, continuing on
indefinitely. To estimate Phi, select a number from the Fibonacci sequence and divide by the
previous number (Fn/Fn-1≈1.618). The estimation of Phi increases in accuracy as the numbers
selected grow larger. For instance, 21 divided by 13 better estimates Phi as 1.615 than does 5
divided by 3, which equals 1.667. In this way, Phi is the limit of this manipulated sequence
based on Fibonacci numbers.
The golden ratio is the ratio of Phi to one, or 1.618:1. The proportion can also be found
through the quadratic equation based on Figure 1 (Markowsky 2).
Figure 1
1
1-X
X
…
…
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Many geometric applications employ the golden ratio and Phi, ranging from rectangles to
triangles to spirals. For the construction of a golden rectangle (Fig. 2), the ratio a:b must equal
the ratio (a+b):a. These ratios must be equivalent to the golden ratio, 1.618:1. Thus,
… (Falbo 3).
a
Figure 2
b
a
a+b
A similar construction results in the “golden triangle.”
The
triangle
contains some qualities of the golden ratio, further explored in the outstanding paper
“Misconceptions about the Golden Ratio” by George Markowsky, an experienced
mathematician.
For the creation of the phi spiral (Fig. 3), squares of dimensions related to the Fibonacci
sequence spiral outward. The two squares in the center contain dimensions 1x1. The subsequent
squares build off of the previous two squares, similar to the construction of the numerical
sequence. Drawing in quarter circles for each square results in the iconic phi spiral.
Figure 3
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The etymology of the golden ratio indicates the many variations on the term “golden
ratio.” One of the original mentions of the golden ratio comes from Euclid’s The Elements,
Book 6, Proposition 30. Euclid instructs to “divide the line AB by an extreme and mean
proportion” (Euclid bk. 6 prop. 30). Extreme and mean proportion was the original name for the
1.618:1 ratio. In more recent times, mathematicians have labeled this ratio as the “divine
proportion” and the “golden ratio.” Variations include the “golden section” or “golden mean.”
The Parthenon
The Parthenon, located in Athens, Greece, represents the epitome of golden ratio
misconceptions.
While poor photography and inaccurate and imprecise measurements
deceptively depict the Parthenon in seemingly golden ratio form, more accurate analyses of the
building indicates that, while aware of the golden ratio, the Greeks did not intend to create a
golden ratio structure. In the end, these arbitrary measurements do not equal the 1.618:1 ratio
needed to label the Parthenon as a golden ratio work.
Mathematical scrutiny of the Parthenon does not support this building as a golden ratio
work of any kind. For an analysis of the Parthenon’s closeness to the golden ratio, Marvin
Trachtenberg and Isabelle Hyman, veteran art historians, list the height measurement as 45 feet 1
inch and the width as 101 feet 3.75 inches. Stuart Rossiter, renowned philatelist and scholar of
geography, lists the height of the apex as 59 feet in his book, Greece. Thus, the ratio of the
width to the height is 2.247 and the ratio of the width to the apex height is 1.717, neither close to
1.618.
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The precision of measurement is deficient as well, as noted by mathematician George
Markowsky in his brilliant paper titled “Misconceptions about the Golden Ratio” (9). This
disparity in measurement results from a variety of false measurements.
Essentially, the
settlement of the structure over time accounts for some necessary inaccuracies, as the Greeks
completed the Parthenon in 432 BCE. The degeneration of the structure’s apex (Fig. 4) prompts
differences in measurement, as the structure today stands much shorter than that of over 2440
years prior. Additionally, many different measurement techniques and reference points result in
a variety of heights and widths.
Thus, Phi enthusiasts can select certain convenient
measurements to achieve the 1.618:1 ratio, influenced by a partial point of view. As a result,
lack of precision in measurement accounts for the creation of the golden ratio in a building
without true golden ratio dimensions.
Figure 4
Despite a lack of accuracy or precision in golden ratio proportions of the Parthenon,
enthusiasts continually link the structure to the ratio. Often, misguided photography supports the
existence of the ratio in the Parthenon’s design, sparking a craze of erroneous information that
falsely overrules structure’s true proportions. Many online photographs of the Parthenon seem to
be taken from a ground position. These tilted photographs deceive viewers into believing the
occurrence of the golden ratio by superimposing graphics over the distorted images, often
including thick lines that may be adjusted to additionally produce a desired outcome. The
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thicker the lines, the more room there is for adjusting. Other online photographs are blurry and
may not be effectively analyzed. Frequently, the arbitrarily-selected, superimposed shapes cut
off stairs or sides of columns to force the golden ratio onto the building. These features appear
throughout websites such as Phi: The Golden Number (Meisner). Therefore, by simply altering
the nature of the photograph, the photographer may produce a falsely convincing image of the
golden ratio in the Parthenon.
Ultimately, had the Greeks intended to employ the golden ratio in the construction of the
Parthenon, that singular fact would overrule all of the uncertainties regarding measurements and
photography. However, a lack of explicit textual indication of the golden ratio’s utilization
indicates that while the Greeks may have understood the golden ratio, they did not intend to
create a golden ratio work. Euclid’s The Elements, a profound mathematical treatise written in
300 BCE, first displayed Greek knowledge of the golden ratio over one hundred years after the
completion of the Parthenon. However, Mario Livio, astrophysicist and author of The Golden
Ratio: The Story of Phi, the World's Most Astonishing Number, asserts that “considerable
knowledge existed among the Pythagoreans prior to [The Elements]” (74). Livio concludes that
while an analysis of Greek knowledge of the time remains uncertain, the Greeks probably did not
employ the golden ratio in the Parthenon in a major fashion. A lack of texts on the construction
or blueprints and with golden ratio calculations supports this conclusion.
Flaws of measurement, precision, mathematics, analysis, and photography, and a lack of
supporting texts indicate that the Greeks did not intentionally employ the golden ratio in the
Parthenon. Deceptive photography ultimately fuels the unsubstantiated notion that the golden
ratio exists in the Parthenon.
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The Great Pyramid of Khufu
The fallacy regarding the golden ratio in The Great Pyramid of Khufu, located in El Giza,
Egypt, bears striking resemblance to that of the Parthenon. Both contain some mistaken aspect
of golden ratio dimensions, both draw conclusions based on arbitrary reasoning, both came from
civilizations with insufficient depth of mathematical knowledge of the golden ratio, and both
ultimately do not employ the golden ratio. While The Great Pyramid of Khufu contains a more
valid aspect of golden ratio mathematics, the calculation is arbitrary, indicating that the
Egyptians had no intention of creating a golden ratio work.
Figure 5
s
h
b
The mathematically-substantiated existence of the golden ratio in The Great Pyramid of
Khufu distinguishes this structure from other unauthenticated, alleged golden ratio works.
However, these measurements seem coincidental considering other factors such as a lack of
interest in creating such a mathematical work and a flawed historical account from Herodotus.
The Great Pyramid of Khufu stands 481.4 ft tall (labeled ‘h’ in Fig. 5) with a 755.79 ft base
(Markowsky 6). Impressive precision from a variety of sources exists among the measurements,
especially when compared to the Parthenon. In calculating the golden ratio, half of the base,
labeled ‘b,’ is 377.895 ft. A Pythagorean Theorem calculation results in a slant height of 612.0
ft, labeled ‘s.’ Thus, the ratio of the slant height to half the base is 612.0/377.895 = 1.619,
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extremely close to 1.618. Indeed, the mathematics supports the existence of the golden ratio
with less than 0.001% error. These calculations immediately stand out, ostensibly suggesting
that the Egyptians intentionally utilized golden ratio dimensions with incredible accuracy.
Upon further examination, these calculations become less prominent. The ratio of slant
height to the half base seems particularly arbitrary, and no obtainable historical documents
reference the selection of this or other proportions. The ratio of base to height would be much
more observable and simpler to calculate. Beyond the irrationality for the selection of these
measurements, doubt arises in the error of the building’s settlement and the uncertainty of the
Egyptian motives. The structure, like the Parthenon, has settled tremendously over time. The
building stood much taller and in a much better state in 2560 BCE than it does today.
Additionally, a building of such mathematical significance would serve little purpose to the
Egyptians, who strove primarily for physical enormity and greatness in their pyramids rather
than a mathematical concept. In fact, further evidence, provided by Corinna Rossi, architect and
Egyptologist, suggests that Egyptians probably did not aspire to create a golden ratio work or
even understand notion.
Egyptian knowledge at the time of construction advises against the intentional existence
of the golden ratio in the Pyramid. Corinna Rossi, an expert in the mathematical knowledge of
the Ancient Egyptians, wrote in her book, Architecture and Mathematics in Ancient Egypt, that
“in general, all the … cases mentioned … do not seem to provide enough evidence to suggest
that the Egyptian artists showed a marked preference for simple Golden Section-based
geometrical figures. The same applies to architecture” (85). Rossi therefore concludes that the
Ancient Egyptians simply would not have preferred a building with golden ratio dimensions.
Furthermore, the Egyptians completed the Pyramid over 2000 years before Euclid wrote The
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Elements, indicating that the Egyptians never discovered or encountered a source of knowledge
to introduce them to the golden ratio. Rossi expands on this claim, asserting that “concepts like
φ or π did not belong to ancient Egyptian mathematics and therefore could not be used by the
ancient Egyptian architects” (87). Without a preference for complex mathematics or knowledge
of Phi, the Egyptians could not employ the golden ratio with purpose.
A flawed quote translation from Herodotus supports the notion of a mathematical
coincidence. George Markowsky enlightens the roots of the wrongly translated Herodotus
quotation, which related the area of the square base to the areas of the slanted triangles,
eventually falsely resulting in the golden ratio (7).
The true quotation reads: “The actual
pyramid took twenty years to build. Each of its sides, which form a square, is eight plethra long
and the pyramid is eight plethra high as well. It is made of polished blocks of stone, fitted
together perfectly; none of the blocks is less than thirty feet long” (Herodotus 145). The 1913
Webster's Revised Unabridged Dictionary defines a plethra as 100 Greek feet, or 101 English
feet. Thus, 8 plethra equals to about 808 feet. Given these estimations, the base would be 808
feet, half of the base would be 404, the height would be 808 feet, and the slant height would be
903.37 feet, found by utilizing the Pythagorean Theorem. The ratio of slant height to half base
would be 903.37/404 = 2.236, far from 1.618. These confusing estimations in no way reflect the
current measurements of the Pyramid, even when considering degeneration over time. Thus,
while modern mathematical interpretations suggest the existence of the golden ratio in the
construction of the Pyramid, accounts from the time conflict with the actual measurements of the
monument and refute any intentional utilization of the mathematical principle.
Illogical measurements, lack of Ancient Egyptian knowledge, and flawed accounts
substantiate a solely mathematical coincidence, concluding that the Ancient Egyptians did not
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intend on the application of the golden ratio in the construction of the Great Pyramid of Khufu.
While the Pyramid does exhibit one arbitrary mathematical characteristic of the golden ratio, in
the end the lack of logic behind its presence undermines any possibility for intentional
utilization.
The Works of Claude Debussy
Composer Claude Debussy’s musical pieces La Mer, Estampes, and Images similarly
contain mathematical references to the golden ratio; however, unlike the Parthenon, which
exhibited neither mathematical nor deliberate employment of the proportion, and the Great
Pyramid of Khufu, which exhibited only mathematical application, the works of Claude Debussy
exhibit both mathematical operation and distinct purpose. An investigation of movement lengths
in the works of Claude Debussy and intriguing details regarding the alteration of a musical
manuscript demonstrate that knew of the golden section and assert deliberately implemented the
ratio in these works.
An analysis of the works of Debussy by Anthony J. Bushard, scholar of music history,
reveals a variety of golden sections within La Mer, Estampes, and Images, often based upon
Fibonacci sequence expressions. There are “55 bars in the introduction to La Mer, 21 bars of the
introduction in Rondes de Printemps, and 55 bars until the beginning of the climax in Reflets
dans l’eau’” (3). These are “unusual numbers musically speaking,” asserts Bushard, but the
numbers are important due to their relationship to the Fibonacci sequence, which demonstrates a
close connection to the golden ratio. The uniqueness of these numbers increases suspicion of
their operation. Additionally, the “dynamic peak” of Reflets dans l’eau’ occurs at bar 58 of the
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94 bar song, reducing to 47/29 = 1.621. The numbers 29 and 47 are in the Lucas Sequence, a
sequence similar to the Fibonacci sequence in its construction. The 58th bar also is the closest
whole bar to the golden section division.
Such proportions evidently occur frequently
throughout many of Debussy’s works.
The effect of these ratios seems intentional and deliberate.
Bushard asserts that
symmetry works in conjunction with the golden section, a characteristic supported in golden
ratio art. He writes: “These new points of symmetry introduce an idea that is prevalent in the
analyzed works of Debussy – points that mark the GS are usually analogous to tension, whereas
points of exact symmetry are associated with balance” (8). Bushard also notes a wave-like
pattern in Reflets dans l’eau’. Possibly, Debussy intended to include such areas of tension and
balance with the golden ratio in mind to create a wave-like feeling of rising and falling with
crests of tension and troughs of balance. Debussy would have strove for such an effect, made
evident by the selection of the title, which means “Reflections in the Water.” Debussy’s other
orchestral composition, La Mer, translates to “the sea,” indicating that Debussy aspired to
reconstruct such a wave-like effect, fashioned by the careful selection of golden ratios, in his
water-related compositions. Clearly, the presence of the Fibonacci numbers appears frequently
and effectively throughout various pieces, each time creating a purposeful effect.
This deliberate desire to create golden ratio manifests in bar 123 of Jardins sous la pluie,
a piece from Estampes. Claude Debussy altered a blank measure of the original manuscript,
adding an unnecessary measure repetition to the print version of the composition and creating a
golden section in the structure of the composition. This alteration suggests Debussy’s conscious
knowledge of the mathematical notion and its application. Roy Howat notes that Debussy added
a repetition of the previous bar to the printed version of Jardins sous la pluie, completing the
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golden ratio of that piece where previously it had not existed (7). The repetition of the previous
bar was “not essential in the music’s grammatical case,” increasing suspicion as to the motives of
the change. The piece also contains the sense of symmetry noted by Bushard, broaching the
topic of harmony between the golden ratio and symmetry. In a letter to Jacque Durand, his
publisher, accompanying Jardins sous la Pluie, Debussy discusses the missing bar. He writes:
“However, it’s necessary, as regards number; the divine number” (Livio 191). Debussy clearly
references the golden ratio through the pseudonym “divine number.” Accordingly, Debussy
knew of and employed the golden ratio in Jardins sous la Pluie, among other possible works.
Additionally, Claude Debussy likely had a personal interaction with the golden ratio,
supplying Debussy with the knowledge of the proportion that the Greeks and Egyptians lacked in
their respective constructions. Debussy constantly maintained contact with artists and painters,
among which the passion for the golden ratio was “endemic” (Howat 7). He may have attended
an exhibition held in Paris in 1912 called the Section d’or, meaning “golden section” in English,
or developed ideas from French Symbolist artists.
Furthermore, Debussy had an unusual
fascination with esotericism in his music. While he did not note any particular bar as the golden
section throughout his works, he did mark climactic and important points. It is natural that
because of Debussy’s mission for esotericism, he wrote nothing on the golden ratio (Goldman
131). Though he never explicitly proclaimed his application of the golden ratio, Debussy likely
interacted with the ratio and maintained mathematical knowledge below an esoteric shell.
Due to music’s regimented rhythms, usually of four, and strict structures, the golden ratio
may seem difficult to apply to compositions. While the implementation of the golden ratio in
music may seem crude due to the imprecise estimations of the Fibonacci numbers, music poses
challenges in creating a mostly tangible application in an intangible medium.
Debussy’s
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utilization of the Fibonacci sequence when dealing with whole measures creatively includes the
golden ratio in music.
Debussy’s search for esotericism, knowledge of the mathematical
property, Fibonacci-related movement lengths, and interesting addition of a bar which completed
the golden ratio indicate that, with minimal uncertainty, Claude Debussy intentionally employed
the golden ratio.
Le Corbusier’s Le Modulor
While artist and architect Le Corbusier expanded Le Modulor, an artistic movement, into
a sort of “modulor empire” including architecture, books, and a system of measurement, this case
study focuses primarily on the prevalence of the golden ratio in one lithograph, Le Modulor
1955, as a symbol of Le Corbusier’s influence. Through scrupulous measurement, allusions to
Phi, personal notes, and written proof of implementation, this lithograph represents the power of
the golden ratio under Le Corbusier mathematic and artistic expertise.
Corbusier intentionally applied the golden ratio to the
work Le Modulor 1955.
Many online images of a basic version of the
Le Modulor 1955 exist, some from Le Corbusier’s
own notes, with units on the sides of the image. No
matter the image of the Modulor Man, the ratios of
these numbers are equivalent. Ratios between these
numbers demonstrate the golden ratio aspect of the
work by the creation of precise and deliberate
Figure 6
Undoubtedly, Le
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numerical evidence. One image from Icon Eye (Fig. 6) has the number 1,130 on the portion
between the man’s navel and the ground, the number 698 on the portion between his head and
his navel, and the number 432 on the portion between his raised hand and the top of his head
(Wiles 1). Thus, the ratio of the man’s total height to the height of his navel nearly equals Phi:
(1130 + 698) / 1130 = 1.617699. The ratio of the man’s navel height to the height of his upper
body also nearly equals Phi: 1130 / 698 = 1.618911. Additionally, the ratio of his navel height
to the height between his hand and his navel is perfectly symmetric: 1130 = (698 + 432). Other
examples of squares in perfect symmetry exist throughout the work. This 1:1 ratio emphasizes
the importance of symmetry in relation to the golden ratio, seen also in the works of Claude
Debussy. Clearly, the various proportions and measurements of Le Modulor 1955 enrich its
mathematical complexity, particularly in regards to the golden ratio.
An art analysis of the lithograph further supports the calculated application of the golden
ratio through allusions to the mathematical property. In the top-right corner of Le Modulor 1955,
Le Corbusier includes a nautilus shell. The nautilus shell has long represented Phi, often
depicted with a Phi spiral superimposed upon of the image. Ironically, mathematicians have
since found that a logarithmic function better fits the nautilus shell (Devlin). Directly to the right
of the Modulor Man, Le Corbusier includes a spiral. Perhaps an allusion to the “Phi spiral,” the
spiral could also refer to a strand of DNA. A DNA molecule measures 34 angstroms long by 21
angstroms wide, two numbers from the Fibonacci sequence and numbers which create the ratio
34 / 21 = 1.619, approximately Phi (Gupta & Saxena 5). Such allusions further confirm the
implementation of the golden ratio in Le Modulor 1955.
Le Corbusier explains his utilization of the golden ratio in The Modulor: A Harmonious
Measure to the Human Scale, Universally Applicable to Architecture and Mechanics, his book
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on the Modulor system. He declares the application of the golden ratio quite plainly: “The
Modulor is a measuring tool based on the human body and mathematics. The man-with-armupraised provides, at the determining points of his occupation of space-foot, solar plexus, head,
tips of fingers of the upraised arm – three intervals which gave rise to a series of golden sections,
called the Fibonacci series” (Le Corbusier 55). In The Modulor, Le Corbusier defines his
“golden rule” based upon three mathematical principles: (a) the initial value (the unit), (b) the
double unit, and (c) its Golden Section (Le Corbusier 84). Throughout the book, Le Corbusier
explains the proper utilization of the golden ratio among other techniques in his Modulor system.
Thus, Le Corbusier not only demonstrates his understanding of the golden ratio but also reveals
its various applications in his work.
Indubitably, Le Corbusier consciously and intentionally used the golden ratio in Le
Modulor 1955 and throughout the Modulor system, made evident by his personal notes and
measurements and comprehensive analyses of his work. The Modulor system and Le Modulor
1955 in particular epitomize both the mathematical and verified utilization of the golden ratio to
a level of certainty not found in the Parthenon, the Great Pyramid of Khufu, and the works of
Claude Debussy.
Salvador Dalí’s The Sacrament of the Last Supper
When examining artist Salvador Dalí’s painting The Sacrament of the Last Supper, the
golden ratio canvas size, the apparent central dodecahedron, and several allusions to symmetry
and divinity indicate that Salvador Dalí consciously employed the golden ratio. Similar to Le
Corbusier’s Le Modulor 1955, The Sacrament of the Last Supper, supported by Dalí’s
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knowledge of the proportion, exemplifies proper mathematical application of the golden ratio
supplemented by written verification of its utilization.
The most striking golden ratio characteristic of The Sacrament is the painting’s immense
Phi dimensions. The National Gallery of Art, where The Sacrament is displayed, lists the overall
size as 166.7 by 267 cm. Thus, the division of length by height produces the ratio 267:166.7, or
about 1.602:1, significantly close to the golden ratio with only a 1.011% error. At first glance,
this aspect seems to be the only manifestation of the golden ratio in The Sacrament; however, an
analysis of the content of the painting reveals other materializations of the proportion.
Figure 7
The central dodecahedron behind Jesus Christ in The
Sacrament contains significant meaning in relation to the
C
1
golden ratio and indicates that Dalí deliberately employed the
A
108
36
36
y
golden ratio. A dodecahedron is a twelve sided shape made
Figure 8
up of pentagon faces, though only four faces of the shape are
1
B
C
visible in The Sacrament. The pentagon (Fig. 8) contains a
special mathematical quality in the geometry of the shape.
A
X
The line segment AB contains properties similar to a golden
segment. For the pentagon to exhibit the golden ratio AB:XB
must equal XB:AX, which must equal Phi, 1.618. If each side
equals 1, and each angle of a regular pentagon equals 108 degrees, consider ΔACB (Fig. 7).
Implementing the sine formula,
, and y = AB = 1.618. In the original pentagon,
ΔCBX is isosceles, so XB = 1. Thus, AX = AB – XB = 1.618 – 1 = 0.618. Returning to the
B
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original proposition, AB:XB = XB:AX. Inputting the found numbers, 1.618:1 = 1:0.618, and
both ratios equal 1.618:1, the golden ratio.
Subtle references to symmetry, divinity, and the golden ratio suggest that Dalí invoked
the golden ratio to create a larger theme. Dalí creates an incredible sense of symmetry, which
admittedly works in harmony with the golden ratio in the works of Debussy and Le Corbusier.
Each of the twelve disciples bows his head in the same manner as the man in the same position
opposite Jesus. If Jesus represents the y-axis of the painting, the disciples’ hair styles, shadows,
and poses reflect across the axis with surprising accuracy. Only slight color differences obstruct
full symmetry. The bread in front of Jesus Christ also represents a strong sense of symmetry.
The bread splits into perfect halves, spread apart for emphasis. Divinity plays another primary
role in this piece. The divine image of a spirit above Jesus Christ, the praying disciples, and
Jesus Christ himself clearly suggest the Christian faith. The mathematical incorporation into this
particular piece may suggest the name “divine proportion.” Similar to the idea of divinity in the
work, the gold light and the golden dodecahedron suggest the name “golden ratio.” Similar to
the wave-like movements Claude Debussy’s works, in The Sacrament, Salvador Dalí employs
the golden ratio to create effects of balance, divinity, and radiance.
In Salvador Dalí’s 50 Secrets of Magic Craftsmanship, a book on the profession of
artistry, Dalí describes his painting secrets, including his implementation of the golden section,
which strongly suggests that he applied golden ratio techniques in The Sacrament. A labeled
image of a pentagon, identical to that of The Sacrament, next to a paragraph explaining how to
derive the golden ratio confirms Dalí’s applied knowledge of the golden ratio (179).
Additionally, Dalí advises to “make use of the properties unique and of a natural magic, derived
from the wise use of the golden section, and called the divina proporzione by Luca Pacioli,”
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suggesting a more than basic understanding of the golden ratio (177). While Salvador Dalí never
explicitly ties the golden ratio to any of his works, his strong understanding of the golden ratio
and its applications suggests that he exercised the proportion throughout The Sacrament.
Similar to the works of Claude Debussy and Le Corbusier, Salvador Dalí’s The
Sacrament of the Last Supper contains sufficient mathematical support of the proportion while
outside literary evidence substantiates Dalí’s intention to apply the mathematics. Canvas size,
the existence of the pentagon, references to symmetry and divinity, and knowledge of the golden
ratio strongly support the conjecture that Salvador Dalí employed the golden ratio in The
Sacrament of the Last Supper.
Conclusion
The golden ratio exists in the realm of human expression as an esoteric and widely
misunderstood principle. Its reality may not properly be suppressed into simply a misconception
or simply a ubiquitous fact. The true extent of the golden ratio lies somewhere in between
inexistent and ever-present, exemplified by the juxtaposition of the above five works in
architecture, art, and music, some with intentional utilizations of the golden ratio, and some
without. However, these confinements of the candid golden ratio are in no way incorrect. In
fact, these views assist in the ongoing challenge to fit the golden ratio into a global context.
From ubiquity, mathematics supports the golden ratio in a variety of mediums throughout time
and across the world. From misconception, analysis of the creator’s intentions confirms or
denies initial suspicions. In perfect harmony, a balance between ubiquity and misconception
defines the true extent of the golden ratio.
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Let the golden ratio act as a stepping stone into a world of interdisciplinary study,
pursuing a more profound understanding of mathematics’ role in everyday actions. Explore
mathematics within another field. Appreciate mathematics for its largely unappreciated role in
society.
Likewise, see the other fields included in mathematics.
Observe the rhythmic
undulations of a sign wave or the artistic curvature of a parabolic arch. Viewing life in such a
multidimensional manner builds a sense of gratefulness not only for mathematics but also for the
functions of life that assist the world to work in harmony or even the little idiosyncrasies that
make it special. Consider this quote from Manil Suri, a mathematics professor and awardwinning writer:
“Despite what most people suppose, many profound mathematical ideas don’t require
advanced skills to appreciate. One can develop a fairly good understanding of the power
and elegance of calculus, say, without actually being able to use it to solve scientific or
engineering problems. Think of it this way: you can appreciate art without acquiring the
ability to paint, or enjoy a symphony without being able to read music. Math also
deserves to be enjoyed for its own sake.”
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Works Cited
Bushard, Anthony. “Debussy, Bartok, and the Golden Section.” College of St. Benedict and St.
John’s University, 1996.
Dalí, Salvador. 50 Secrets of Magic Craftsmanship. Trans. Haakon Chevalier. Courier Dover
Publications, 1992.
Devlin, Keith. “Math Encounters - Fibonacci & the Golden Ratio Exposed.” 2011. Educational.
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