*This is an excerpt from Master Fibonacci: The Man Who Changed Math. All citations are catalogued on the Citations page.*

(Previous Section: *Chart of Terms*)

Fibonacci believed that calculation was an art form; to him, it was a “marvelous” thing of beauty. He considered the art of calculation with Hindu-Arabic numerals to be appealing because their use facilitates the creation of harmonious, orderly, proportionate dimensions. To a businessman like Fibonacci, order was beautiful. His proclivities were not uncommon either in his day or in ours. Modern neuroscientific research supports the ancient assumption that humans favor the aesthetic appeal of order and symmetry. Evidence suggests that “humans can detect symmetry within about 0.05 of a second. This stimulus duration is too brief for eye movements to be completed.” Architect Don Ruggles, in his book titled *Beauty, Neuroscience & Architecture, Timeless Patterns and Their Impact on Our Well Being*, concludes, “this implies that human symmetry processing is a global, hard-wired activity of the brain” (Miller). The desire for harmony – one of the most ancient and primal aesthetic cravings – still exists; Fibonacci’s sequence helps people objectify the subjective components of beauty (“As Easy”).

Objective beauty can be more complex than bilateral symmetry or mirroring; special number sequencing and ratios are evident in diverse applications, such as literary texts (Euclid’s *Elements* and Shakespearean sonnets) and architecture (the Parthenon and the Taj Mahal), botany (red rose) and sculpture (Polycleitus’ *Doryphoros*). The classic opinion is that beauty exists when integral parts are arranged into a coherent whole exhibiting proportion, harmony, symmetry, unity, and order. Aristotle believed the mathematical sciences uniquely demonstrate the chief forms of beauty, which are order, proportionality, symmetry, and definitude (size limitation) (*Metaphysics* vol. 2, 1705 [1078a36]) (Stakhov 40). More precisely, he proposes that “a living creature, and every whole made up of parts, must … present a certain order in its arrangement of parts” to be considered beautiful (*Poetics*, vol. 2, 2322 [1450b34]) (Sartwell).

Aristotle’s mentor, the Greek philosopher, Plato (427-347 BC), proposed a tripartite theory of soul harmony (*Republic*, c. 380 BCE), which “recognized that the highest beauty of perfect figures and proportions was based upon the principle of the division in extreme and mean ratio” (the Golden Section). This ancient Harmony theory greatly influenced the development of science and art in European culture (Stakhov 41).

For example, Roman architect and military engineer Marcus Vitruvius Pollio, better known simply as Vitruvius, wrote a treatise on the history of ancient architecture and engineering which also emphasized the importance of structural harmony (*De Architectura, On Architecture*, c. 20 BCE). Because the book is the only such work to survive intact from antiquity, it is an invaluable resource on Greek and Roman architecture, but also on a wide range of other topics such as “science, mathematics, geometry, astronomy, astrology, medicine, meteorology, philosophy, and the importance of the effects of architecture, both aesthetic and practical, on the everyday life of citizens” (Cartwright). Importantly, Vitruvius characterized architecture as embodying beauty in its complexity constrained by underlying unity. Architecture, he said, consists of order (Greek: *taxis*), arrangement (Greek: *diathesis*), proportion, symmetry, décor, and distribution (Greek: *oeconomia*, economy) (Sartwell). Astronomer Johannes Kepler (1571-1630) expressed a similar opinion, asserting that “the chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics” (Stakhov 42). As mathematical instruments of investigation, the Fibonacci sequence and the Golden Ratio have been used often to measure the order and harmony of some classical oggetti d’arte e musica (objects of art and music). Since antiquity, objects having measurable harmonious and symmetrical proportions relative to the Golden Ratio have been considered especially appealing, graceful, and beautiful.

According to Pythagoras, the most beautiful and pleasing proportion is created when a line is divided so that the ratio between the larger and smaller of the two parts is identical to that between the original line and the larger of its subdivisions, alternately called the Golden Mean or the Golden Ratio (“As Easy”) or the Golden Section.

The properties of the Golden Section can be appropriated in temporal as well as spatial patterns of mathematical series and geometrical patterns (Akhtaruzzaman and Shafie). A wide survey of the relationship between Fibonacci numbers and the spheres of art and music reveals that some of the world’s most outstanding artistic works incorporate design based upon the Fibonacci sequence and/or the Golden Ratio. Besides the beautiful objects mentioned previously, these include Khufu’s Great Pyramid of Cheops at Giza, the sculptural Bust of Nefertiti crafted by Tuthmose, the majority of Greek sculptural monuments, the magnificent *Mona Lisa* by Leonardo da Vinci, *The School of Athens* and other works by Raphael, paintings by Shishkin and Konstantin Vasiliev, Chopin’s etudes, the musical works of composers Beethoven, Tchaikovsky, Debussy and Béla Bartók, and the Моdulor of Corbusier (Stakhov 59, Sinha).

As previously explained, the numbers generated by Leonardo of Pisa’s “rabbit problem” in Chapter 12 of *Liber Abaci *comprise a sequence that is astonishingly connected to the Golden Ratio. Ratios of successive numbers in the Fibonacci sequence (wherein each subsequent number is the sum of the previous two) become rational approximations of the Golden Mean φ with ever increasing accuracy. The greater the magnitude of the numbers, the more a graph of their development produces the Golden Ratio (Posamentier and Lehmann; “As Easy”). Therefore, Fibonacci numbers are used as dimensions in construction or design to “circumvent the difficulty of using the irrational number (φ)” (Posamentier and Lehmann 232). The “modern cult of Fibonacci’s numbers dates from 1877” when the French mathematician Edouard Lucas saw their significance and assigned them the name “Fibonacci sequence” (“As Easy”). Modern artists and architects are fascinated by the Fibonacci sequence, but “this is as nothing” compared to the “obsession with the Golden Ratio” in past centuries. The Golden Ratio was applied, most famously, by the architects of the Parthenon, but also by every Renaissance engineer who tried to “rediscover the lost harmony of Pythagoras” in every classical edifice emulating Phidias’ magnum opus (“As Easy”).

For example, evidence suggests that builders of Gothic castles used bricks formed in a special kind of rectangular parallelepiped shape which was based upon the Golden Rectangle; these were called Golden Bricks. There is speculation that the surprising strength and durability of gothic style architectural monuments is attributable to the use of the Golden Bricks (Stakhov 22). Renaissance architects, artists and designers frequently employed Golden Section proportions in eminent works of art, sculptures, paintings and architectures (Akhtaruzzaman and Shafie). According to many historians, they were nowhere near the first. In fact, many architects through the ages have either intuitively or deliberately used the Golden Section “in their sketches and construction plans, either for the entire work or for the apportionment of parts” (Posamentier and Lehmann 231). Pythagoras, the Greek father of mathematics, is also considered to be the “father of aesthetics.” Aristotle’s *Metaphysics* provide proof of his mentor’s beliefs concerning “a pure, clean cosmos concealed by the chaos of appearances.” Pythagoras believed numbers “revealed this hidden order.” It was he (or one of his followers) who first proved that “the natural notes of a plucked string only and always occur at regular intervals – when the string is subdivided at ratios of 2:1, 3:1, and so on.” The same harmonies existed at “the grandest cosmic levels,” Pythagoreans believed; “the music of the spheres” is a metaphysical principle derived from such Pythagorean theories (“As Easy”).

Pythagoras employed arithmetic, geometric and harmonic proportions, and the law of the Golden Section. He gave exceptional consideration to the Golden Section by choosing the pentagram as the distinctive symbol of the “Pythagorean Union.” Plato analyzed the five regular polyhedrons (the so-called Platonic solids) and emphasized their ideal beauty, further developing Pythagorean theories on harmony (Stakhov 40).

Not long after Pythagoras, Greek mathematician Phidias (Gr. Φειδίας) (490-430 BC) appears to have applied Phi while designing the Parthenon sculptures. Two millennia before Phidias and across the Mediterranean Sea, the Egyptian engineers of the pyramids were ingenious architects who used both Pi (π) and Phi (φ) in the structural design of the Great Pyramids (Akhtaruzzaman and Shafie). The pyramids served not only as vaults of a Pharaoh’s mortal remains, they were also a tribute to his majesty and power, and a monument to the riches of the country, its history and its culture. The pyramids clearly demonstrate deep “scientific knowledge” embodied in their forms, sizes, and orientation of terrain. “Each part of a pyramid and each element of its form were selected carefully to demonstrate the high level of knowledge of the pyramid creators.” (Stakhov 34).

The pyramids were constructed to endure for millennia, “for all time.” As the Arabian proverb says: “All in the World are afraid of a Time. However, a Time is afraid of the Pyramids” (Stakhov 34). The mathematical principle that the square of the hypotenuse of a right triangle (the side opposite the right angle) is equal to the sum of the squares of the other two sides was well-known to the Egyptians long before Pythagoras proved it (Pythagorean Theorem). They selected the golden right triangle as the geometric calibration instrument for the construction of Cheops’ Pyramid. The initial height of Cheops’ Pyramid is calculated to have been equal to H = (L/2) x τ = 148.28 m. The ratio of the external area of the pyramid to its base is then equal to the Golden Mean (Stakhov 36).

Three to five thousand years ago, around the same time the pyramids were built, Arkaim and Stonehenge were built; in the Chernihiv region, near the city of Ichnia, Ukraine, yet another manmade tool for solar observation was built using the Golden Ratio. Bezvodovka is an ancient Bronze Age architectural land monument spanning nearly twenty square kilometers. The ancient mounds of earth were thought to be burial mounds of nomadic tribes until recently, when aerial photography records and computer applications made it possible to determine their true purpose.

Oleksandr Klykavka, an agrochemist and soil scientist from the National Agricultural University in Kyiv, Ukraine, believes Bezvodovka was an ancient solar observatory like the other more famous prehistoric monuments. It is a scientific “instrument of incredible scale, the components of which are land, sky and cosmic objects,” he says. At the group of mounds in the Bezvodovka plateau, the regularities of the movements of the sun and other celestial bodies are discernible on the horizon. Klykavka explains that Arkaim is located at Latitude 52°39′ North, Stonehenge is located at Latitude 51°11′ North, and Bezvodovka is located at Latitude 50°31′ North. At a distance of nearly two thousand kilometers between them, “the three observatories are located within a single belt where the real shape of the Earth (non-ideal sphere) intercrosses with the imaginable correct shape.” All three observatories have a Northern-eastern view of the sunrise on June 22 – the longest day of the year (Klykavka).

Moreover, Klykavka says, “The “distance between the center and the western distant site is specifically 830m rather than 700 or 1000m. This is significant because, if an imaginary giant Fibonacci ‘Golden Spiral’ could be transposed over the observatory so that “its beginning comes from the center, the spiral would proceed through a few close sites and then some distant ones.” This, he says, explains why there is a distance of 2960m to some distant sites.” It is a plausible explanation for why the eight original mounds (five extant) were built so distant from each other (Klykavka).

Architectural design using the Golden Ratio or Fibonacci numbers was prevalent in Renaissance architecture. One such example is the Santa Maria del Fiore Cathedral in Florence, whose dome was constructed in 1434 by Filippo Brunelleschi (1337-1446). The rough sketch of the dome by Giovanni di Gherardo da Prato (1426) exhibits the Fibonacci numbers, 55, 89, and 144, as well as 17 (half the Fibonacci number 34) and 72 (half of 144) in the projection of curvature of the vault segments (”cells” or “sails”) (Posamentier and Lehmann 239).

“The Apollo Project of the Golden Renaissance,” by Nora Hamerman and Claudio Rossi, notes: “The pointed-fifth curve happens to be an arc of a circle whose radius is in the ratio of 8:5 to the radius of the circumscribing circle of the internal octagon base – a ratio in the Fibonacci series that closely approximates the famous Golden Section, the self-similar growth ratio . . . Similarly, the pointed-fourth cur- vature yields a ratio of 3/2 of the radius of the external base octagon to the radius of the vault curvature, another proportion in the Fibonacci growth series.” (30)

Hidden Fibonacci numbers were recently discovered in the façade of the Church of San Nicola in southern Italy, which was already centuries old by the time the Augustinians enlarged it in the late thirteenth century, on the cusp of the Renaissance. Another eight hundred years passed before restoration efforts in 2015 made it possible to discover a series of circular and rectangular inlays in one of the church’s lunettes in the portal. Professor and petrology expert at the University of Pisa, Pietro Armienti, recognized that the arrangement of the geometric figures in the marble intarsia contained a coded message while closely observing the process of marble cleaning. Professor Armienti published his research report in the “Journal of Cultural Heritage” explaining that the formerly- indecipherable artifact embodies an explicit reference to the findings of Leonardo Fibonacci (“Fibonacci Numbers”).

The first nine numbers of the Fibonacci sequence, 1, 2, 3, 5, 8, 13, 21, 34 and 55, denote the radii of the various circles in the design. To Professor Armienti, the inlaid tiles “can be used as an abacus to draw sequences of regular polygons inscribed in a circle of given radius” and was “made to calculate with good approximation the sides of the regular polygons inscribed” in the largest circle (“Fibonacci Numbers”).

The arabesque is inscribed within a circle, which is inscribed within a square, which is inserted in a rectangle whose ratio is the Golden Ratio. Armienti explains, “The Golden Ratio recurs in the grid of the background, now fully visible after the restoration, and it is precisely the background that provides the key to understanding the meaning of the lunette and its prominent position on the façade” (Armienti).

Armienti’s article, “The Medieval Roots of Modern Scientific Thought. A Fibonacci Abacus on the Facade of the Church of San Nicola in Pisa,” is a fascinating study providing many detailed descriptions of the various Fibonacci- related components of the design with explanations of how they ultimately served as a tool for “the education of elites, in tune with the aims of scholastic philosophy: a precious gift of the wisdom of the ancients whose heritage must be valued.”

Here are just a few of the details he provides:

*The concentric coronas forming a girdle in the intarsia have maximum and minimum radii of 21 and 13 respectively and have their centers on the circle of diameter 34. This arrangement describes the property that FN/FN–2 = 2 with the advance of FN-3. In fact, all the circles of size 21 in the girdle are tangent both to the circle 55 and to the circle of size 13 in the center. With reference to the diameters, this implies: 110 = 42 + 42 + 26 that is equivalent to: 55 = 21 + 21 + 13 (or 55/21 = 2 advance = 13). The same rule applies to circles of radii 34 and 13, in fact 34 = 13 + 13 + 8 (there are four coronas of size 2, between the two circles of radius 13, whose centers lie at the extremes of the radius of circle 34).
*

*The size of the line borders in this way is necessarily 2, as required by the fact that the difference between 21 and 13 is 8 and has to be distributed on four coronas of equal size.
* *
* *The other linear elements of the intarsia are inscribed in circles of 55–2 = 53 respectively (2 is the width of all the coronas), and 34–2 = 32.
*

*Fifty-three and 32 are the sums of the N–2 Fibonacci numbers that precede 55 and 34 in the series. This arrangement is related to the second property stated above for the Fibonacci series.
*

*Lower limits of the battlements are arranged on the circle of radius 34–2, while their cusps are defined by a kink marked by the circle of radius 42 = 55–13 along which the circles of radii 1 and 2 are also aligned.
* (Armienti)

Other details, linked to the grid in the lunette around the intarsia, are proof that the author of the design was aware of the relations between the Golden Ratio and the Fibonacci series. This allowed him to find and represent to a very good approximation, regular polygons inscribed in a circle of a given radius **r** (Armienti).

The conclusion must be made that artists, theologians, mathematicians, and artisans worked closely together to create this masterpiece, “following a common code based on the insights of Fibonacci, and with utter dedication to their art.” Armienti believes he has deciphered only a small part of this code, and there are many historiographic and mathematical problems which remain to be solved. For example, he says, “the intarsia shows that the artists were fully aware of the connections that existed between their plan, the Fibonacci sequence and the Golden Ratio, even though, until today, the discovery of these connections has been attributed to Luca Pacioli, a mathematician of the early seventeenth century” (Armienti). Clearly, Pacioli had only rediscovered what others had known – and revered – hundreds of years before.

Polyclitus and Phidias are regarded as the most famous and authoritative masters of ancient Greek sculpture of the Classical era. Their statues were long considered the standard of beauty and harmonious construction. Polyclitus’ statue of *Doryphorus* (*Spear Bearer*) (late fifth/early fourth century BCE) is considered one of the greatest achievements of classical Greek art. This statue is an archetype for the proportions of the ideal human body established by ancient Greek sculptors. The name ascribed to this sculpture is especially important because the “*Canon*” (sometimes spelled “*Kanon*”) was not only a statue which deigned to display perfect human proportion, but was, in fact, a physical representation of what Polyclitus had described in his treatise on beauty, also titled “Canon.” While both his written treatise and the original statue are now lost, marble copies of the statue remain and text records of observations by ancient researchers and historians enable modern vicarious examination of what some considered to be the most exquisite model of the perfect human form (Sartwell).

The ancient physician Aelius or Claudius Galenus (often Anglicized as Galen and better known as Galen of Pergamum) (130-210 CE), characterized the *Canon* as specifying the perfect symmetries of the body; he describes the statue’s proportions as perfect, in “the finger to the finger, and of all the fingers to the metacarpus, and the wrist, and of all these to the forearm, and of the forearm to the arm, in fact of everything to everything… just as it is written in the *Canon* of Polyclitus” (Sartwell). Some attribute such “perfection” to the fact that Polyclitus applied the principles of the Golden Mean to his creation. Russian architect G.D. Grimm analyzed the harmonic dimensions of the *Doryphorus* and presented the following connections between the *Canon* and the Golden Mean (*Proportionality in Architecture* 1933):

1. First golden cut division: at the navel 2. Second division: lower part of the torso, passes through his knee 3. Third division: passes through the line of his neck (Stakhov 41)

In addition to the Parthenon, Phidias created enormous statues of Athena, including one in bronze (Winner in Battle) and another in ivory and gold *Athena Parthenos (The Virgin)* in commemoration of the Athenian victory over the Persians. He also created the statue of Zeus for the Olympian temple of Zeus (about 430 B.C.), which is considered one of the Seven Wonders of the Ancient World.

Despite the unprecedented monumental sizes of his sculptures (9m *Athena Parthenos* and 13m *Zeus*), Phidias constructed them with strict adherence to the principles of harmony based on the Golden Mean (Stakhov 5-6).

“Phi is more than an obscure term found in mathematics and physics.” It appears to inform not only design and construction, but even our aesthetic preferences. When research subjects (who were not mathematicians or physicists familiar with phi) were asked to view random faces, those consistently deemed to be most attractive were those which exhibited “Golden Ratio proportions between the width of the face and the width of the eyes, nose, and eyebrows.” Researchers conclude “the Golden Ratio elicited an instinctual reaction” (Hom).

Attempts to create an ideal model of a harmoniously developed human body continued during the age of the Renaissance. The ideal human figure created by Leonardo da Vinci (1452-1519) is widely known. His drawing of “Vitruvian Man” is said to illustrate the Golden Ratio (Hom). The ratio of the square side to the circle radius corresponds to the value of Phi with a deviation of just 1.7 percent (Posamentier and Lehmann 257). It clearly shows “pentagonal” or “five-fold” symmetry which is characteristic for flora and animals (Stakhov 43). The man’s head, two hands, and two legs are positioned in the form of a pentagram, as if they are beams of a pentagonal star. Da Vinci based the measurements of his ideal man on the proportions described by Vitruvius (chapter 1 of Book III) (Cartwright).

Leonardo da Vinci illustrated the book *De Divina Proportione* (1509) by Franciscan monk Fra Luca Pacioli (c. 1445-1517), in which the latter referred to the number Phi as the “Divine Proportion;” Da Vinci later called this *sectio aurea*, or the **Golden Section**. Many assume that Da Vinci was therefore “consciously guided by this magnificent ratio” and used the Golden Ratio in all (or most) of his work (Posamentier and Lehmann 260). Some claim that he used it to define all of the proportions in his painting, *The Last Supper*, “including the dimensions of the table and the proportions of the walls and backgrounds.” The Golden Ratio also appears in his iconic portrait, Mona Lisa. Many other famous artists are believed to have employed the Golden Ratio, including Michelangelo, (*Madonna Doni*) Raphael (*Sistine Madonna*), Rembrandt (*A Self-portrait*), Seurat (*Circus Parade*), and Salvador Dali (*Half a Giant Cup Suspended with an Inexplicable Appendage Five Meters Long*) (Posamentier and Lehmann; Hom).

It is uncommon for an artist to explicitly testify to the conscious use of Fibonacci numbers as the basic structure of their work. However, one such artist is the German Rune Mields (b. 1935) who explained that her piece titled *Evolution: Progression and Symmetry III and IV* is “subject to the laws of symmetry” in that, in “‘an ascending line, a progression of triangles is generated, with the help of the famous mathematical series of the Leonardo Pisano, called Fibonacci’” (Posamentier and Lehmann 266).

Fibonacci’s influence remains pervasive in Modern Architecture, where the sequence itself has become a feature of the design. The smokestack of the power station in Turku, Finland, has become a midtown landmark because it showcases the first ten numbers of the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, and 55) in a playful display of bright neon numbers, two meters high. The “Fibonacci Chimney” was created in 1994 by Italian artist Mario Merz as an environmental art project (Lobo). It is just one of his many “conceptual” works which incorporate the Fibonacci sequence. His “Fibonacci Naples” (1970) “consists of ten photographs of factory workers, building in Fibonacci numbers from a solitary person to a group of fifty-five.” Merz featured Fibonacci’s numbers because his desire was to “protest against a dehumanized, consumer-driven society” by creating art inspired by a sequence “which underlies so many growth patterns of natural life” (Livio 176).

Originally planned to function as a Jewish synagogue, the *Mole Antonelliana* (1863-1889) is used today as a movie museum; the five-floor building in Turin, Italy, is believed to be the tallest museum in the world. It is also Europe’s tallest brick structure with the tallest dome. On one side of the four-faced dome today, the first Fibonacci numbers are illuminated by red neon lights. *Il Volo Dei Numeri (Flight of the Numbers)* (1998) was designed by Mario Merz (“Fibonacci – Flight”).

Like Fibonacci before him, 20th-century architect Charles-Edouard Jeanneret (known as Le Corbusier) (1887–1965) became fascinated by mathematical concepts while traveling; he had journeyed throughout Europe and learned principles of proportion while investigating ancient buildings everywhere he went, particularly from German architects (Cohen). Decades later, Le Corbusier published *Le Modulor: A Harmonious Measure to the Human Scale Universally Applicable to Architecture and Mechanics* and insisted his work was a unique, universally- applicable measuring system that would give architecture a mathematical order oriented to a human scale. “Le Corbusier developed his doctrine for the proportions of construction by combining the imperial measuring system based on the foot with the metric decimal system and relating this to human body measurements. He started from an assumed standard size of the human body and marked three intervals related to each other in the proportion of the Golden Ratio” (“Le Corbusier”). Specifically, he explained, “A man with a raised arm provides the main points of space displacement – the foot, solar plexus, head, and fingertip of the raised arm – three intervals which yield a number of sections that are determined by Fibonacci’ (Posamentier and Lehmann 241). It was “a tool of linear or optical measures, similar to musical script,” with which he was familiar (Cohen). His systematic “tool” for “planning architecture and industrial products gained worldwide currency and was applied by countless practitioners” (“Le Corbusier”).

Built in the shape of a sunflower and the size of a spaceship, the Core was first built in 2005 and re-imagined in 2017-2018. Home to the Invisible Worlds exhibition at the Cornwall Education Center, the building was designed using natural forms (biomimicry) and sustainable construction and patterns based on Fibonacci numbers (“How”). Architect Jolyon Brewis explains: “We decided that the structure of the building itself should be derived from the double spiral, and we looked to the mathematics behind these spirals in nature to generate the design. We were delighted to discover that this produced an efficient and elegant network of timber beams” (“Journey” 9).

San Francisco landscape photographer Mike Spinak recounts some of the many ways modern artists, including photographers, “derive a wide variety of mathematical constructs from the Golden Mean, for the sake of composition guidelines.” He says: *They divide a line segment according to ~1.618 (the Golden Section). They make a rectangle where the long sides are ~1.618 times as long as the short sides (the Golden Rectangle). They make an isosceles triangle where the two long sides of the triangle are ~1.618 times the length of the short side (the Golden Triangle). They make a triangle where the longest side is ~1.618 times as long as the second longest side, which is ~1.618 times as long as the shortest side (Kepler Triangle). They make a logarithmic spiral which gets wider by a factor of ~1.618 for every quarter turn of its rotation (the Golden Spiral). And so on, with numerous others, such as the golden rhombus, Bakker’s Saddle, Saint Andrew’s Cross, and Bouleau’s Armature of the rectangle. Some artists also like more oblique and esoteric constructs, such as division of the visible light spectrum by the Golden Mean, or segments thereof* (Spinak).

Dallas photographer James Brandon offers a few suggestions for ways that amateur photographers can use the Golden Ratio to compose a photograph. According to Brandon, the software program Adobe Lightroom 3 has a Golden Ratio overlay option for cropping an image. Lines of interest or points in a photograph are lined up to coincide with a grid of the Golden Ratio. By taking the “sweet spot” of the Fibonacci Ratio and duplicating it four times into a grid, the result looks to be a rule-of-thirds grid. However, upon closer inspection it is evident that the grid does not split the frame precisely into three pieces. Instead of a 3-piece grid that divides the frame 1+1+1, there is a grid dividing the frame vertically and horizontally 1+.618+1 (Brandon). A popular way to apply the Fibonacci spiral to a composition is to position “the primary element of the picture approximately where the tightly curled ‘end’ of the golden spiral would fit into the frame.” The photo is “considered even more aesthetically pleasing” if the picture subject can be arranged so that “some of the picture’s lines roughly follow the spiral’s lines.” The various other constructs listed by Spinak are occasionally “used for choosing relative proportions – such as composing with the background building ~1.618 times as tall, in the picture, as the person in the foreground. Or, they’re used to choose color combinations for a picture’s palette” (Spinak). Spinak says the practice of applying the Golden Mean to composition has “seemingly become elevated to established orthodoxy.” A Google Search on the topic, he adds, “will bring up more than one and a half million listings.” Moreover, “most basic photography instruction books discuss composing with the Golden Mean.” Adobe Lightroom has “several Golden Section overlays built into the program,” including overlays for a Golden Ratio grid, a golden spiral, and a Saint Andrew’s cross. Visitors to other websites can see their pictures “superimposed with Golden Mean overlays. There are also software applications available for overlaying anything on your computer screen with a variety of Golden Mean- derived visual constructs.” Finally, Golden Mean calipers can be purchased online from hundreds of sources.

The Golden Mean is popular not only for composing, but also for analyzing compositions. “Analysts deconstruct pictures by drawing various line and pattern overlays upon pictures” and determining whether and how closely a particular picture conforms to some derivative of the Golden Mean (Spinak).

Brandon insists, “Fibonacci’s Ratio is a powerful tool for composing your photographs, and it shouldn’t be dismissed as a minor difference from the rule of thirds. While the grids look similar, using Phi can sometimes mean the difference between a photo that just clicks, and one that doesn’t quite feel right.” He considers Phi a far superior composition tool to use “and [a] much more intelligent and historically-proven method for composing a scene” (Brandon).

Author Jess McNally describes fractals as “patterns formed from chaotic equations [that] contain self-similar patterns of complexity increasing with magnification.” Nearly identical but smaller copies of the whole are created when you divide a fractal pattern into parts. By duplicating or repeating relatively simple fractal- generating equations, infinite complexity is formed. Unique but recognizable patterns are created. Remarkably, the number of particular geometric shapes of a specific size formed within the patterns often turn out to be Fibonacci numbers! (McNally).

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