Math and the Mona LisaMath and the Mona Lisa

A Study of the Golden RatioA Study of the Golden RatioBy: M. Mendoza and C. GinsonBy: M. Mendoza and C. Ginson

StatementStatement

Mathematical “truths” have always Mathematical “truths” have always been “discovered” therefore we may been “discovered” therefore we may conclude by definition of the word conclude by definition of the word ‘discover’ that all of these “truths” ‘discover’ that all of these “truths” are already “divinely” in place just are already “divinely” in place just waiting for the mathematician to waiting for the mathematician to “find” it.“find” it.

M. MendozaM. Mendoza

Introduction: History of the Introduction: History of the Golden RatioGolden Ratio

While the proportion known as the While the proportion known as the Golden MeanGolden Mean has always existed in has always existed in mathematics and in the physical mathematics and in the physical universe, it is unknown exactly when it universe, it is unknown exactly when it was first discovered and applied by was first discovered and applied by mankind. It is reasonable to assume mankind. It is reasonable to assume that it has perhaps been discovered and that it has perhaps been discovered and rediscovered throughout history, which rediscovered throughout history, which explains why it goes under several explains why it goes under several names.  names. 

Uses in architecture date to Uses in architecture date to the ancient Egyptians and the ancient Egyptians and Greeks Greeks • It appears that the Egyptians may have It appears that the Egyptians may have

used both pi and phi in the design of the used both pi and phi in the design of the Great Pyramids. The Greeks based the . The Greeks based the design of the design of the Parthenon on this proportion. on this proportion.

• Phidias (500 BC - 432 BC), a Greek Phidias (500 BC - 432 BC), a Greek sculptor and mathematician, studied phi sculptor and mathematician, studied phi and applied it to the design of sculptures and applied it to the design of sculptures for the Parthenon.for the Parthenon.

• Euclid (365 BC - 300 BC), in "Elements," referred Euclid (365 BC - 300 BC), in "Elements," referred to dividing a line at the 0.6180399... point as to dividing a line at the 0.6180399... point as "dividing a line in the extreme and mean ratio." "dividing a line in the extreme and mean ratio." This later gave rise to the use of the term mean This later gave rise to the use of the term mean in the golden mean.  He also linked this number in the golden mean.  He also linked this number to the construction of a pentagram.to the construction of a pentagram.

The Fibonacci Series was The Fibonacci Series was discovered around 1200 AD discovered around 1200 AD

• Leonardo Fibonacci, an Italian born in 1175 , an Italian born in 1175 AD discovered the unusual properties of the AD discovered the unusual properties of the numerical series that now bears his name, numerical series that now bears his name, but it's not certain that he even realized its but it's not certain that he even realized its connection to phi and the Golden Mean. His connection to phi and the Golden Mean. His most notable contribution to mathematics most notable contribution to mathematics was a work known as Liber Abaci, which was a work known as Liber Abaci, which became a pivotal influence in adoption by became a pivotal influence in adoption by the Europeans of the Arabic decimal system the Europeans of the Arabic decimal system of counting over Roman numerals. of counting over Roman numerals.

Fibonacci numbersFibonacci numbers

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,... 233,...

The Fibonacci numbers form what we call a The Fibonacci numbers form what we call a recurrent sequence. Beginning with 1, recurrent sequence. Beginning with 1, each term of the Fibonacci sequence is the each term of the Fibonacci sequence is the sum of the sum of the two preceding numberstwo preceding numbers..

0+1=1     1+1=2     1+2=3     2+3=5     0+1=1     1+1=2     1+2=3     2+3=5     3+5=8    5+8=13….3+5=8    5+8=13….

It was first called the "Divine It was first called the "Divine Proportion" in the 1500'sProportion" in the 1500's

• Da Vinci provided illustrations for a Da Vinci provided illustrations for a dissertation published by Luca Pacioli dissertation published by Luca Pacioli in 1509 entitled "De Divina in 1509 entitled "De Divina Proportione", perhaps the earliest Proportione", perhaps the earliest reference in literature to another of its reference in literature to another of its names, the "Divine Proportion."  This names, the "Divine Proportion."  This book contains drawings made by book contains drawings made by Leonardo da Vinci of the five Platonic Leonardo da Vinci of the five Platonic solids.  It was probably da Vinci who solids.  It was probably da Vinci who first called it the "sectio aurea," which first called it the "sectio aurea," which is Latin for golden section.is Latin for golden section.

• The Last SupperThe Last Supper

Mona LisaMona Lisa

• If I could write the beauty of your eyesIf I could write the beauty of your eyes And in fresh numbers, number all your gracesAnd in fresh numbers, number all your graces The age to come would say, “This poet lies;”The age to come would say, “This poet lies;” His heavenly touches ne’er touch’d earthly faces.His heavenly touches ne’er touch’d earthly faces. -William Shakespeare-William Shakespeare

PhiPhi

What is Phi?What is Phi?

Phi ( = 1.618033988749895... ), most Phi ( = 1.618033988749895... ), most often often pronounced fi like "fly," is ," is simply an irrational number like pi simply an irrational number like pi

( p = 3.14159265358979... ), but one ( p = 3.14159265358979... ), but one with many unusual mathematical with many unusual mathematical properties.  Unlike pi, which is a properties.  Unlike pi, which is a transcendental number, phi is the transcendental number, phi is the solution to a quadratic equation.solution to a quadratic equation.

The quadratic equation : The quadratic equation : ll22 – lw – w – lw – w22 = 0 = 0

• One property of golden rectangles is that their• length (length + width)• ------ = ----------------• width length• When we cross multiply the above proportion, we get• l2 = lw + w2 or• l2 - lw - w2 = 0.• Solving the equation gives us• l = w(1 + sqrt5)• ------------• 2• Divide both sides by w:• l = (1 + sqrt5)• --- ---------• w 2• Hence; Φ = 1.61803398874989484820....................

Phi and the Fibonacci Series Phi and the Fibonacci Series

• Leonardo Fibonacci discovered the series which Leonardo Fibonacci discovered the series which converges on phiconverges on phi

• In the 12th century, Leonardo Fibonacci discovered a simple In the 12th century, Leonardo Fibonacci discovered a simple numerical series that is the foundation for an incredible numerical series that is the foundation for an incredible mathematical relationship behind phi.mathematical relationship behind phi.

• Starting with 0 and 1, each new number in the series is Starting with 0 and 1, each new number in the series is simply the sum of the two before it. simply the sum of the two before it.

• 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .• The ratio of each successive pair of numbers in the series The ratio of each successive pair of numbers in the series

approximates phi (1.618. . .) , as 5 divided by 3 is 1.666..., approximates phi (1.618. . .) , as 5 divided by 3 is 1.666..., and 8 divided by 5 is 1.60.and 8 divided by 5 is 1.60.

• The table below shows how the ratios of the successive The table below shows how the ratios of the successive numbers in the Fibonacci series quickly converge on Phi.  numbers in the Fibonacci series quickly converge on Phi.  After the 40th number in the series, the ratio is accurate to After the 40th number in the series, the ratio is accurate to 15 decimal places.15 decimal places.

• 1.618033988749895 . . . 1.618033988749895 . . .

The Golden Rectangle thru The Golden Rectangle thru Fibonacci’s sequence:Fibonacci’s sequence:

Determining the nth number of Determining the nth number of the Fibonacci series the Fibonacci series • You can use phi to compute the nth number in the You can use phi to compute the nth number in the

Fibonacci series (fFibonacci series (fnn):):• ffnn =  Phi =  Phi nn / 5 / 5½½

• As an example, the 40th number in the Fibonacci As an example, the 40th number in the Fibonacci series is 102,334,155, which can be computed as:series is 102,334,155, which can be computed as:

• ff4040   =   Phi   =   Phi 4040 / 5 / 5½½   =  102,334,155   =  102,334,155 • This method actually provides an estimate which This method actually provides an estimate which

always rounds to the correct Fibonacci number.always rounds to the correct Fibonacci number.• You can compute any number of the Fibonacci You can compute any number of the Fibonacci

series (fseries (fnn) exactly with a little more work:) exactly with a little more work:• ffnn = [ Phi = [ Phi nn - (-Phi) - (-Phi)-n-n ] / (2Phi-1) ] / (2Phi-1)• Note:  2Phi-1 = 5Note:  2Phi-1 = 5½½= The square root of 5 = The square root of 5 

Here a phi, there a phi, Here a phi, there a phi, everywhere a phi, phi…everywhere a phi, phi…

• In Architecture:In Architecture:It was used it in the design of Notre Dame in Paris, which was It was used it in the design of Notre Dame in Paris, which was

built in the 1163 and 1250.built in the 1163 and 1250.

In India, it was used in the construction of In India, it was used in the construction of the Taj Mahal, which was completed in the Taj Mahal, which was completed in

1648.1648.

In Art In Art ""Without mathematics thereWithout mathematics there is is no artno art."."

--Luca Pacioli--Luca Pacioli • The French impressionist painter Georges Pierre Seurat is The French impressionist painter Georges Pierre Seurat is

said to have "attacked every canvas by the golden section," said to have "attacked every canvas by the golden section," as illustrated below.  as illustrated below. 

• In "The Sacrament of the Last Supper," Salvador Dali In "The Sacrament of the Last Supper," Salvador Dali framed his painting in a golden rectangle.  Following Da framed his painting in a golden rectangle.  Following Da Vinci's lead, Dali positioned the table exactly at the golden Vinci's lead, Dali positioned the table exactly at the golden section of the height of his painting.  He positioned the two section of the height of his painting.  He positioned the two disciples at Christ's side at the golden sections of the width disciples at Christ's side at the golden sections of the width of the composition.  In addition, the windows in the of the composition.  In addition, the windows in the background are formed by a large dodecahedron.  background are formed by a large dodecahedron.  Dodecahedrons consist of 12 pentagons, which exhibit phi Dodecahedrons consist of 12 pentagons, which exhibit phi relationships in their proportions relationships in their proportions

In Nature: In Nature: "Nature hides her secrets because of her "Nature hides her secrets because of her essential loftiness, but not by means of ruse."essential loftiness, but not by means of ruse."--Einstein, Albert (1879-1955)--Einstein, Albert (1879-1955)

• sunflowers have a Golden Spiral seed arrangement. This sunflowers have a Golden Spiral seed arrangement. This provides a biological advantage because it maximizes the provides a biological advantage because it maximizes the

number of seeds that can be packed into a seed headnumber of seeds that can be packed into a seed head

• many flowers have a Fibonacci number of petals. Some, like many flowers have a Fibonacci number of petals. Some, like this rose, also have Fibonacci, or this rose, also have Fibonacci, or Golden SpiralGolden Spiral, petal , petal arrangement arrangement

• Every key body feature of the angel fish falls at golden Every key body feature of the angel fish falls at golden sections of its width and length sections of its width and length

• The body sections of an ant are defined by the golden The body sections of an ant are defined by the golden sections of its length. Its leg sections are also golden sections of its length. Its leg sections are also golden sections of its length.sections of its length.

• The shell of the chambered Nautilus has Golden proportions. It is a The shell of the chambered Nautilus has Golden proportions. It is a logarithmic spiral.logarithmic spiral.

• "[The universe] cannot be read until we have learnt the language and "[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly other geometrical figures, without which means it is humanly impossible to comprehend a single word."impossible to comprehend a single word."--Galilei, Galileo (1564 - 1642), Opere Il Saggiatore p. 171.--Galilei, Galileo (1564 - 1642), Opere Il Saggiatore p. 171.

• Humans exhibit Fibonacci Humans exhibit Fibonacci characteristics, too. The characteristics, too. The Golden RatioGolden Ratio is seen in the proportions in the is seen in the proportions in the sections of a finger.sections of a finger.

In Beauty: In Beauty: "The good, of course, is always beautiful, "The good, of course, is always beautiful, and the beautiful never lacks proportion."and the beautiful never lacks proportion."--Plato--Plato

"Beauty is in the phi of the "Beauty is in the phi of the beholder."beholder."

• It has long been said that beauty is in the eye of the It has long been said that beauty is in the eye of the beholder and thought that beauty varies by race, beholder and thought that beauty varies by race, culture or era.  The evidence, however, shows that culture or era.  The evidence, however, shows that our perception of physical beauty is hard wired into our perception of physical beauty is hard wired into our being and based on how closely one's features our being and based on how closely one's features reflect phi in their proportions.reflect phi in their proportions.

• A template for human beauty is found in A template for human beauty is found in phi and the pentagon phi and the pentagon

Take another look at beauty Take another look at beauty through the eyes of medical through the eyes of medical sciencescience

ConclusionConclusion

• We now have a deeper appreciation We now have a deeper appreciation of the things around us as a result of of the things around us as a result of our study. It is our hope that your our study. It is our hope that your senses may also be opened through senses may also be opened through the mathematics of science and art.the mathematics of science and art.

Sources used in this study:Sources used in this study:

• http://goldennumber.net/http://goldennumber.net/• http://jwilson.coe.uga.edu/EMAT6680/Parveen/http://jwilson.coe.uga.edu/EMAT6680/Parveen/

nikhat.htmlnikhat.html• http://mathcentral.uregina.ca/index.phphttp://mathcentral.uregina.ca/index.php• http://www.beautyanalysis.com/http://www.beautyanalysis.com/• http://rusmp.rice.edu/curriculum/http://rusmp.rice.edu/curriculum/

geometrymodule/TableofContents.htmgeometrymodule/TableofContents.htm• http://www6.district125.k12.il.us/~solver/http://www6.district125.k12.il.us/~solver/• Atalay, Bulent (2006). Math and the Mona Lisa: Atalay, Bulent (2006). Math and the Mona Lisa:

The Art and Science of Leonardo Da Vinci. New The Art and Science of Leonardo Da Vinci. New York: Smithsonian BooksYork: Smithsonian Books