dr. oleg artamonov - lecture slides 16th april, 2015

De Divina Proportione

Oleg Artamonov

Assistant ProfessorSSE, Habib University

[email protected]

April 16, 2015

Outline

1 Golden Ratio

2 Golden Figures

3 Applications

4 Divine Proportion in Nature

Oleg Artamonov De Divina Proportione April 16, 2015

Johannes Kepler about the Divine Proportion

Geometry has two great treasures: one is thetheorem of Pythagoras, the other the division of a lineinto mean and extreme ratio. The first we maycompare to a mass of gold, the second we may call aprecious jewel.


Bertrand Russell about Mathematics

Mathematics, rightly viewed, possesses not onlytruth, but supreme beauty – a beauty cold and austere,like that of sculpture, without appeal to any part of ourweaker nature, without the gorgeous trappings ofpainting or music, yet sublimely pure, and capable of astern perfection such as only the greatest art can show.The true spirit of delight, the exaltation, the sense ofbeing more than Man, which is the touchstone of thehighest excellence, is to be found in mathematics assurely as poetry.


Divine ProportionIntroduction and Definition

Pic: Portrait of Luca Pacioli, traditionally attributed to Jacopo de’Barbari, 1495 .


Divine ProportionIntroduction and Definition

Pic: De Divina Proportione, title page of 1509 edition.

the whole is the longer part

plus the shorter part

the whole is to the longer part

as the longer part is to the

shorter part


Golden RatioIntroduction and Definition

Two quantities are in the golden ratio if their ratio is the same as theratio of their sum to the larger of the two quantities.

Fig: Line segments in the golden ratio.

Fig: Approximate numerical expression of ϕ.


Golden RatioIntroduction and Definition

ϕ =ab

=a + b

a= 1 +

ba

= 1 +1ϕ

=⇒

ϕ2 − ϕ− 1 = 0 =⇒ two roots :1±√

52

,

where ϕ =

√5 + 12

and −ϕ−1 = 1− ϕ =1−√

52

;

Φ = ϕ−1 = ϕ− 1 =

√5− 12

is the golden ratio conjugate.


Golden RatioExpressions and Properties

ϕ =1 +√

5

2≈ 1.618034

Important partition of unit:

0.3820.618

≈ Φ2

Φ≈ 0.618

Properties:

ϕ2 = ϕ + 1

1ϕ

= ϕ− 1

1Φ

= Φ + 1



Numerical Expressions:

ϕ =

√1 +

√1 +√

1 + · · ·

ϕ = 1 +1

1 +1

1 +1

1 + · · ·

ϕ =138

+∞∑

n=0

(−1)(n+1) (2n + 1)!

(n + 2)! n! 4(2n+3)

ϕ =

√5 +√

5

5−√

5



Relations to π:

F

ϕ = 1 + 2 sin“ π

10

”=

1

2 sin“ π

10

”F

ϕ = 2 cos“π

5

”F

2 sin“π

5

”=p

3− ϕ

F

ϕ = 2 sin

„3π

10

«


Golden RatioRelation to Fibonacci Numbers

Fig: Leonardo Bonacci, 1170− 1250

Fig: A page of the Liber Abaci (1202) showing thenumbers of the Fibonacci sequence.



Fig: Leonardo Bonacci, 1170 - 1250

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .

Fig: Pascal’s triangle and Fibonacci numbers.

Fn =Fn−1 + Fn−2, F0 =0, F1 =1



limn→∞

Fn+1

Fn= ϕ

Fig: Series of golden rectangles and progressiveapproximation to golden ratio by dividing successive pairs ofFibonacci numbers.

Another relations:

Fn =ϕn − (−ϕ−n)√

5

ϕn = Fnϕ + Fn−1

limn→∞

Fn+α

Fn= ϕα


Golden RatioGeometric Constructions

Fig: Construction by compass and ruler of a line segment which length is ϕ. Fig: Golden ratio caliper.



Fig: Construction by compass and straightedge thatdivides a line segment into two line segments where theratio of the longer to the shorter line segment is ϕ.

Having a line segment AB, constructa perpendicular BC at point B, withBC half the length of AB. Draw thehypotenuse AC.

Draw an arc with center C and radiusBC. This arc intersects thehypotenuse AC at point D.

Draw an arc with center A and radiusAD. This arc intersects the originalline segment AB at point S. Point Sdivides the original segment AB intoline segments AS and SB withlengths in the golden ratio.



Fig: In the equilateral 4DEF, where AE = AF and BE = BD. Extend AB to meet the circumcircle DEF at

C. We have‖AB‖‖BC‖ =

‖AC‖‖AB‖ = ϕ.


Golden Figures

Fig: Leonardo da Vinci’s illustrations to the ”De Divina Proportione”.


Golden FiguresGolden Rectangle

Fig: Golden rectangle, a+ba = a

b = ϕ. Fig: Construction by compass and ruler.


Golden FiguresGolden Rectangle Division and Fibonacci Spiral

Fig: A distinctive feature of this shape is thatwhen a square section is removed, the remainder isanother golden rectangle; that is, with the sameaspect ratio as the first. Square removal can berepeated infinitely. . .

Fig: Corresponding corners of the squares form an infinitesequence of points on the golden spiral, the unique logarithmicspiral with this property.


Golden FiguresGolden Spiral


Golden FiguresGolden Rhombus

Fig: Golden Rhombus.Fig: Rhombic hexecontahedron is a stellation of therhombic triacontahedron. It is nonconvex with 60golden rhombic faces with icosahedral symmetry.


Golden FiguresGolden Triangle

Fig: Golden triangle,a

b= ϕ.

Fig: θ = 2 sin−1“

b2a

”= 2 sin−1

“1

2ϕ

”=

1

5π,

h =q`

b ϕ´2−

`12b´2

= bqϕ2 − 1

4=

= 12bp

5 + 2√

5, three angles are in 2 :2 :1

proportions.


Golden FiguresGolden Triangle

Fig: The triangles at the tips of a pentagram andobtained by dividing a decagon by connecting oppositevertices are golden triangles.

Fig: Golden triangle is bisected in Robinson triangles: agolden triangle and a golden gnomon ( AX

AC = Φ, three

angles are in 1 :1 :3 proportion).


Golden FiguresGolden Spirals

Fig: Fibonacci Spiral. The bisection process of base angles canbe continued infinitely, creating an infinite number of goldentriangles. A golden spiral can be drawn through the vertices.


Golden FiguresPentagon and Pentagram

Fig:diagonal

side=

b

a= ϕ.

Fig: The four lengths are in golden ratio to one another:

red

green=

green

blue=

blue

magenta= ϕ.


Golden FiguresPentagon and Pentagram

Fig: Golden proportions in pentagon and pentagram.Fig: The first printed illustration of arhombicuboctahedron, by Leonardo da Vinci, published in”De Divina Proportione”.


Golden FiguresSurface Pavement.

Areas can be filled completely and symmetrically with tiles of 3, 4 and 6 sides, but it was long believed that it wasimpossible to fill an area with 5-fold symmetry.

Fig1: 3 sides

Fig2: 4 sides

Fig3: 5 sides leaves gaps

Fig4: 6 sides


Golden FiguresPenrose Tiling.

A surface can be completely tiled in an asymmetrical, non-repeating manner in five-fold symmetry with just twoshapes based on ϕ.

Fig: Penrose tiling is a nonperiodic infinite tiling of a plane. Itmade from kites and darts; a kite is made from the goldentriangle, and a dart is made from two gnomons.

Fig: Roger Penrose standing on a floor with aPenrose tiling.


Golden FiguresPenrose Tiling.

Penrose tile is accomplished by creating a set of two symmetrical tiles, each of which is the combination of the twoRobinson triangles (found also in the geometry of the pentagon).

Fig1: The relationship of the sides ofthe pentagon, and also the tiles, isϕ, 1, and 1/ϕ.

Fig2: One creates a set of tiles, called“kites” and “darts” like this.

Fig3: The other creates a set ofdiamond tiles like this.

Fig4: The ratio of the two types oftiles in the resulting patterns is alwaysϕ!


Applications


ApplicationsArchitecture

Fig: Melancolia, Albrecht Durer, 1514 (engraving).



Fig: a=612.01 cubits, b=377.9 cubits, h=481.4 cubits

Fig: Great pyramid of Giza.

Fig: A Kepler triangle is a right triangle formed by threesquares with areas in geometric progression according to thegolden ratio.



Fig: Hagia Sophia, Constantinople, 537.Fig: St. Mark’s Basilica, Venice, 1117.



Fig: Cathedrale Notre-Dame de Laon.



Fig: The superimposed regulator lines show that the cathedral hasgolden proportions.



Fig: The southern facade of Notre-Dame de Paris. Fig: The western facade illuminated at night.



Fig: Taj Mahal, Agra, Uttar Pradesh, India.



Fig: Naqsh-e Jahan Square and the adjacent Sheikh Loftollah mosque, Isfahan, Iran.



Fig: Metropolitan Cathedral of the Assumption of Mary, Mexico City, Mexico.



Fig: Alhambra, Granada, Andalusia, Spain.



Fig: Le Corbusier sectioned his model human body’sheight at the navel with the two sections in golden ratio,then subdivided those sections in golden ratio at theknees and throat; he used these golden ratio proportionsin the Modulor system – an anthropometric scale ofproportions.

Fig: Le Corbusier and golden proportion on a Swiss bank-note.



Fig: Headquarters of the United Nations. The shape of the facade of thesecond is the result of three golden rectangles; however, each of the threerectangles that can be appreciated have different heights.

Fig: Modern building.


ApplicationsArt

Fig: A traveller puts his head under the edge of the firmament, original printing of the Flammarion engraving.


ApplicationsArt

Fig: The Last Supper , Leonardo da Vinci, 1498.


ApplicationsArt

Fig: All the key dimensions of the room, the table and ornamental shields were based on the golden ratio.


ApplicationsArt

Fig: The Virgin, Child, and St. Anne, Leonardo da Vinci, 1510. The golden ratio in this painting leads your eyefrom the baby then up through the two women, and then sets itself on the women’s face in perfect ratio.


ApplicationsArt

Fig: Rectangles superimposed on the Mona Lisa demonstratinggolden ratio at work. In Mona Lisa, the mismatch between the leftand right backgrounds creates the illusion of perspective and depth.A golden rectangle whose base extends from her right wrist to herleft elbow and reaches the very top of her head can be subdividedinto smaller golden rectangles to produce a golden spiral. The edgesof the new rectangles come to intersect the focal points of MonaLisa: chin, eye, nose, and upturned corner of her mouth. The overallshape of the woman is a triangle with her arms as the base and herhead as the tip, drawing attention to her face.

Fig: The face of the Mona Lisa outlines agolden rectangle.


ApplicationsArt

Fig: Isleworth Mona Lisa (1505) and Mona Lisa (1517).


ApplicationsArt

Fig: Mona Lisa (1517) and Mona Lisa, Prado (1517) .


ApplicationsArt

Fig: Creation of Adam, Michelangelo, Sistine Chapel’s ceiling, 1512. The finger of God touches the finger ofAdam precisely at the golden ratio point of the width and height of the area that contains them both.


ApplicationsArt

Fig: In his La Grande Odalesque, the 19th century neoclassical artist Jean Auguste Dominique Ingres used asimilar method for establishing the placement of the woman’s hand.


ApplicationsArt

Fig: The death of Socrates, Jacques-Louis David, 1787.


ApplicationsArt

Dali also incorporated in the painting a huge dodecahedron engulfing the supper table. The dodecahedron, whichaccording to Plato is the solid ”which the god used for embroidering the constellations on the whole heaven”,consist of 12 pentagons, which exhibit ϕ relationships in their proportions.

Fig: The Sacrament of the Last Supper , Salvador Dali , 1956.


ApplicationsArt

Fig: Dali framed his painting in a golden rectangle and also positioned the table exactly at the golden section ofthe height of his painting. He positioned the two disciples at Christ’s side at the golden sections of the width of thecomposition.


ApplicationsArt

As you let your eyes wander throughout the picture, examining each golden rectangle and its sub-sections, you willdiscover how Dali manage to create a dynamic and interesting interplay between shapes, along with numerousmini-paintings.

Fig: Living Still Life, Salvador Dali, 1956.


ApplicationsPhotography


ApplicationsPhotography

Fig: Scene in the Ukrainian parliament.


ApplicationsWeb Design

Using the Divine Proportion as a guide to your compositions can improve thecommunication of your design. -Mark Boulton

Fig: Example of Twitter using thegolden ratio in their 2010 redesign.


ApplicationsForex

Fig: Golden mean gauge and markets’ behavior.Fig: Fibonacci’s net is a powerful instrument of analysisin the foreign exchange market terminals.


ApplicationsDesign

Fig: Automobiles and their logos design.


ApplicationsDesign

Fig: Mercedes-Benz’s logotype.Fig: Honda’s logotype.


ApplicationsDesign

Fig: Red and Blue Chair designed by Gerrit Rietveld in 1917. Someworks in the Dutch artistic movement called De Stijl, orneoplasticism, exhibit golden ratio proportions.

Fig: Fibonacci spiral earrings.


ApplicationsDesign

Fig: ϕ, golden rectangles, and goldenspiral in jewellery’s design.


ApplicationsDesign


Divine Proportion in Nature

Fig: The golden spiral in nature.

Fig: The drawing of a man’sbody in a pentagram suggestsrelationships to the golden ratio.


Divine Proportion in NaturePlants

Fig: Even succulent plants like aloe display good approach tothe golden spiral. Plants grow new cells in spirals, which is howthis pattern appears. It works to the plants advantage bypreventing new leaves from blocking older leaves access tosunlight, directing the maximum amount of rain and dew tothe roots.

Fig: The golden spiral is highlighted in this image of aleaf from a bromeliad plant.



Fig: Aloe.Fig: Sago.



Fig: It is the golden spiral in a repeating pattern at thecenter of a sunflower. Growing in this manner creates themost compact pattern possible with no gaps frombeginning to end.

Fig: Just as with sunflowers, the pattern of seeds on apine cone can be found in repeating in either clockwise orcounter-clockwise motion.


Divine Proportion in NatureSpiderwebs and Sea Shells

Fig: Certain species of spiders form their webs in spirals that closelyapproximate the golden spiral.

Fig: When cut in half, a nautilus shelldisplays its chambers and its spiral structurebecomes even more apparent.


Divine Proportion in NatureContinent and Hurricane

Fig: Africa. Fig: Hurricane Isabel.


Divine Proportion in NatureGalaxies

Fig: Here is the Fibonacci spiral in the swirl of a galaxy, just as itappears in so many other natural forms.


Divine Proportion in NatureScale of Temperature Comfort and DNA

Fig: ϕ proportions are in a scale of temperature comfort.

Fig: Golden ratio in DNA.


Divine Proportion in NatureFauna

Fig: Batterfly.

Fig: Ram’s horn.

Examples of nature coinciding with the golden proportionin profusion can be found in fauna.

Fig: Eaglel.


Divine Proportion in NatureHuman Body

Fig: Vitruvian man. Golden ratio proportions of thehuman body.

The following sections of the modern human body are allin ϕ proportion:

♣ finger tip to elbow . . . wrist to elbow

♣ shoulder line/top of head . . . head length

♣ navel/top of head . . . shoulder line/top of head

♣ navel to knee . . . knee to end of foot



Fig: Roman copy of Doryphoros, originally byPolykleitos. Fig: The golden ratio person.



Fig: Manipulations with the proportion of Classical and Renaissance sculptures’ features by violating the goldenratio.



Fig: Leonardo da Vinci’s illustration of a human headfrom the Pacioli’s book.


Divine Proportion in NatureHuman Body Let us speak about women!

Fig: The Birth of Venus, Sandro Botticelli, 1482.



Fig: Fibonacci spirals in a woman’s face from the side.



Fig: Example faces with different length and width ratios. Faces with an average length or width ratio are:



Fig: Golden proportion and Hollywood actresses.


Divine Proportion in NatureHuman Body Golden ratio is not the last measure of beauty!

Fig: Jennifer Aniston and Angelina Jolie. Golden ratio verdict? Ugly!



Fig: Studies on the proportions of the female body,Albrecht Durer, 1528.

Fig: Divine proportions.



Fig: Tastes are different, ideal is one . . .



Fig: Women provide applied mathematics with few more additionalparameters. . .


Mathematics is a beautiful science!


thank youfor your attention

dr. oleg artamonov - lecture slides 16th april, 2015

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