Dr. Oleg Artamonov - lecture slides 16th April, 2015
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Transcript of Dr. Oleg Artamonov - lecture slides 16th April, 2015
De Divina Proportione
Oleg Artamonov
Assistant ProfessorSSE, Habib University
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
Outline
1 Golden Ratio
2 Golden Figures
3 Applications
4 Divine Proportion in Nature
Oleg Artamonov De Divina Proportione 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
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
Divine ProportionIntroduction and Definition
Pic: Portrait of Luca Pacioli, traditionally attributed to Jacopo de’Barbari, 1495 .
Oleg Artamonov De Divina Proportione April 16, 2015
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
Oleg Artamonov De Divina Proportione April 16, 2015
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
Oleg Artamonov De Divina Proportione April 16, 2015
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
Oleg Artamonov De Divina Proportione April 16, 2015
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
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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
Oleg Artamonov De Divina Proportione April 16, 2015
Golden RatioExpressions and Properties
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
Oleg Artamonov De Divina Proportione April 16, 2015
Golden RatioExpressions and Properties
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
«
Oleg Artamonov De Divina Proportione April 16, 2015
Golden RatioRelation to Fibonacci Numbers
Fig: Leonardo Bonacci, 1170− 1250
Fig: A page of the Liber Abaci (1202) showing thenumbers of the Fibonacci sequence.
Oleg Artamonov De Divina Proportione April 16, 2015
Golden RatioRelation to Fibonacci Numbers
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
Oleg Artamonov De Divina Proportione April 16, 2015
Golden RatioRelation to Fibonacci Numbers
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
Oleg Artamonov De Divina Proportione April 16, 2015
Golden RatioGeometric Constructions
Fig: Construction by compass and ruler of a line segment which length is ϕ. Fig: Golden ratio caliper.
Oleg Artamonov De Divina Proportione April 16, 2015
Golden RatioGeometric Constructions
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.
Oleg Artamonov De Divina Proportione April 16, 2015
Golden RatioGeometric Constructions
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.
Oleg Artamonov De Divina Proportione April 16, 2015
Golden Figures
Fig: Leonardo da Vinci’s illustrations to the ”De Divina Proportione”.
Oleg Artamonov De Divina Proportione April 16, 2015
Golden FiguresGolden Rectangle
Fig: Golden rectangle, a+ba = a
b = ϕ. Fig: Construction by compass and ruler.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
Golden FiguresGolden Spiral
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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).
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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”.
Oleg Artamonov De Divina Proportione April 16, 2015
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
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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!
Oleg Artamonov De Divina Proportione April 16, 2015
Applications
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArchitecture
Fig: Melancolia, Albrecht Durer, 1514 (engraving).
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArchitecture
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.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArchitecture
Fig: Hagia Sophia, Constantinople, 537.Fig: St. Mark’s Basilica, Venice, 1117.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArchitecture
Fig: Cathedrale Notre-Dame de Laon.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArchitecture
Fig: The superimposed regulator lines show that the cathedral hasgolden proportions.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArchitecture
Fig: The southern facade of Notre-Dame de Paris. Fig: The western facade illuminated at night.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArchitecture
Fig: Taj Mahal, Agra, Uttar Pradesh, India.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArchitecture
Fig: Naqsh-e Jahan Square and the adjacent Sheikh Loftollah mosque, Isfahan, Iran.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArchitecture
Fig: Metropolitan Cathedral of the Assumption of Mary, Mexico City, Mexico.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArchitecture
Fig: Alhambra, Granada, Andalusia, Spain.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArchitecture
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.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArchitecture
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.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArt
Fig: A traveller puts his head under the edge of the firmament, original printing of the Flammarion engraving.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArt
Fig: The Last Supper , Leonardo da Vinci, 1498.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArt
Fig: All the key dimensions of the room, the table and ornamental shields were based on the golden ratio.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArt
Fig: Isleworth Mona Lisa (1505) and Mona Lisa (1517).
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArt
Fig: Mona Lisa (1517) and Mona Lisa, Prado (1517) .
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsArt
Fig: The death of Socrates, Jacques-Louis David, 1787.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsPhotography
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsPhotography
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsPhotography
Fig: Scene in the Ukrainian parliament.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsForex
Fig: Golden mean gauge and markets’ behavior.Fig: Fibonacci’s net is a powerful instrument of analysisin the foreign exchange market terminals.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsDesign
Fig: Automobiles and their logos design.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsDesign
Fig: Mercedes-Benz’s logotype.Fig: Honda’s logotype.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsDesign
Fig: ϕ, golden rectangles, and goldenspiral in jewellery’s design.
Oleg Artamonov De Divina Proportione April 16, 2015
ApplicationsDesign
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NaturePlants
Fig: Aloe.Fig: Sago.
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NaturePlants
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureContinent and Hurricane
Fig: Africa. Fig: Hurricane Isabel.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureScale of Temperature Comfort and DNA
Fig: ϕ proportions are in a scale of temperature comfort.
Fig: Golden ratio in DNA.
Oleg Artamonov De Divina Proportione April 16, 2015
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.
Oleg Artamonov De Divina Proportione April 16, 2015
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
Oleg Artamonov De Divina Proportione April 16, 2015
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
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body
Fig: Roman copy of Doryphoros, originally byPolykleitos. Fig: The golden ratio person.
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body
Fig: Manipulations with the proportion of Classical and Renaissance sculptures’ features by violating the goldenratio.
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body
Fig: Leonardo da Vinci’s illustration of a human headfrom the Pacioli’s book.
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body Let us speak about women!
Fig: The Birth of Venus, Sandro Botticelli, 1482.
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body
Fig: Fibonacci spirals in a woman’s face from the side.
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body
Fig: Example faces with different length and width ratios. Faces with an average length or width ratio are:
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body
Fig: Example faces with different length and width ratios. Faces with an average length or width ratio are:
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body
Fig: Golden proportion and Hollywood actresses.
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body Golden ratio is not the last measure of beauty!
Fig: Jennifer Aniston and Angelina Jolie. Golden ratio verdict? Ugly!
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body
Fig: Studies on the proportions of the female body,Albrecht Durer, 1528.
Fig: Divine proportions.
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body
Fig: Tastes are different, ideal is one . . .
Oleg Artamonov De Divina Proportione April 16, 2015
Divine Proportion in NatureHuman Body
Fig: Women provide applied mathematics with few more additionalparameters. . .
Oleg Artamonov De Divina Proportione April 16, 2015
Mathematics is a beautiful science!
Oleg Artamonov De Divina Proportione April 16, 2015
thank youfor your attention