De Divina Proportione Oleg Artamonov Assistant Professor SSE, Habib University Oleg.Artamonov@sse.habib.edu.pk

April 16, 2015

Outline

1 Golden Ratio

Oleg Artamonov

De Divina Proportione

April 16, 2015

Outline

1 Golden Ratio 2 Golden Figures

Oleg Artamonov

De Divina Proportione

April 16, 2015

Outline

1 Golden Ratio 2 Golden Figures 3 Applications

Oleg Artamonov

De Divina Proportione

April 16, 2015

Outline

1 2 3 4

Oleg Artamonov

De Divina Proportione

Golden Ratio Golden Figures Applications Divine Proportion in Nature

April 16, 2015

Johannes Kepler about the Divine Proportion

Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into mean and extreme ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.

Oleg Artamonov

De Divina Proportione

April 16, 2015

Bertrand Russell about Mathematics

Mathematics, rightly viewed, possesses not only truth, but supreme beauty â€“ a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.

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De Divina Proportione

April 16, 2015

Divine Proportion Introduction and Definition

Pic: Portrait of Luca Pacioli, traditionally attributed to Jacopo deâ€™Barbari, 1495 . Oleg Artamonov

De Divina Proportione

April 16, 2015

Divine Proportion Introduction and Definition

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

De Divina Proportione

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Divine Proportion Introduction and Definition

the whole is the longer part plus the shorter part

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

De Divina Proportione

April 16, 2015

Divine Proportion Introduction and Definition

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

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

De Divina Proportione

April 16, 2015

Divine Proportion Introduction and Definition

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

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

De Divina Proportione

April 16, 2015

Golden Ratio Introduction and Definition

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

Fig: Line segments in the golden ratio.

Fig: Approximate numerical expression of Ď•.

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De Divina Proportione

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Golden Ratio Introduction and Definition

ϕ =

a+b b 1 a = = 1+ = 1+ b a a ϕ

2

ϕ −ϕ−1 = 0 √ where

ϕ =

5+1 2

and √

Φ = ϕ

Oleg Artamonov

−1

= ϕ−1 =

=⇒

5−1 2

two roots :

−ϕ

−1

=⇒ √ 1± 5 , 2

√ 1− 5 = 1−ϕ = ; 2

is the golden ratio conjugate.

De Divina Proportione

April 16, 2015

Golden Ratio Expressions and Properties

Properties:

ϕ2 = ϕ + 1

√

ϕ =

1+ 5 ≈ 1.618034 2

1 = ϕ−1 ϕ

Important partition of unit:

0.382 Φ2 ≈ ≈ 0.618 0.618 Φ

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De Divina Proportione

1 = Φ+1 Φ

April 16, 2015

Golden Ratio Expressions and Properties

Numerical Expressions:

∞

r ϕ =

q √ 1 + 1 + 1 + ···

13 X (−1)(n+1) (2n + 1)! + 8 (n + 2)! n! 4(2n+3) n=0

1

ϕ = 1+ 1+

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ϕ =

s

1 1 1+ 1 + ···

De Divina Proportione

ϕ =

5+ 5−

√ √

5 5

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Golden Ratio Expressions and Properties

Relations to π: F ϕ = 1 + 2 sin

“π” 1 “π” = 10 2 sin 10

F ϕ = 2 cos F 2 sin

“π” 5

=

F

5 p 3−ϕ

„ ϕ = 2 sin

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“π”

3π 10

«

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April 16, 2015

Golden Ratio Relation to Fibonacci Numbers

Fig: Leonardo Bonacci, 1170 âˆ’ 1250 Oleg Artamonov

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

De Divina Proportione

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Golden Ratio Relation to Fibonacci Numbers

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 Ratio Relation to Fibonacci Numbers

lim

n→∞

Fn+1 = ϕ Fn

Another relations:

Fn =

ϕn − (−ϕ−n ) √ 5

ϕn = Fn ϕ + Fn−1

lim

n→∞

Fn+α = ϕα Fn

Fig: Series of golden rectangles and progressive approximation to golden ratio by dividing successive pairs of Fibonacci numbers.

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De Divina Proportione

April 16, 2015

Golden Ratio Geometric Constructions

Fig: Construction by compass and ruler of a line segment which length is Ď•.

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De Divina Proportione

Fig: Golden ratio caliper. April 16, 2015

Golden Ratio Geometric Constructions

Having a line segment AB, construct a perpendicular BC at point B, with BC half the length of AB. Draw the hypotenuse AC. Draw an arc with center C and radius BC. This arc intersects the hypotenuse AC at point D.

Fig: Construction by compass and straightedge that divides a line segment into two line segments where the ratio of the longer to the shorter line segment is Ď•.

Oleg Artamonov

De Divina Proportione

Draw an arc with center A and radius AD. This arc intersects the original line segment AB at point S. Point S divides the original segment AB into line segments AS and SB with lengths in the golden ratio.

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Golden Ratio Geometric Constructions

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

kACk

C. We have kBCk = kABk = Ď•.

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De Divina Proportione

April 16, 2015

Golden Figures

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

De Divina Proportione

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Golden Figures Golden Rectangle

a = Ď•. Fig: Golden rectangle, a+b = b a

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De Divina Proportione

Fig: Construction by compass and ruler.

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Golden Figures Golden Rectangle Division and Fibonacci Spiral

Fig: A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle; that is, with the same aspect ratio as the first. Square removal can be repeated infinitely. . . Fig: Corresponding corners of the squares form an infinite sequence of points on the golden spiral, the unique logarithmic spiral with this property.

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De Divina Proportione

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Golden Figures Golden Spiral

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De Divina Proportione

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Golden Figures Golden Rhombus

Fig: Golden Rhombus.

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De Divina Proportione

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

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Golden Figures Golden Triangle

“ ” “ ” 1 b 1 θ = 2 sin−1 2a = 2 sin−1 2ϕ = π, 5 q` ´ ` ´ q 2 2 h= b ϕ − 12 b = b ϕ2 − 14 = p √ = 12 b 5 + 2 5, three angles are in 2 : 2 : 1 Fig:

Fig: Golden triangle,

Oleg Artamonov

a b

= ϕ.

proportions.

De Divina Proportione

April 16, 2015

Golden Figures Golden Triangle

Fig: The triangles at the tips of a pentagram and obtained by dividing a decagon by connecting opposite vertices are golden triangles.

Oleg Artamonov

Fig: Golden triangle is bisected in Robinson triangles: a golden triangle and a golden gnomon ( AX = ÎŚ, three AC angles are in 1 : 1 : 3 proportion).

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Golden Figures Golden Spirals

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

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De Divina Proportione

April 16, 2015

Golden Figures Pentagon and Pentagram

Fig:

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diagonal side

=

b a

Fig: The four lengths are in golden ratio to one another: = Ď•.

red green blue = = = Ď•. green blue magenta

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Golden Figures Pentagon and Pentagram

Fig: Golden proportions in pentagon and pentagram.

Oleg Artamonov

Fig: The first printed illustration of a rhombicuboctahedron, by Leonardo da Vinci, published in â€?De Divina Proportioneâ€?.

De Divina Proportione

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

Oleg Artamonov

Fig1: 3 sides

Fig3: 5 sides leaves gaps

Fig2: 4 sides

Fig4: 6 sides De Divina Proportione

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Golden Figures Penrose Tiling. A surface can be completely tiled in an asymmetrical, non-repeating manner in five-fold symmetry with just two shapes based on Ď•.

Fig: Penrose tiling is a nonperiodic infinite tiling of a plane. It made from kites and darts; a kite is made from the golden triangle, and a dart is made from two gnomons. Oleg Artamonov

De Divina Proportione

Fig: Roger Penrose standing on a floor with a Penrose tiling. April 16, 2015

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

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

Fig4: The ratio of the two types of tiles in the resulting patterns is always ϕ!

Fig3: The other creates a set of diamond tiles like this.

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

De Divina Proportione

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Applications

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De Divina Proportione

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Applications Architecture

Fig: Melancolia, Albrecht DÂ¨ urer, 1514 (engraving). Oleg Artamonov

De Divina Proportione

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Applications Architecture

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

Fig: Great pyramid of Giza. Oleg Artamonov

Fig: A Kepler triangle is a right triangle formed by three squares with areas in geometric progression according to the golden ratio.

De Divina Proportione

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Applications Architecture

Fig: Hagia Sophia, Constantinople, 537.

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De Divina Proportione

Fig: St. Markâ€™s Basilica, Venice, 1117.

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Applications Architecture

Fig: CathÂ´ e drale Notre-Dame de Laon. Oleg Artamonov

De Divina Proportione

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Applications Architecture

Fig: The superimposed regulator lines show that the cathedral has golden proportions. Oleg Artamonov

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Applications Architecture

Fig: The southern facade of Notre-Dame de Paris. Oleg Artamonov

De Divina Proportione

Fig: The western facade illuminated at night. April 16, 2015

Applications Architecture

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

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De Divina Proportione

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Applications Architecture

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

De Divina Proportione

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Applications Architecture

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

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Applications Architecture

Fig: Alhambra, Granada, Andalusia, Spain. Oleg Artamonov

De Divina Proportione

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Applications Architecture

Fig: Le Corbusier sectioned his model human bodyâ€™s height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system â€“ an anthropometric scale of proportions. Fig: Le Corbusier and golden proportion on a Swiss bank-note. Oleg Artamonov

De Divina Proportione

April 16, 2015

Applications Architecture

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

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Applications Art

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

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Applications Art

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

De Divina Proportione

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Applications Art

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

Applications Art

Fig: The Virgin, Child, and St. Anne, Leonardo da Vinci, 1510. The golden ratio in this painting leads your eye from the baby then up through the two women, and then sets itself on the womenâ€™s face in perfect ratio. Oleg Artamonov

De Divina Proportione

April 16, 2015

Applications Art

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

De Divina Proportione

Fig: The face of the Mona Lisa outlines a golden rectangle.

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Applications Art

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

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Applications Art

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

De Divina Proportione

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Applications Art

Fig: Creation of Adam, Michelangelo, Sistine Chapelâ€™s ceiling, 1512. The finger of God touches the finger of Adam 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

Applications Art

Fig: In his La Grande Odalesque, the 19th century neoclassical artist Jean Auguste Dominique Ingres used a similar method for establishing the placement of the womanâ€™s hand. Oleg Artamonov

De Divina Proportione

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Applications Art

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

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De Divina Proportione

April 16, 2015

Applications Art Dali also incorporated in the painting a huge dodecahedron engulfing the supper table. The dodecahedron, which according 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

Applications Art

Fig: Dali framed his painting in a golden rectangle and also positioned the table exactly at the golden section of the height of his painting. He positioned the two disciples at Christâ€™s side at the golden sections of the width of the composition. Oleg Artamonov

De Divina Proportione

April 16, 2015

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

Fig: Living Still Life, Salvador Dali, 1956. Oleg Artamonov

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Applications Photography

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De Divina Proportione

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Applications Photography

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De Divina Proportione

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Applications Photography

Fig: Scene in the Ukrainian parliament.

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De Divina Proportione

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Applications Web Design

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

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

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Applications Forex

Fig: Golden mean gauge and marketsâ€™ behavior.

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Fig: Fibonacciâ€™s net is a powerful instrument of analysis in the foreign exchange market terminals.

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Applications Design

Fig: Automobiles and their logos design.

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De Divina Proportione

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Applications Design

Fig: Mercedes-Benzâ€™s logotype.

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De Divina Proportione

Fig: Hondaâ€™s logotype.

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Applications Design

Fig: Fibonacci spiral earrings. Fig: Red and Blue Chair designed by Gerrit Rietveld in 1917. Some works in the Dutch artistic movement called De Stijl, or neoplasticism, exhibit golden ratio proportions. Oleg Artamonov

De Divina Proportione

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Applications Design

Fig: Ď•, golden rectangles, and golden spiral in jewelleryâ€™s design.

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De Divina Proportione

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Applications Design

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De Divina Proportione

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Divine Proportion in Nature

Fig: The drawing of a manâ€™s body in a pentagram suggests relationships to the golden ratio. Fig: The golden spiral in nature. Oleg Artamonov

De Divina Proportione

April 16, 2015

Divine Proportion in Nature Plants

Fig: Even succulent plants like aloe display good approach to the golden spiral. Plants grow new cells in spirals, which is how this pattern appears. It works to the plants advantage by preventing new leaves from blocking older leaves access to sunlight, directing the maximum amount of rain and dew to the roots. Oleg Artamonov

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

De Divina Proportione

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Divine Proportion in Nature Plants

Fig: Aloe.

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Fig: Sago.

De Divina Proportione

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Divine Proportion in Nature Plants

Fig: It is the golden spiral in a repeating pattern at the center of a sunflower. Growing in this manner creates the most compact pattern possible with no gaps from beginning to end.

Oleg Artamonov

Fig: Just as with sunflowers, the pattern of seeds on a pine cone can be found in repeating in either clockwise or counter-clockwise motion.

De Divina Proportione

April 16, 2015

Divine Proportion in Nature Spiderwebs and Sea Shells

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

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De Divina Proportione

Fig: When cut in half, a nautilus shell displays its chambers and its spiral structure becomes even more apparent.

April 16, 2015

Divine Proportion in Nature Continent and Hurricane

Fig: Hurricane Isabel.

Fig: Africa.

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Divine Proportion in Nature Galaxies

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

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De Divina Proportione

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Divine Proportion in Nature Scale of Temperature Comfort and DNA

Fig: Ď• proportions are in a scale of temperature comfort. Fig: Golden ratio in DNA.

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April 16, 2015

Divine Proportion in Nature Fauna

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

Fig: Batterfly.

Fig: Eaglel. Fig: Ramâ€™s horn. Oleg Artamonov

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April 16, 2015

Divine Proportion in Nature Human Body

Fig: Vitruvian man. Golden ratio proportions of the human body.

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De Divina Proportione

April 16, 2015

Divine Proportion in Nature Human Body

The following sections of the modern human body are all in 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: Vitruvian man. Golden ratio proportions of the human body.

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De Divina Proportione

April 16, 2015

Divine Proportion in Nature Human Body

Fig: Roman copy of Doryphoros, originally by Polykleitos. Oleg Artamonov

De Divina Proportione

Fig: The golden ratio person. April 16, 2015

Divine Proportion in Nature Human Body

Fig: Manipulations with the proportion of Classical and Renaissance sculpturesâ€™ features by violating the golden ratio.

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De Divina Proportione

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Divine Proportion in Nature Human Body

Fig: Leonardo da Vinciâ€™s illustration of a human head from the Pacioliâ€™s book.

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De Divina Proportione

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Divine Proportion in Nature Human Body

Let us speak about women!

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

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Divine Proportion in Nature Human Body

Fig: Fibonacci spirals in a womanâ€™s face from the side.

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De Divina Proportione

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Divine Proportion in Nature Human Body

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

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De Divina Proportione

April 16, 2015

Divine Proportion in Nature Human Body

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

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De Divina Proportione

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Divine Proportion in Nature Human Body

Fig: Golden proportion and Hollywood actresses.

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De Divina Proportione

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Divine Proportion in Nature Human Body

Oleg Artamonov

De Divina Proportione

April 16, 2015

Divine Proportion in Nature Human Body

Golden ratio is not the last measure of beauty!

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

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De Divina Proportione

April 16, 2015

Divine Proportion in Nature Human Body

Fig: Studies on the proportions of the female body, Albrecht DÂ¨ urer, 1528. Oleg Artamonov

De Divina Proportione

Fig: Divine proportions. April 16, 2015

Divine Proportion in Nature Human Body

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

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De Divina Proportione

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Divine Proportion in Nature Human Body

Fig: Women provide applied mathematics with few more additional parameters. . .

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De Divina Proportione

April 16, 2015

Mathematics is a beautiful science!

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De Divina Proportione

April 16, 2015

thank you for your attention