The mystery novel
‘The Da Vinci Code’
revolves around a series of clues based on the works of the great artist and Renaissance Man Leonardo Da Vinci
The 1
stclue: 13 - 3 - 2 - 21 -1-1- 8 – 5 o draconian devil
oh lame saint
1-1-2-3-5-8-13-21
“This is it? All you did was put
the numbers in increasing order?”
“Exactly! …
The sequence of numbers you have in your hand happens to be one of the most famous mathematical progressions in history.”
- Sophie Neveau, cryptologist and one of the protagonists
(from: Da Vinci Code)
Leonardo Da Vinci
(1452 – 1519)
Leonardo Da Vinci was an artist whose works were heavily influenced by mathematics
We will see the role played by math in the arts – even in the concept of beauty through the ages
The Fibonacci Numbers
LEONARDO FIBONACCI
(1180-1250)
1 , 1 , 2 , 3, 5 , 8 , 13 , 21 , 34, 55, …
Start with the numbers 1, 1:
Each succeeding number is the sum of the two numbers before it
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
1/1 = 1 2/1 = 2 3/2 = 1.500 5/3 =1.666…
8/5 =1.600 13/8 =1.625 21/13 =1.615…
34/21 =1.619…
55/34 =1.618…
89/55 = 1.618…
The ratio between one number and its predecessor in the Fibonacci series approaches 1.618… as the numbers increase, and so on…
This limiting value (approximately 1.618…) of ratios of successive Fibonacci numbers is a number called the golden ratio (or divine proportion)
The Golden Ratio
The Golden Ratio
(phi) = 1.618033988749895… is an
irrational number (non-repeating, non- terminating decimal) , just like another famous irrational, (pi) = 3.14159265358979…
Shapes proportioned according to the golden ratio have long been considered aesthetically pleasing (especially in Ancient Greece). Many artists, including Da Vinci have incorporated this ratio in their works
How to get the golden ratio
Divide any line segment (say one with unit length 1) into two parts, in the following way:
The ratio of the whole (1) to the longer part (X) equals the ratio of the longer to the shorter part.
How to get the golden ratio
Divide any line segment (say one with unit length 1) into two parts, in the following way:
The ratio of the whole (1) to the longer part (X) equals the ratio of the longer to the shorter part.
x 1 - x
= 1 = x x 1 - x
Then solve the equation:
The value of is 1.618… (curiously, the value of x is 0.618…).
The Golden Rectangle
The golden rectangle is a rectangle whose sides are in the proportion of the golden ratio
How to draw a golden rectangle
Start with any base, and get the midpoint of the base.
Draw an arc whose radius is the length of the midpoint to a corner of the rectangle
A
C
B
D
Form a rectangle whose corners are determined by the tip of the arc. The big rectangle formed is a golden rectangle
A
C
B
D
GOLDEN TRIANGLE
A
B C
AB:BC = The Golden Ratio and the
Golden Rectangle in Art
The Vitruvian Man by Da Vinci Mona Lisa
Mondrian
Bathers by Seurat (1859-1891)
Pyramids of Giza
The ratio of the slant height of the pyramid (hypotenuse of the triangle) to the distance from ground center (half the base dimension) is 1.61804…
the sides of the right triangle are in the proportion 1:sqrt(phi):phi and the pyramid has a height of sqrt(phi)
The Golden Proportion in Ancient Egypt
The Parthenon (Acropolis, Athens)
Chartres Cathedral and the
Seattle Tower (Space Needle) Le Corbusier’s designs
United Nations Building
Apart from the golden ratio, there are other elements of mathematics that have influenced artists through the ages.
For example, the basic shapes of the triangle (3) and the pentagon or pentagram (5) – which are Fibonacci numbers, were instrumental in many works of important artists
Michelangelo’s Holy Family
Raphael’s Crucifixion
Concepts of beauty vary from individual to individual and across cultures, and also change through time. Perhaps one reason why so many people found the golden rectangle and the golden ratio pleasing and natural is because these objects occur in nature …
The Fibonacci Sequence and the
Golden Ratio: Nature’s Numbers
Plants illustrate the Fibonacci numbers:
Numbers of leaves
Arrangements of leaves around stem
Positioning of leaves, sections, seeds
Spiralling
Branching
5 PETALS 8 PETALS
Head of a Giant Sunflower showing curves of which the seeds are planted. (from Colman, S. and Coan, C.A.
Proportional Form, New York, Putnam 1920).
34 spirals to left – 55 spirals to right 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 – FIBONACCI SEQUENCE
Fibonacci numbers in plant sections
Bananas have 3 Apples have 5
Many flowers have petals that total a number in, or very close to, the Fibonacci series:
Fibonacci numbers in flower petals
.
Each successive level of branches is often based on a progression through the Fibonacci series.
Fibonacci nos. in plant branching
There are 13 radial lines in the web of this species of spider.: “A History of Geometrical Methods.” Coolidge, J.L. New York: Dover Publications Inc.
The eyes, beak, wing and key body markings of the penguin all fall at golden sections of its height.
The spiral growth of sea shells provide a simple, but beautiful, example . . .
The eye-like markings of this moth fall at golden sections of the lines that mark its width and length.
Every key body feature of the angel fish falls at golden sections of its width and length.
The eye, fins and tail all fall at golden sections of the length of a dolphin's body.
The DNA spiral is a golden section
The DNA molecule, the program for all life, is based on the golden section. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.
34 and 21, of course, are numbers in the Fibonacci series and their ratio, 1.6190476 closely approximates phi, 1.6180339.
DNA in the cell appears as a double-stranded helix referred to as B-DNA.
This form of DNA has a two groove in its spirals, with a ratio of phi in the proportion of the major groove to the minor groove, or
roughly 21 angstroms to 13 angstroms.
Saturn's rings are divided at two phi points The Cassini division in the rings of Saturn falls at the Golden Section of the width of the ring.
A closer look at Saturn's rings reveals a darker inner ring which exhibits the same golden section proportion as the brighter outer ring.
Where else can we find the golden ratio?
The Human Form
To the ancient Greeks, the human form which exhibited the golden ratio was considered the most beautiful
Michaelangelo’s statue, David.
The great creations of the Greek sculptors are considered as the standards of human beauty and the samples of a harmonic body.
In those works the principle of the Golden Proportion are used.
The figure of the young David expresses the unity of beauty and valor underlying the Greek art principles. In the same way Afrodita's sculpture created by Agesander is considered to be the masterpiece of woman's beauty.
The ancient Egyptians also valued the golden proportion in the design of their sarcophagus and other structures
The Vitruvian Man, by Leonardo da Vinci
The Vitruvian Man, Redux
In the face
a : b
ab
Where can you find the golden ratio?
a = From head to chin b = Width of face
a = longer part, b = shorter part
In the face ( a = longer, b = shorter)
b a
Where can you find the golden ratio?
(Nose-chin): (Mouth-chin)= a : b
In the body
a : b
ab
Where can you find the golden ratio?
a = From head to sole
b = From waist to sole
In the body
a : b
a bWhere can you find the golden ratio?
a = From head to waist b = From head to neck
In the body a : b
b
a
Where can you find the golden ratio?
a = From shoulder to fingertip b = From shoulder to
elbow
In the body
a : b
ba
Where can you find the golden ratio?
a = From shoulder to fingertip
b =
Fromshoulder to elbow
In cows and bovines
a : b
a
b
Where can you find the golden ratio?
a = Snout to tail
b = Neck to tail
b = distance ofconsecutive spikes
Where can you find the golden ratio?
In the Starfish a : b
a = distance of alternate spikes
b
Are your
proportions also
divine?
Curiously enough, you also have 2 hands, each with 5 digits, and your 8 fingers are each comprised of 3 sections. All Fibonacci numbers!
The ratio of your forearm to hand is Phi
Your hand creates a golden section in relation to your arm, as the ratio of your forearm to your hand is also 1.618, the Divine Proportion.
THE MARQUARDT
MASK:
•Decagons and pentagons use the divine proportion in all their dimensions
•The more a face fits the mask, the more beautiful it is.
“All life is biology.
All biology is physiology.
All physiology is chemistry.
All chemistry is physics.
All physics is math."
- Dr. Stephen Marquardt
Other influences of math in art
Luca Pacioli's (1445-1514)
Golden Secret
Lorenzetti, The Presentation in the Temple, c1342
Notice how the tiles get smaller
Giotto, The Flight into Egypt, c1313
Notice how the trees are the same size
Perspectivity and the Development of Projective Geometry
CD=distance from picture to eye
Where’s the best view point?
174cm above, 770cm away
Masaccio, Trinity, 1427
One of the first perspective pictures
Leonardo, The Last Supper, c1497
The perspective is an integral part of the painting
Dürer at the Singapore Art Museum
Dürer, 1527
Anamorphic art
Holbein, The Ambassadors, 1533
False viewpoints
Pozzo’s ceiling (1694) and cupola (1685) in St.
Ignatius, Rome
Cubism: interpreting other dimensions
The Sacrament of The Last Supper by Salvador Dali (1904-1989)
SYMMETRY:
a fundamental object of study in mathematics
Patterns in Islamic art
Fez, Morocco, 1325
Polygons and polygrams Patterns in Islamic art
Isfahan, Iran, end of 15th century
What is an isometry?
An isometry of the plane is a mapping that preserves distance, and therefore shape
Four types of plane isometries
Translation
Reflections
Rotations
Glide reflections
Translation
A translation moves a fixed distance in a fixed direction
Reflection
A reflection flips across an axis of reflection
Rotation
A rotation has a centre of rotation and an angle of rotation
Rotational symmetry
6 60°
120°
3
180°
2
Symmetry Regions Figure
Angle of Rotation Order of Rotation
Glide reflection
A glide reflection is a combination of a reflection and a translation
Symmetric patterns
A plane pattern has a symmetry if there is an isometry of the plane that preserves it.
There are three types of symmetric patterns.
Rosette patterns (finite designs)
Frieze patterns
Wallpaper patterns
Sumerian symmetry Examples of rosette patterns
Frieze patterns
Frieze patterns are patterns that have translational symmetry in one direction
We imagine that they go on to infinity in both directions or wrap around
Frieze patterns on cloth
Wallpaper floor tilings Wallpaper cloth
The 17 types of wall paper groups
M.C. Escher
Escher: Angels and Devils
Fractals: modern math and modern art
Guess what this is?
There’s a lot more math in literature, music, dance …
Lewis Carrol was a mathematics professor
Mathematics as a Creative Discipline
“Good mathematics must be both beautiful and serious.
Its beauty is derived from its precision, elegance of its results and proofs, and patterns – which, like the painter’s or the poets - is capable of stirring emotions…
There is no place in the world for ugly mathematics.”
- G. H. Hardy
Is math the cause, or the effect?
Is math a catalyst, or a coincidence?
Is math created or is it discovered?
Outline
1.
What’s math got to do with art?
The Da Vinci Code and Fibonacci Numbers
Fibonacci Numbers, the Golden Ratio and the Golden Rectangle
Math in nature
Math in modern art, literature, music
2.