Mathematics and art have a long historical relationship. The Golden ratio, Geometric patterns, Fractals are all fascinating mathematical ideas that have inspired artists and architects for centuries, I am just exploring these ideas in this presentation

1. Mathematics
in
Art,
Design &
Architecture
Numan Sheikh

2. MATHEMATICAL IDEAS THAT
CONTRIBUTE TO ART
Patterns, Symmetry , Tiling
Geometry & Islamic Art
Higher Order Geometry and Topology: Escher
Fractals
Fibonacci Numbers and Golden Ratio

3. Patterns: Tiling
• Occur in many settings
• Have a rich
mathematical structure
• Examples
– Using Regular Polygons
– Penrose Tiling
– Escher Tessellations

4. Patterns: Symmetry
• A mathematical operation, or transformation,
• Occurs in
– the sciences
– the arts
– Architecture
– Nature
– our everyday life.
• The term symmetry is used both in the arts
and in the sciences.
• In art
– often used as an aesthetic element
– a kind of balance in which the
corresponding parts are not
necessarily alike but only similar.
– Generally is a balance between
various parts of an object.
– Several examples of symmetry
in painting.

5. Symmetry: Architecture

6. Symmetry: African Culture
(Right) Kente Cloth
(Left) African Masks
(Top) Totem Pole

7. Symmetry: Manadalas
• Spiritual and ritual
symbol in Hinduism and
Buddhism
• Represents the Universe.
• The basic form of most
mandalas is a square with
four gates containing a
circle with a center point.

8. Geometry: Perspective
• originated in the
Renaissance
• changed the way
we represent the world.

9. Geometry: 3D
• The five Platonic solids and polyhedra
– inspired people throughout the ages
Tetrahedron
Faces are all
equilateral
triangles
Hexahedron
Faces are all
squares
Octahedron
Faces are all
equilateral
triangles
Dodecahedron
Faces are all
pentagons
20 vertices
Icosahedron
Faces are all
equilateral
triangles

10. GEOMETRY IN
ISLAMIC
ART &
ARCHITECTURE
Nothing in the World
Exceeds the Use of
Symmetry in the
Art and Architecture of the
Islamic World

11. Islamic Art
• Prohibition from making
representations of people in holy sites
• Developed an instantly recognizable
aesthetic based on
– Calligraphy
– Arabesque (vegetal, plants patterns)
– Geometrical shapes (repeated tiling)

12. Arabesque
• Symbolic representation of plants.
• Used in
– Tiles
– Ceramics
– Domes
– Carpets
– Miniature
Painting

13. Geometric Shapes
Repeated Tiling
• The mathematical elegance of these
designs is that no matter how
elaborate they are, they are always
based on grids constructed using
only a ruler and a pair of compasses.

14. Tiles in the Alhambra - Spain

15. Textile fragment (14th century) - Spain

16. Dome of the shrine of Shah Nematollah Vali in Mahan, Iran

17. Jali (pierced screen)
16th century
Mughal India
Used extensively in
Indian architecture as
•Windows
•room dividers
and railings around
•thrones
•Platforms
•Terraces
•balconies.

18. Dado panel
15th century
Mamluk, Egypt
•Penrose Tiling
•non-periodic tiling
•generated by an
aperiodic set of
prototiles.
•named after
mathematician and
physicist Roger Penrose
who investigated these
sets in the 1970s
•Use of Golden Ratio

19. M. C. ESCHER
DUTCH
GRAPHIC ARTIST
(1898 - 1972)
For me it remains an open
question whether [this work]
pertains to the realm of
mathematics or to that of art.
- M.C. Escher

20. M. C. Escher
• Obsessed with the depiction of infinity
• Fascinated with paradox and “impossible” figures
• Used an idea of Roger Penrose’s to develop many intriguing works of art.
• Escher’s work encompasses two broad areas:
– the geometry of space,
– the logic of space.
• Gödel, Escher, Bach (a Pulitzer Prize-winning book) explores the
relationships between the works of
– mathematician Kurt Gödel,
– artist Escher,
– and composer Johann Sebastian Bach

21. Escher: Tesselations
A tessellation is a covering of the plane by shapes, called tiles, so
that there are no empty spaces and no overlapped tiles.

22. Escher: The shape of space
Escher’s interest in shaping of space manifested itself in his work throughout his career.
It exemplifies his concern with the dimensionality of space, and with the mind’s ability to discern
three-dimensionality in a two-dimensional representation.

23. Escher: The logic of space
The work of Escher is rich in mathematical content.
Much of it is related to hyperbolic geometry.

24. Escher: Self-reference & Information
A central concept Escher captured is that of self-reference, which many believe lies near the
heart of the enigma of consciousness—and the brain’s ability to process information in a way
that no computer has yet mimicked successfully.

25. Mathematics in Art,
Design & Architecture
• Other beautiful applications
of geometry
– mazes and labyrinths
– kaleidoscopes,
– the fourth dimension and optical
illusions.

26. FRACTALS
Self Symmetric

27. Fractals
Infinitely complex patterns
that are self-similar across
different scales.
Mandelbrot set
Koch snowflake

28. Fractals in Nature

29. Fractal Art
• Mostly algorithmic art created by calculating
fractal objects and representing the results as
– still images
– Animations
– and other media
• Developed from the mid-1980s onwards.

30. Fractal
Art

31. Fractal
Architecture
•The mathematics of
fractals has been used to
show that the reason why
existing buildings have
universal appeal and are
visually satisfying is
because they provide the
viewer with a sense of
scale at different viewing
distances.
•In Hindu temples such as
the Virupaksha temple at
Hampi, the parts and the
whole have the same
character.

32. Fractal Architecture
European Cathedrals

33. Fractal Architecture
Eiffel Tower
The
Basic
Building
Block

34. Fractal Art in Islamic Architecture
The interior side view of the main dome of Selimiye Mosque in Edirne, Turkey, which
contains some self-similar patterns.

35. Fractals in African Architecture
Aerial view of a Ba-ila settlement in southern Zambia.

36. Fractal pattern in Ba-ila settlement.
Fractal generation of Ba-ila simulation.
- First iteration is similar to a single house
- Second to a family ring
- Third to village as whole

37. Fractals and African Cultures
• Such architectural fractals abound in African village structures
– some rectangular rather than circular
– some much more diffuse than coherent
• Fractal characteristics can also be seen in African:
– textiles, paintings, sculpture, masks, religious icons, cosmologies, and
social structures.
• Religious symbols include recursively nested calabashes, snakes of
“infinite length” coiled into a finite space, crosses-within-crosses-
within-crosses
• Even numeric systems and games in African can have fractal
characteristics.

38. THE GOLDEN
RATIO
A Mathematics Inspiration
Probably More Than Any Other

39. The Golden Ratio is
Also Known As:
The Golden Mean
The Golden Section
The Golden Rectangle
The Golden Number
The Golden Spiral
The Divine Proportion
ɸ: Greek Letter PHI
The Sacred Ratio

40. Golden Ratio
• In mathematics, 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.
where the Greek letter 𝝓
represents the golden
ratio. Its value is:

41. 8
5
21
13
1 1
23
1 1 2 3 5 8 13 21 34 55 89 …
1 2 1.5 1.66… 1.6 1.625 … … … … 1.618…
The Fibonacci Sequence

42. Golden Ratio: Other Formulations
The formula 𝜑 = 1 + 1/𝜑 can be used to expressed 𝜑 as a
continued fraction:
Other Formulations:
A pentagram colored
to distinguish its line
segments of different
lengths. The four
lengths are in golden
ratio to one another.

43. Golden Spiral &
Nature
Animate
Detail of
Aeonium
tabuliforme
showing the
multiple
spiral
arrangement
In 2010, the
journal Science
reported that the
golden ratio is
present at the
atomic scale in
the magnetic
resonance of
spins in cobalt
niobate crystals.

44. Golden Ratio in Logo Design
… and many more

45. Mozart’s sonatas tend to divide
in parts exactly at the Golden
Section of total time of the
work.
In Beethoven’s 5th Symphony
the opening motto is repeated
at exactly the Φ point through
the Symphony (Bar 372) and
also at the start of the
recapitulation 1-Φ of the way
through
Musical scales themselves are based on Fibonacci numbers
There are 13 notes in the span of any note through its octave. A scale is composed of 8
notes, of which the 5th and 3rd notes create the basic foundation of all chords, and are
based on whole tone which is 2 steps from the root tone that is the 1st note of the
scale.
In Music

46. GOLDEN RATIO
&
ARCHITECTURE
Phi (Φ) the Golden
Section, has been used by
mankind for centuries in
architecture.

47. Great Pyramid
of Egypt at Giza
• The Ahmes papyrus of Egypt gives
an account of the building of the
Great Pyramid of Giza in 4700 B.C.
with proportions according to a
sacred ratio.
MORE FUN FACTS
• In cubits (the first recorded unit of length), the pyramid’s perimeter
is 365.24 – the number of days in the year
• Pyramid’s perimeter divided by twice its height is equal to pi
(3.1416)
• King’s Chamber measurements are based on a Pythagorean triangle
(3, 4, 5)

48. The Parthenon
(447–432 BC)
• … temple built on the Acropolis
in the 5th century BC for
the Greek goddess Athena.
• It is the most important surviving building of Classical Greece.
• It appears to use golden ratio in some aspects of its design to
achieve beauty and balance its design.
• The extent of usage of 𝝓 however remains disputed amongst
researchers.
OTHER FUN FACTS
• The width to height ratio of 9:4 governs the vertical and horizontal
proportions of the temple as well as other relationships of the
building, for example the spacing between the columns.

49. Great Mosque of Kairouan (Tunisia)
• The oldest mosque in North Africa, built by Uqba ibn Nafi in 670 A.D.
• Dimensions reveals a very consistent application of the golden ratio in its
design.
(Floor plan)

50. Notre Dame, Paris
•Built in between 1163 and
1250 (Gothic era) appears
to have golden ratio
proportions in a number
of its key proportions of
design.

51. The Taj Mahal (completed in 1648)
Exquisite Symmetry, A Mathematical Property
Some claims that Divine Proportion was used in the construction of the Taj Mahal
Unfortunately no quantitative reference could be found, validating this claim

52. GOLDEN
RATIO &
ART

53. Arabic Calligraphy
• Multiple styles
– Kufi, oldest
– Naskh (Thulus, Riqa, Muhaqqaq)
– Other Regional (Nastaliq, Diwani, Sini)
• Evolved over ages
• The shape of each character is governed by
strict rules
• Often the proportions between the various
parts of the characters correspond to the
golden ratio

54. Arabic Calligraphy
• Ibn Muqla
(Baghdad, 885-940 AD)
• Invented a mathematically
proportioned cursive script, (‫)نسخ‬
al-khatt al-mansub
using the Golden Ratio
• This would enable subsequent
generations to practice the art of
calligraphy in a manner that was
both free and rational.

55. Michelangelo’s painting of
“The Creation of Adam”
on the ceiling of the Sistine Chapel

56. The Vetruvian Man
"(The Man in Action)"
by Leonardo Da Vinci
•From navel to foot is
1.6 times the distance
from navel to head.
•Each one set for the
head area, the torso,
and the legs.
•There are a lot more
golden-ratio elements
in this image.

57. The Golden Section was used extensively by Leonardo Da Vinci.
Note how all the key dimensions of the room, the table and ornamental shields in Da
Vinci’s “The Last Supper” were based on the Golden Ratio

58. MONALISA
by Leonardo Da Vinci
•More myth than reality
•No claims by DaVinci himself
•However…
•The width of her face is very
close to a golden ratio of the
width of the canvas.
•Her eye is rather precisely
aligned with the center of the
canvas.
•Golden ratio lines from the
center of the painting to the
sides of the canvas align nicely
with the width of her hair.
•Golden ratios in positioning of
her head, the garment neck line
and her arm.

59. SOME MORE
INTERESTING
MATHEMATICS IN
ARCHITECTURE
Its not Over Yet

60. Chichen Itza,
Mexico
• Built by the
Maya Civilization
• Fifty two panels
on each side of
the pyramid represent the number of years in the
Mayan cycle
• the stairways dividing the eighteen tiers correspond to
the Mayan calendar of eighteen months and the steps
within El Castillo mirror the solar year, with a total of
365 steps, one step for each day of the year.

61. Sagrada Familia
Barcelona, Spain
•Gaudi used
hyperbolic paraboloid
structures which is a
quadric surface, in this case
a saddle-shaped doubly-
ruled surface, that can be
represented by the equation
𝒛 = 𝒙𝟐/𝒂𝟐 – 𝒚𝟐/𝒃𝟐,
which can be seen within
particular façades.
•Passion façade has a Magic
Square an arrangement
where the numbers in all
columns, rows and diagonals
add up to the same sum: in
this case, 33.

62. Philips Pavilion,
Brussels, Belgium
•Commissioned by
electronics company
Philips.
•Intended to be used as a
venue to showcase
technological progress
after the Second World
War in Expo’58.
•The Pavilion was a mind-
boggling collection of
asymmetric hyperbolic
paraboloids and steel
tension cables.

63. References
• About Golden Ratio Being Myth
– Misconceptions about the Golden Ratio - George Markowsky
– An In-depth Investigation of the Divine Ratio - Birch Fett
– Golden Ratio Myth, Fact and Misunderstanding (for Debunkers)
http://www.goldennumber.net/golden-ratio-myth/
• Wikipedia Articles
– http://en.wikipedia.org/wiki/Mathematical_beauty
– http://en.wikipedia.org/wiki/Mathematics_and_art
– http://en.wikipedia.org/wiki/Mathematics_and_architecture
– http://en.wikipedia.org/wiki/Mathematics_and_fiber_arts
– http://en.wikipedia.org/wiki/Fourth_dimension_in_art
– http://en.wikipedia.org/wiki/Great_Mosque_of_Kairouan
– http://en.wikipedia.org/wiki/Music_and_mathematics
– http://en.wikipedia.org/wiki/Patterns_in_nature
– http://en.wikipedia.org/wiki/Mandala

64. References
• Fractals
– The Fractal Geometry of Nature - Mandelbrot (1982)
– Fractals in Architecture
http://classes.yale.edu/fractals/panorama/Architecture/Arch/Arch.html
– African Fractals - Eglash, R. (1999)
– Architecture without architects - Bernard Rudolfsky (1965)
– Low-Rise, High-Density Housing
http://tajvedelem.hu/Tankonyv/AI_en/AI_book.html
• Islamic Art
– Art of Islam, Language and Meaning - Titus Burckhardt (2009)
– The Use of the Golden Section in the Great Mosque of Kairouan - Kenza Boussora and Said Mazouz, Nexus
Network Journal (Spring 2004),
– A Golden age of Arab culture - The UNESCO Courier (Dec 1977)
– Leaving His Mark on an Ancient Art: Arabic Calligrapher Honda Kōichi
http://www.nippon.com/en/people/e00028/
– Ibn Muqla Ratio - Calligraphy Gallery
http://calligraphygallery.com/product/ibn-muqla-ratio/
• Other Architecture
– 9 Most Mathematically Interesting Buildings in the World
http://www.tripbase.com/blog/9-most-mathematically-interesting-buildings-in-the-world