A VizMath presentation featuring videos by Neil Currie on the golden ratio and by Rostom Kouyoumdjian on drawing with one point perspective. Illustrations of the use of math in art through the ages.
2. The Art of Math
Compiled by Carol Yeager
with original works by
Neil Currie
and
RostomKouyoumdjian
3. • Most images and copy are taken from Wikipedia and
resource listings are contained within the Wikipedia
descriptions.
• http://en.wikipedia.org/wiki/Mathematics_and_art
• Text is available under the Creative Commons Attribution-
ShareAlike License; additional terms may apply. See Terms
of Use for details.
Wikipedia® is a registered trademark of the Wikimedia
Foundation, Inc., a non-profit organization.
4. Mathematics and art
From Wikipedia, the free encyclopedia
Mathematics and art have a long historical relationship. The
ancient Egyptians and ancient Greeks knew about the
golden ratio, regarded as an aesthetically pleasing
ratio, and incorporated it into the design of monuments
including the Great Pyramid,[1] the Parthenon, the
Colosseum. There are many examples of artists who have
been inspired by mathematics and studied mathematics as
a means of complementing their works. The Greek sculptor
Polykleitos prescribed a series of mathematical proportions
for carving the ideal male nude. Renaissance painters
turned to mathematics and many, including Piero della
Francesca, became accomplished mathematicians
themselves.
5. Pyramid of Kufu
If we divide the slant height of the pyramid by half its base length, we get
a ratio of 1.619, less than 1% from the golden ratio. This would also
indicate that half the cross-section of the Khufu’s pyramid is in fact a
Kepler’s triangle. Debate has broken out between prominent
pyramidologists, including Temple Bell, Michael Rice, and John
Taylor, over whether the presence of the golden ratio in the pyramids is
due to design or chance.
6. Pyramidologists, Martin Gardner, Herbert Turnbull, and David Burton contend that:
Possible base:hypotenuse(b:a) ratios for the Pyramid of Khufu: 1:φ (Kepler’s Triangle), 3:5 (3-4-5
Triangle), and 1:4/π
Herodotus related in one passage that the Egyptian priests told him that the dimensions of the Great
Pyramid were so chosen that the area of a square whose side was the height of the great pyramid
equaled the area of the triangle.[7]
7. Great Mosque of Kairouan
The geometric technique of construction of the golden section seems to have determined the major
decisions of the spatial organisation. The golden section appears repeatedly in some part of the
building measurements. It is found in the overall proportion of the plan and in the dimensioning of the
prayer space, the court and the minaret. The existence of the golden section in some parts of Kairouan
mosque indicates that the elements designed and generated with this principle may have been realised
at the same period.[13]” Because of urban constraints, the mosque's floor plan is not a perfect rectangle.
Even so, for example, the division of the courtyard and prayer hall is almost a perfect golden ratio.
8. Polykleitos
Polykleitos gives us a mathematical approach towards sculpturing the
human body. The influence of the Canon of Polykleitos is immense both in
Classical Greek, Roman, and Renaissance sculpture, with many sculptors
after him following Polykleitos’ prescription. While none of Polykleitos’
original works survive, Roman copies of his works demonstrate and
embody his ideal of physical perfection and mathematical precision.
9. Renaissance
The Renaissance saw a rebirth of Classical Greek and
Roman culture and ideas, among them the study of
mathematics as a relevant subject needed to
understand nature and the arts. Two major reasons
drove Renaissance artists towards the pursuit of
mathematics. First, painters needed to figure out
how to depict three-dimensional scenes on a two-
dimensional canvas. Second, philosophers and artists
alike were convinced that mathematics was the true
essence of the physical world and that the entire
universe, including the arts, could be explained in
geometric terms.[17] In light of these
factors, Renaissance artists became some of the best
applied mathematicians of their times.
10. Pierodella Francesca
Piero della Francesca (c.1415-1492), an early Renaissance artist from
Italy, exemplified this new shift in Renaissance thinking. Though chiefly
appreciated for his art, he was an expert mathematician and geometer
and authored many books on solid geometry and the emerging field of
perspective
11. .
DaVinci
Woodcut from De DivinaProportione illustrating the golden ratio as applied to the human face.
Leonardo da Vinci (1452–1519) was an Italian
scientist, mathematician, engineer, inventor, anatomist, painter, sculptor, and architect. Leonardo has often been described
as the archetype of the Renaissance man.[37][38]
Renowned primarily as a painter, Leonardo incorporated many mathematical concepts into his artwork despite never
having received any formal mathematical training. It was not until the 1490s that he trained under Luca Pacioli and
prepared a series of drawings for De DivinaProportione. Leonardo studied Pacioli'sSumma, from which he copied tables of
proportions and multiplication tables.[39]
Notably in Mona Lisa and The Last Supper, Leonardo’s work incorporated the concept of linear perspective. By making all
of the lines in the painting converge on a single, invisible point on the horizon, a flat painting can appear to have depth. In
creating the vanishing point, Leonardo creates the illusion that the painting is an extension of the room itself. [40]
12. M. C. Escher
Circle Limit III by M.C. Escher (1959)
A renowned artist born in 1898 and died in 1972, M.C. Escher was known for his
mathematically inspired work.[49] Escher’s interest in tessellations, polyhedrons, shaping of
space, and self-reference manifested itself in his work throughout his career.[50] In the
Alhambra Sketch, Escher showed that art can be created with polygons or regular shapes such
as triangles, squares, and hexagons.
13. Salvador Dalí
Salvador Dalí (1904–1989) incorporated mathematical themes in several of his later works.
His 1954 painting Crucifixion (Corpus Hypercubus) depicts a crucified figure upon the net of
a hypercube.
14. Fractals
Main article: Fractal art
The processing power of modern computers allows mathematicians and
non-mathematicians to visualise complex mathematical objects such as
the Mandelbrot set. In the modern industry of computer
animation, fractals play a key role in modelling mountains, fire, trees and
other natural objects.
15. Platonic solids in art
The Platonic solids and other polyhedra are a recurring theme in Western art.
Examples include:
A marble mosaic featuring the small stellated dodecahedron, attributed to Paolo
Uccello, in the floor of the San Marco Basilica in Venice.[18]
Leonardo da Vinci's outstanding diagrams of regular polyhedra drawn as
illustrations for Luca Pacioli's book The Divine Proportion.[18]
A glass rhombicuboctahedron in Jacopo de Barbari's portrait of Pacioli, painted in
1495.[18]
A truncated polyhedron (and various other mathematical objects) which feature in
Albrecht Dürer's engraving Melancholia I.[18]
Salvador Dalí's painting The Last Supper in which Christ and his disciples are
pictured inside a giant dodecahedron.
16. Neil Currie, of Manchester, UK
discusses
The Golden Ratio in math and
artistic pursuits
17. Immediately following is
a film demonstrating one point
perspective drawing in an original
composition by the Armenian artist
living in Beirut …
RostomKouyoumdjian
18. Thank you for spending some time
with us in an overview of the Art of
Mathematics.
My special thanks to Neil Currie and
RostomKouyoumdjian who accepted the
challenge to show the use of maths in their
artistic pursuits
This passage, if true, would undeniably prove the intentional presence of the golden ratio in the pyramids. However, the validity of this assertion is found to be questionable.
His work on geometry influenced later mathematicians and artists, including Leonardo of Vinci
