INTRODUCTION
For most of the
people, mathematics is a school course with symbols and rules, which they
suffer understanding and find difficult throughout their educational life, and
even they find it useless in their daily life. Mathematics and arts, in
general, are considered and put apart from each other. Mathematics represents
truth, while art represents beauty. Mathematics has theories and proofs, but
art depends on personal thought. Even though mathematics is a combination of
symbols and rules, it has strong effects on daily life and art.
It is obvious that those people
dealing with strict rules of positive sciences only but nothing else suffer
from the lack of emotion. Art has no meaning for them. On the other hand,
artists unaware of the rules controlling the whole universe are living in a
utopic world. However, a man should be aware both of the real world and
emotional world in order to become happy in his life. In order to achieve that
he needs two things: mathematics and art. Mathematics is the reflection of the
physical world in human mind. Art is a mirror of human spirit. While
mathematics showing physical world, art reveals inner world.
For some people, mathematics
and arts are two different field of study which have no relations at all. In
reality, this is not so. Mathematics has aesthetic, and art in itself and it
has interrelation with the different fields or arts such as architecture,
painting, poetry and music, dance, sculpture and even in nature. While
investigating the relation between mathematics and art, we frequently face with
some mathematical concepts such as Golden Ratio, Fibonacci sequence, and
Pythagoras Comma etc. Then it is not wrong to say that they have strong
relations since they both have great effects on human beings. In this study,
the relation between art and mathematics are shown by the examples of, Golden
Ratio, Fibonacci sequence, and music.
In art, mathematics is not
always visible, unless we are looking for it. But there is much symmetry,
geometry, and measurement involved in creating beautiful art. As well,
many artists take advantage of mathematical findings, such as the golden ratio
to make their artwork realistic and beautiful. Angles and perspective can
also be described using math. Perhaps math and art are quite intricately
linked.
Mathematics and art have a long
historical relationship. Artists have used mathematics since the 5th century BC when the Greek
sculptor Polykleitos wrote his Canon, prescribing proportions based
on the ratio 1:√2 for the ideal male nude. Persistent popular claims have been
made for the use of the golden ratio in ancient art
and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the
influential treatise De DivinaProportione (1509),
illustrated with woodcuts by Leonardo da Vinci, on the use of the
golden ratio in art. Another Italian painter, Pierodella Francesca, developed Euclid's ideas on perspective in treatises such as De ProspectivaPingendi, and in his
paintings. In modern times, the graphic artist M. C. Escher made intensive
use of tessellation and hyperbolic geometry. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well
as kilim.
Mathematics produces art
At the most practical level,
mathematical tools have always been used in an essential way in the creation of
art. Since ancient times, the lowly compass and straightedge, augmented by
other simple draftsmen's and craftsmen's tools, have been used to create
beautiful designs realized in the architecture and decoration of palaces,
cathedrals, and mosques. The intricate Moorish tessellations in tile, brick,
and stucco that adorn their buildings and the equally intricate tracery of
Gothic windows and interiors are a testament to the imaginative use of ancient
geometric knowledge.
Today's mathematical tools
are more sophisticated, with digital technology fast becoming a primary choice.
In the hands of an artist, computers can produce art, powered by unseen complex
internal mathematical processes that provide their magical abilities.
Mathematical transformations provide the means by which an image or form in one
surface or space is represented in another. Art is illusion, and
transformations are important in creating illusion. Isometries, similarities,
and affine transformations can transform images exactly or with purposeful
distortion, projections can represent three (and higher)-dimensional forms on
two-dimensional picture surfaces, even curved ones. Special transformations can
distort or unscramble a distorted image, producing anamorphic art. All these
transformations can be mathematically described, and the use of guiding grids
to assist in performing these transformations has been replaced today largely
by computer software. Compasses, rulers, grids, mechanical devices, keyboard
and mouse are physical tools for the creation of art, but without the power of
mathematical relationships and processes these tools would have little creative
power.
Pattern is a fundamental concept in both
mathematics and art. Mathematical patterns can generate artistic patterns.
Often a coloring algorithm can produce "automatic art" that may be as
surprising or aesthetically pleasing as that produced by a human hand. Other fractals, as well as images based on
attractors, are also produced by iteration and coloring according to rules. The
intricacy of these images, their symmetries, and the endless (in theory)
continuance of the designs on ever-smaller scales, makes them spellbinding.
Recursive algorithms
applied to geometric figures can generate attractive self-similar patterns. Transformations and symmetry are also fundamental
concepts in both mathematics and art. Mathematicians actually define symmetry
of objects (functions, matrices, designs or forms on surfaces or in space) by
their invariance under a group of transformations. Conversely, the application
of a group of transformations to simple designs or spatial objects
automatically generates beautifully symmetric patterns and forms. In 1816,
Brewster's newly-invented kaleidoscope demonstrated the power of the laws of
reflection in automatically generating eye-catching rosettes from jumbles of
colored shards between two mirrors.
Art
illuminates mathematics
When mathematical patterns
or processes automatically generate art, a surprising reverse effect can occur:
the art often illuminates the mathematics. Who could have guessed the
mathematical nuggets that might otherwise be hidden in a torrent of symbolic or
numerical information? The process of coloring allows the information to take
on a visual shape that provides identity and recognition. Who could guess the
limiting shape or the symmetry of an algorithmically produced fractal? With
visual representation, the mathematician can exclaim "now I see!"
Since periodic tessellations can be generated by groups of isometries, they can
be used to illuminate abstract mathematical concepts in group theory that many
find difficult to grasp in symbolic form: generators, cosets, stabilizer
subgroups, normal subgroups, conjugates, orbits, and group extensions, to name
a few.
In the examples above,
illumination of mathematics is a serendipitous outcome of art created for other
reasons. But there are examples in which the artist'smain purpose is to
express, even embody mathematics. Several prints by M.C. Escher are the result
of his attempts to visually express such mathematical concepts as infinity,
duality, dimension, recursion, topological morphing, and self-similarity.
Perhaps the most striking examples of art illuminating mathematics are provided
by the paintings of Crockett Johnson and the sculptures of Helaman Ferguson.
From 1965 to 1975, Johnson produced over 100 abstract oil paintings, each a
representation of a mathematical theorem. Ferguson's sculptures celebrate
mathematical form, and have been termed "theorems in bronze and
stone." Each begins with the idea of capturing the essence of a
mathematical theorem or relationship, and is executed by harnessing the full
power of mathematically-driven and hand-guided tools.
Mathematics inspires art
Patterns, designs, and forms
that are the "automatic" product of purely mathematical processes
(such as those described in "Mathematics generates art") are usually
too precise, too symmetrical, too mechanical, or too repetitive to hold the art
viewer's attention. They can be pleasing and interesting, and are fun to create
(and provide much "hobby-art") but are mostly devoid of the subtlety,
spontaneity, and deviation from precision that artistic intuition and
creativity provide. In the hands of an artist, mathematically-produced art is
only a beginning, a skeleton or a template to which the artist brings imagination,
training, and a personal vision that can transform the mathematically perfect
to an image or form that is truly inspired.
Pure mathematical form, often
with high symmetry, is the inspiration for several sculptors who create
lyrical, breathtaking works. With practiced eye and hand, relying on their
experience with wood, stone, bronze, and other tactile materials, the artists
deviate, exaggerate, subtract, overlay, surround, or otherwise change the form
into something new, often dazzlingly beautiful. With the advent of digital
tools to create sculpture, the possibilities of experimentation without
destruction of material or of producing otherwise impossible forms infinitely
extends the sculptor's abilities.
Mathematics
constrains art
We often hear of "artistic
freedom" or "artistic license,"which imply the rejection of
rules in order to have freedom of expression. Yet many mathematical constraints
cannot be rejected; artists ignorant of these constraints may labor to realize
an idea only to find that its realization is, indeed, impossible. Euler's
theorem (v + f = e + 2) and Descartes' theorem (the sum of the vertex defects of
every convex polyhedron is 720°) govern the geometry of polyhedra. Other
theorems govern the topology of knots and surfaces, aspects of symmetry and
periodicity on surfaces and in space, facts of ratio, proportion, and
similarity, the necessity for convergence of parallel lines to a point, and so
on. Rather than confining art or requiring art to conform to a narrow set of
rules, an understanding of essential mathematical constraints frees artists to
use their full intuition and creativity within the constraints, even to push
the boundaries of those constraints. Constraints need not be negative -they can
show the often limitless realm of the possible.
Voluntary mathematical
constraints can serve to guide artistic creation. Proportion has always been
fundamental in the aesthetic of art, guiding composition, design, and form.
Mathematically, this translates into the observance of ratios. Whether these be
canons of human proportion, architectural design, or even symbols and letter
fonts, ratios connect parts of a design to the whole, and to each other. Repeated
ratios imply self-similarity, hardly a new topic despite its recent
mathematical attention.
OBJECTIVE
The aim of this study is to
develop into the relationship between mathematics and art, and prove that they
are more linked than they seem. Talking about the relation between mathematics and
arts, we frequently coincide some mathematical facts such as golden ratio,
symmetry, Fibonacci sequence, etc. In this study, I will try to investigate the
relation between mathematics and art and these concepts in detail. That is this
project is aim to study the applications of mathematics in art. This study also
helps us to know how math is used in the development of the art and to know how
math directly and indirectly affect art.
HYPOTHESIS
At first glance, mathematics and the arts seem to have nothing
in common. Mathematics is above all logical and about calculations and proofs,
whereas the arts are more emotional and creative. However, once you scratch the
surface of each, you find they are linked in more ways than you would imagine. Project
was stated on the assumption that mathematics has wide application in art. We assume that there should be a
connection with math and art. That is both art and math are interrelated.
METHODOLOGY
The history of the study of
mathematics and art is interrelated, so it is only natural to begin this
project briefly outlining this relationship. Then the report examines the
relationship between mathematics and art from different points of view. Then the study is concentrated on different
mathematical concepts related to various art forms and also collects the
relationship between math and art. Finally the report will conclude with the
analysis of the artistic aspects of mathematics, a subject traditionally deemed
to be a science. Collected examples shows application
of mathematics in different fields. Then make an analysis based on the
collected examples.
DATA COLLECTION
Mathematics and art are related in a variety of ways.
Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in
arts such as music, dance, painting, architecture, sculpture, and textiles. Art and Math may at first seem to
be very differing things, but people who enjoy math tend to look for
mathematics in art. They want to see the patterns and angles and lines of
perspective. This is why artists like M.C. Escher appeal to mathematicians
so much. There is a large amount of math involved in art, not to mention
basic things like measuring and lines, but the intricacies of art can often be
described using math.
Some of the mathematical concepts that
connect math with art
For some people, mathematics
and arts are two different field of study which have no relations at all. In
reality, this is not so. Mathematics has aesthetic, and art in itself and it
has interrelation with the different fields or arts such as architecture,
painting, poetry and music, and even in nature. While investigating the
relation between mathematics and art, we frequently face with some
mathematical concepts such as Golden Ratio, Fibonacci sequence, and Pythagoras
Comma etc.
GOLDEN RATIO
The golden ratio, indicated by φ, is a geometric and numeric ratio
of the parts of a whole that gives the most appropriate dimensions. It can be
also called golden mean, divine proportion or golden section. Two objects are
said to be in golden ratio if the sum of two objects to the larger one equals
the ratio of the larger one to the smaller one. This can be simply explained on
a line as explained in Figure below:
Figure: Golden Ratio on a line.
This explanation of golden
ratio was first given by Euclid ca. in 300 BC. Euclid stated that the total
length of a line divided by the longer part is equal to longer part divided by
shorter part. Thus, a perfect division is generated. This ratio called golden
ratio and is indicated by the Greek letter φ. The reciprocal of golden ratio φ is called “phi” and for larger quantities of golden ratio is
defined by “Phi”.
When we use the same logic on a two dimensional objects such as a
rectangle with dimensions of φ and
1, we get the rectangles
with ratios close to the golden ratio; that is to say, rectangles for which the
height divided by the width (or vice versa) results in a number close to the
golden ratio of φ. such rectangles are called golden rectangle. Successive points
dividing a golden rectangle into squares and smaller rectangle as drawn on
golden rectangle gives a logarithmic spiral that is mostly called the golden
spiral, as shown on Figure below
Figure:
Golden Spiral.
Golden ratio in art
The great German
mathematician and astronomer Johannes Kepler described the golden number as one
of the “two great treasures of geometry.” (The other is the Theorem of
Pythagoras.) Golden number appears in many basic geometric constructions and
applications: The golden ratio, roughly equal to 1.618. The golden ratio has persistently
been claimed in modern times to have been used in art and architecture by the ancients in Egypt, Greece
and elsewhere, without reliable evidence. The claim may derive from confusion
with "golden mean", which to the Ancient Greeks meant "avoidance
of excess in either direction", not a ratio. Pyramidologists since the nineteenth century have argued on dubious
mathematical grounds for the golden ratio in pyramid design. The 5th century
BCE temple, the Parthenon in Athens, has been claimed to use
the golden ratio in its façade and floor plan, but these claims too are disproved by
measurement. The Great Mosque of
Kairouan in
Tunisia has similarly been claimed to use the golden ratio in its design, but
the ratio does not appear in the original parts of the mosque.
FIBONACCI SEQUENCE
Fibonacci numbers are easy
to define, and are the numbers of a sequence discovered by Leonardo da Piza
(otherwise known as Fibonacci). Its first two elements are both 1, and then
each subsequent element is formed by adding the previous two together. Or in other words the sequence of numbers {Fn}defined
as The Fibonacci numbers by the linear recurrence equation,
Fn
= Fn-1 + Fn-2
With F1 = F2 = 1 and as a result of the
definition, it is conventional to define F0 = 0. Thus, the first few
elements are 1, 1, 2, 3, 5, 8, 13,…..
The sum of the
squares of any series of Fibonacci numbers is equal the last Fibonacci number
which is used in the series times the next Fibonacci number. Thus, based on the
progression below and properties of the Fibonacci series, the result is
Fibonacci spiral:
12 + 12 + 22 + 32 + 52 = 5 x 8
Giving the general
formula
12 + 12 + . . . + Fn2 =
Fn x F(n+1)
Although Golden Spiral is based on a series of
identically proportioned golden rectangles, each having a golden ratio of 1.618
of the length of the long side to that of the short side of the rectangle,
Fibonacci spiral is very similar to Golden Spiral. The Fibonacci spiral is
slightly different than the Golden Spiral for the smaller values of n. As n
increases, Fibonacci Spiral gets closer and closer to a Golden Spiral because
of the ratio of each number in the Fibonacci series to the one before it
converges on Phi, 1.618, as the series progresses (e.g., 1, 1, 2, 3, 5, 8 and
13 produce ratios of 1, 2, 1.5, 1.67, 1.6 and 1.625, respectively). ... It can be found in many
patterns in nature, such as flowers and pine cones, and has been used in many
paintings and other works of art.
Fibonacci spirals and Golden spirals appear in nature on many creatures.
But not every spiral in nature is related to Fibonacci numbers or Golden
Number.
Human ear has a shape of Golden
Spiral and Golden Rectangle .On the other hand, the Nautilus shell which can be
found on many parts of the world is often shown as an illustration of the
golden ratio in nature, but the spiral of a nautilus shell is not a golden
spiral, as the following figure shows.
The following figure shows that the growth of Daisy Flower obeys the
Fibonacci Numbers. There are many examples of Fibonacci Spirals in
nature. Sunflowers locate their seeds in Fibonacci Spirals and reveals
Fibonacci Numbers.Pinecones and pineapples show the same
construction as sunflower. The numbers of seeds are arranged in Fibonacci
numbers. If you count the seeds of pinecone shown on the above figure, you will
find that the numbers of seeds are 13 on clockwise and 8 in counterclockwise, which
are Fibonacci Numbers.
Symmetries
in art
Planar symmetries have for millennia been exploited in artworks such as carpets, lattices, textiles and tilings. Many traditional rugs,
whether pile carpets or flatweavekilims, are divided into a central field
and a framing border; both can have symmetries, though in handwoven carpets
these are often slightly broken by small details, variations of pattern and
shifts in colour introduced by the weaver. In kilims from Anatolia, the motifs used are themselves usually
symmetrical. The general layout, too, is usually present, with arrangements
such as stripes, stripes alternating with rows of motifs, and packed arrays of
roughly hexagonal motifs. Weavers certainly had the intention of symmetry,
without explicit knowledge of its mathematics.
Symmetries are prominent in textile arts including quilting,knitting,cross-stitch, crochet, embroidery and weaving.Rotational symmetry is found in circular structures
such as domes; these are sometimes elaborately decorated with symmetric
patterns inside and out, as at the 1619 Sheikh Lotfollah
Mosque in Isfahan. Items of embroidery and lace work such as tablecloths and table
mats, made using bobbins or by tatting, can have a wide variety of reflectional and rotational
symmetries which are being explored mathematically.
Islamic artexploits symmetries in many of its artforms, notably in
girihtilings. These are formed using a set of five tile shapes,
namely a regular decagon, an elongated hexagon, a bow tie, a rhombus, and a
regular pentagon. All the sides of these tiles have the same length; and all
their angles are multiples of 36° (π/5 radians), offering fivefold and tenfold symmetries. The tiles are
decorated with strapwork lines (girih), generally more
visible than the tile boundaries. Elaborate geometric zelligetilework is a distinctive element in Moroccan architecture. Muqarnas vaults are three-dimensional but were designed in two
dimensions with drawings of geometrical cells.
Polyhedra
in art
The Platonic solids and other polyhedra are a recurring theme in Western art. They are found, for
instance, in a marble mosaic featuring the small stellated
dodecahedron,
attributed to Paolo Uccello, in the floor of the San Marco Basilica in Venice; in Leonardo da Vinci's
diagrams of regular polyhedra drawn as illustrations for Luca Pacioli's book The Divine
Proportion;[12] as a glass rhombicuboctahedron in Jacopo de Barbari's portrait of
Pacioli, painted in 1495; in the truncated polyhedron (and various other
mathematical objects) in Albrecht Dürer's engraving Melencolia I; and in Salvador Dalí's painting The
Last Supper in which Christ and his disciples are pictured inside a
giant dodecahedron.
Albrecht Dürer (1471–1528) was a GermanRenaissanceprintmaker who made important contributions to polyhedral literature
in his book, Underweysung der Messung
(Education on Measurement) (1525), meant to teach the subjects of linear perspective, geometry in architecture, Platonic solids, and regular polygons. While the examples of perspective in Underweysung der Messung are
underdeveloped and contain inaccuracies, there is a detailed discussion of
polyhedra. Dürer is also the first to introduce in text the idea of polyhedral nets, polyhedra unfolded to lie flat for
printing.
The mathematics of tessellation, polyhedra, shaping of space, and self-reference provided
the graphic artist M. C. Escher (1898—1972) with a lifetime's worth
of materials for his woodcuts. In the Alhambra
Sketch, Escher showed that art can be created with polygons or regular
shapes such as triangles, squares, and hexagons. Escher used irregular polygons
when tiling the plane and often used reflections, glide reflections, and translations to obtain further patterns.
Some of Escher's many
tessellation drawings were inspired by conversations with the mathematician H. S. M. Coxeter on hyperbolic geometry. Escher was especially interested
in five specific polyhedra, which appear many times in his work. The Platonic solids—tetrahedrons, cubes, octahedrons, dodecahedrons, and
icosahedrons—are especially prominent in Order
and Chaos and Four Regular
Solids.
These stellated figures often reside within another figure which further
distorts the viewing angle and conformation of the polyhedrons and provides a
multifaceted perspective artwork.
The visual intricacy of
mathematical structures such as tessellations and polyhedra has inspired a
variety of mathematical artworks. Stewart Coffin makes polyhedral puzzles in rare and beautiful woods; George W. Hart works on the theory of polyhedra and sculpts objects inspired by them; Magnus Wenninger makes "especially beautiful" models of complex
stellated polyhedra.[131]
Topology in art
The mathematics of topology has inspired several artists in modern times. The sculptor John Robinson (1935–2007) created works such as Gordian Knot and Bands of Friendship, displaying knot theory in polished bronze. Other works by Robinson explore the
topology of toruses. Genesis is
based on Borromean rings – a set of three circles, no two of
which link but in which the whole structure cannot be taken apart without
breaking. The sculptor Helaman Ferguson creates complex surfaces and other topological objects. His works are visual representations of mathematical
objects; The Eightfold Way is
based on the projective
special linear groupPSL(2,7), a finite group of 168 elements. The sculptor Bathsheba Grossman similarly bases her work on
mathematical structures. A liberal arts inquiry project examines
connectionsbetween mathematics and art through the Möbius strip, flexagons, origami and panorama photography. Mathematical objects including the Lorenz manifold and the hyperbolic plane have been crafted using fiber arts including crochet.
Geometry
in art
We know that line, shape,
form, pattern, symmetry, scale, and proportion are the building blocks of both
art and math. Geometry offers the most obvious connection between the two
disciplines. Both art and math involve drawing and the use of shapes and forms,
as well as an understanding of spatial concepts, two and three dimensions,
measurement, estimation, and pattern. Many of these concepts are evident in an
artwork’s composition, how the artist uses the elements of art and applies the
principles of design. Problem-solving skills such as visualization and spatial
reasoning are also important for artists and professionals in math, science,
and technology. By taking an inter disciplinary approach to art and geometry.
Geometry is everywhere. We can train ourselves
to find the geometry in everyday objects and in works of art. Look
around at the buildings, roads, signage, foliage, and other features of our
immediate environment. designers use geometry when they develop patterns used
in buildings.
The pillars in the above figure
were designed by the graphic designer /artist M.C. Escher. The designs were
created using some techniques in geometry . It looks pretty tricky, but if you
know enough geometry you can make designs like this too. Geometry will explain
how to create the flower designs and the animals.Artists use basic components
(often called the elements of art) to create a work of art: color, value, line,
shape, form, texture, and space. The principles of design, such as perspective
and proportion, are used by artists to arrange the elements of their artworks
and to create certain effects. Artists “design” their works to varying degrees
by controlling and ordering the elements of art. Look closely at each artwork
to identify the elements of art and principles of design an art, proportion is
the principle of design concerned with the size relationships of parts of a
composition to each other and to the whole. In math, proportion is the ratio or
relation of one part or another to the whole with respect to size, quantity,
and degree. Look carefully at each artwork we can see the use of geometrical
concepts such as lines, shapes, solids, proportions, etc. Today, artists often
use geometrical elements such as lines, angles, and shapes to create a theme
throughout their artwork. Also, artists started using these geometrical
elements as a way to create the illusion of the third dimension. This art is
known as Optical or Op Art. The following are examples of optical art.
MATHEMATICS IN ARHITECTURE
Pyramids of Egypt
The Egyptian pyramids have
very close constructions to the golden pyramid. Within those pyramids, one, the
Great Pyramids of Giza (also known as Cheops) has a slope of 51.52 degrees
which is very close to the slope of the golden pyramid which is 51.83 degrees.
The other pyramids such as Chephren with a slope of 52.20 degrees and Mycerinus
which has a slope of 50.47 degrees are very close to the golden pyramid also. There are many other buildings of ancient world which
golden ratio, golden rectangle or golden pyramid used widely in their
constructions.
Parthenon
in Athens
The Parthenon in Athens,
built by the ancient Greeks from 447 to 438 BC, is regarded by many people that
posses the Golden Ratio in design. The figure below shows a Golden Rectangle
with a Golden Spiral overlaying to the entire face of the Parthenon. This
illustrates that the height and width of the Parthenon conform closely to
Golden Ratio proportions with an assumption that the bottom of the golden rectangle
should align with the bottom of the second step into the structure and that the
top should align with a peak of the roof that is projected by the remaining
sections. Given that assumption, the top of the columns and base of the roof
line are in a close golden ratio proportion to the height of the Parthenon.
The following figure
shows the golden ratio proportions that appear in the height of the roof
support beam and in the decorative rectangular sections that run horizontally
across it. The gold colored grids below are golden rectangles, with a width to
height ratio of exactly 1.618 to 1.
Modern
Buildings
Not only ancient ones, but also
modern buildings still use Golden Ratio in their construction.
If you take a look to the
United Nations Building in New York, you will observe two golden rectangles,
one from base to the upper intermediate level, and the second rectangle from
the top to the lower intermediate level, as drawn by yellow lines on the above
figure. Another example is the world tallest tower,
CN Tower shown on figure. CN Tower, located in Toronto, has a height of 553.33
meters. The ratio of observation deck which is at 342th meters to the total
height of 553.33 is 0.618 or phi that is the reciprocal of Golden Ratio.
Islamic
World
.
The great architect Sinan
used Golden Ratio in Selimiye Mosque (see the Figure given above) and
Suleymaniye Mosque. The ratio of lighted balconies of the minarets gives Golden
Ratio. There are so many other examples on
architectural constructions of the modern world which use mathematical concepts
as base.
MATHEMATICS IN PAINTING
As the Golden Ratio is found
in the design and beauty of nature, we can say that it can also be used for
beauty and balance in the design of art. The use of Golden Ratio in art is not
a rule, but it is a tool resulting more aesthetic work. The Golden Rectangle is
proposed to be the most aesthetically pleasing of all possible rectangles. For
this reason, it and the Golden Ratio have been used extensively in art and
architecture for many years. The most prominent and well known uses of the
Golden Rectangle in art were created by the great Italian artist, inventor, and
mathematician, Leonardo da Vinci.The Mona Lisa, Leonardo's most famous
painting, is full of Golden Rectangles. If you draw a rectangle whose base
extends from the woman's right wrist to her left elbow and extend the rectangle
vertically until it reaches the very top of her head, you will have a Golden
Rectangle.
You will discover that the edges of these new squares come to all the
important focal points of the woman: her chin, her eye, her nose, and the
upturned corner of her mysterious mouth, if you draw squares inside this Golden
Rectangle. By drawing a rectangle around her face, we can see that it is indeed
golden rectangle. If we divide that rectangle with a line drawn across her
eyes, we get another golden rectangle. It is considered that Leonardo, as a
mathematician, made this painting line up with Golden Rectangles in this
fashion in order to further the incorporation of mathematics into art on
purpose. In addition to that, the overall shape of the woman is a triangle with
her arms as the base and her head as the tip. This is meant to draw attention
to the face of the woman in the portrait
Leonardo's another famous
study of the proportions of man, "The Vetruvian Man" (The Man in
Action), is also full of Golden Rectangles. In case of the Mona Lisa, all the
lines of the Golden Rectangle are assumed by the mathematicians. But in
"The Vetruvian Man", many of the lines of the rectangles are actually
drawn into the image. There are three distinct sets of Golden Rectangles in
this painting: one set for the head area, one for the torso, and one for the
legs. Figure given below shows these details.
The famous painting by Leonardo
da Vinci, “The Last Supper”, contains a lot of Golden Rectangles. In this
painting, successive divisions of each section by the golden section define the
key elements of composition. The table, ceiling, people, windows are full of
Golden Rectangles shown at the Figure 24 below, and the painting itself is a
perfect sample of Golden Ratio.
Da Vinci created other pieces that were also drawn according to the golden
ratio such as Old Man, and The Vitruvian Man. The Vitruvian Man (or Man in Action) is the drawing of a
man inscribed in a circle. The height of the man is in golden proportion
from the top of his head to his navel and from his navel to the bottom of his
feet.
.
Golden Rectangle was used in
their paintings by other famous artists. In “The Sacrament of the Last Supper”
(see the figure given below), it is seen that Salvador Dali’s painting is
framed in a golden rectangle. Dali positioned the table exactly at the golden
section of the height of his painting following Da Vinci’s lead. He positioned
the two disciples at Christ‘s side at the golden sections of the width of the
composition. In addition, the windows in the background are formed by a large
dodecahedron which consists of 12 pentagons which has golden ratio relations in
their proportion.
Escher is a famous artist who
created mathematically challenging artwork. He used only simple drawing
tools and the naked eye, but was able to create stunning mathematical
pieces. He focused on the division of the plane and played with
impossible spaces. He produced polytypes, sometimes in drawings, which
cannot be constructed in the real world, but can be described using
mathematics. His particular drawing, Ascending and Descending, which can be viewed at the M. C. Escher website, was one
of these masterpieces. In this drawing, Escher creates a staircase that
continues to ascend and descend, which is mathematically impossible, but the
drawing makes it seem realistic. The following image, Relativity, is an example.
Sometimes, artists want to create certain
linear perspectives. In order to accomplish this, the artist will pick a
point on the piece such that all the lines in the piece will come together at
that single point. In this way, artists use math to create a certain
perception for their audience, without any special mathematical tools.
Many artists use math without realizing it.
MATHEMATICS IN SCULPTURE
Polykleitos the Elder was a Greeksculptor from the school of Argos, and a contemporary of Phidias. His works and statues consisted mainly of bronze and were
of athletes. According to the mathematician Xenocrates, Polykleitos is ranked as one of the most important
sculptors of Classical antiquity for his work on the Doryphorus and the statue of Hera in the Heraion of Argos. In the Canon
of Polykleitos, a treatise he wrote designed to document the
"perfect"anatomical proportions of the male nude,Polykleitos gives us a
mathematical approach towards sculpturing the human body.
Polykleitos
uses the distal phalanx of the little finger as the basic module for determining the proportions of the
human body.Polykleitos multiplies the length of
the distal phalanx by the square root of two (√2) to get the distance of the second phalanges and multiplies the length again by
√2 to get the length of the third phalanges. Next, he takes the finger length and multiplies that by √2 to get the length of the palm
from the base of the finger to the ulna. This geometric
series of measurements progresses until
Polykleitos has formed the arm, chest, body, and so on.
MATHEMATICS IN DANCE
Mathematics is present in
dance. It is not the mathematics of simple number manipulation; we do not
attempt to add or integrate through movement, instead we would like to employ
abstract mathematics and various methods of analysis to understand dance at a
deeper level. Geometry is perhaps the most apparent subfield of mathematics
present in dance. We can consider the shapes, patterns, angles and symmetry of
many different aspects of dance within a variety of scopes. The analysis could
concern anything from one dancer frozen in a position to a whole ensemble
actively moving in space. In the first case, we would look at the lines of the
body and their relation to each other and to the space in which the dancer
exists. Pieces involving more than one dancer very often use the idea of
translation. To be more specific, if we asked a whole ensemble, or even just a
few dancers within the group, to perform the same movement at the same time, we
introduce translation of that pose.
Since
dancers are three dimensional creatures, their movements and poses exhibit
different geometrical relations depending on the angle at which we are
observing the piece. In addition, with groups, we might need to deconstruct the
formation in order to find relationships. It is possible, and in fact more
interesting, to have a pose that as a whole does not posses simple geometric
properties, but when taken apart exposes their presence. Dancers performing
symmetrical movements make the dance attractive. Geometry in dance is unavoidable.
Above we have presented a few varying examples of the many levels on which one
could look for geometric properties in dance. The moment a dancer enters the
floor, their body and their moves create shapes and patterns that simply wait
to be noticed by the audience.
Geometry
is not the only mathematics concept that has sneaked its way into the world of
dance. Because of the simple fact that dancers change their positions in space
as time passes, the ensemble can be looked at as a multidimensional dynamical
system. We could consider each dancer’s position in space as the elements and
explore the system’s behavior as time goes on. The majority of choreographers,
in fact, perform this exact task intuitively; they look at their formations and
make sure that the arrangements don’t feel heavy on any particular side; they
also make sure the transitions feel fluid that the ensemble as a whole follows
the predetermined progression path. We what proportion of time the dancers were
using suspended movements, movements on the floor, or jumps; we could consider
how many sharp versus fluid movements there were, or how many fast versus slow
movements were used. There is a delicate balance between the proportion of
different types of movements and how interesting the piece appears. Another
interesting concept is patterns of rhythms and the changes within those
patterns. Not all dances follow the simple one through eight. We could have a
dance in which a couple of distinct count patterns get repeated, or even they
themselves come in a pattern.
ANALYSIS
From
this study we can analyses that there is obviously a strong link between
mathematics and the arts. Music, fine art, and literature wouldn't be the same
without mathematics. From Mozart to Escher to Crichton, musicians, artists and
novelists have used mathematics to highlight, improve and develop their work.
This doesn't mean you need a degree in maths to play the piano, paint a
picture, or write a novel - however, it does mean that an understanding of
certain mathematical concepts can make you a better pianist, artist or author.
Mathematics has aesthetic, and
art in itself and it has interrelation with the different fields or arts such
as architecture, painting, poetry and music, dance, sculpture and even in
nature. While investigating the relation between mathematics and art, we
frequently face with some mathematical concepts such as Golden Ratio, Fibonacci
sequence, and Pythagoras Comma, patterns, fractals, topology, geometry,
permutations, combinations, etc. Then it is not wrong to say that they have
strong relations since they both have great effects on human beings. In this
study, the relation between art and mathematics are shown by the examples of,
Golden Ratio, Fibonacci sequence, and music.
Also this study analysis how
various mathematical concepts used in the development of various art forms and
how these concept provide beauty to them. The mathematical ideas like golden
ratio, Fibonacci numbers, patterns, permutations, etc has essential role in the construction of
various art forms. The different branches of mathematics such as geometry,
topology, calculus, arithmetic, etc have large application in the field of art.
From the following table we can easily analyse how math and art are
interrelated.
|
Sl.
No.
|
Different art forms
|
Mathematical concepts/terms used
|
|
1.
|
architecture
|
Golden ratio
Geometrical shapes
patterns
Topology
|
|
2.
|
music
|
Permutations
Combinations
Sequence
Golden ratio
Fibonacci numbers
|
|
3.
|
dance
|
Geometry
Patterns
Translation
rotations
|
|
4.
|
painting
|
Fractals
Golden ratio
Geometrical shapes
length
|
|
5.
|
sculpture
|
Measurements
Patterns
Geometrical shapes
|
|
6.
|
literature
|
Logic
Patterns
sequence
|
MAJOR FINDINGS
Ø There is
strong relationship between mathematics and arts. Music, fine art, and literature wouldn't be the same
without mathematics. Mathematics and art are related in a
variety of ways. Mathematics has itself been described as an art motivated by beauty.
Ø Symmetries are prominent in textile arts including quilting,knitting,cross-stitch, crochet, embroidery and weaving.Rotational symmetry is found in circular structures
such as domes.
Ø Mathematical concepts
such as fractals, geometry, permutations, golden ratio, Fibonacci numbers,
sequence, etc have a great application in art.
Ø Mathematics has vital
role in architecture. There are many buildings of ancient world which golden
ratio, golden rectangle or golden pyramid used widely in their constructions.
Ø Mathematics play a
great role in creation of a good painting. The famous paintings of great
artists are mainly based on the mathematical concepts.
Ø Mathematics and
sculpture are closely related. Mathematics is essential for the creation of good
sculpture.
Ø Mathematics is present in dance.
Geometry is perhaps the most apparent subfield of mathematics present in dance.
CONCLUSION
The above explanations, examples and
proofs show that mathematics and art are inevitably interrelated. Not only art,
but nature and universe reveal mathematical forms very clearly. Mathematics is
not an abstract science, but its effects can be seen everywhere and in every
living organism. Mathematics is based on truth and proof, whereas art is based
on thoughts and imagination. But a wide imagination requires a wide angle of
sight, which can be gained by mathematics. All explanations given above study
clearly indicate that art and mathematics are very closely related to each
other and art without mathematics cannot be considered alone. On the other
hand, mathematics itself is art with its magnificent applications.
In art, mathematics is not always visible,
unless you are looking for it. But there is much symmetry, geometry, and
measurement involved in creating beautiful art. As well, many artists
take advantage of mathematical findings, such as the golden ratio to make their
artwork realistic and beautiful. Angles and perspective can also be
described using math. Mathematics can be discerned in many of the arts,
such as music, dance, painting, architecture, and sculpture. Each of these is richly associated with mathematics. Thus, we come to the
conclusion that mathematics and art constitute an inseparable composition.
Understanding and enjoying the world we live in, we have to understand
mathematics and art, and their undeniable relation and cooperation.
REFERENCE
1.
Devlin, Keith (2000). "Do Mathematicians Have Different
Brains?". The
Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip. Basic Books. p. 140. ISBN 978-0-465-01619-8.
2.
Wasilewska, Katarzyna (2012). "Mathematics in the World of Dance" (PDF). Bridges. Retrieved 1 September 2015.
3.
Malkevitch,
Joseph. "Mathematics and Art".
American Mathematical Society. Retrieved 1 September 2015.
4.
Malkevitch,
Joseph. "Mathematics and Art. 2. Mathematical tools for artists". American Mathematical Society. Retrieved 1 September.
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