ART,
MATHEMATICS AND MUSIC
Benno
Moiseiwitsch
CONTENTS
Preface
Introduction
1.
Primitive Art
2.
Music
3.
Islamic Art
4.
Renaissance Art
5.
Modern Art
6.
Epilogue
Appendix
A. Golden Section
Appendix
B. Spiral Curves
Appendix
C. Mandelbrot Sets
References
Moonlight
Sonata
PREFACE
The
main idea behind the writing of this article was to establish that art,
mathematics and music, as well as science and astronomy, are in many ways quite
closely connected together.
The
drawing and painting of people and animals, an activity of artists since
primitive times, is clearly associated with symmetry, and thus geometry, since
the human form and the forms of living creatures in general are obviously
highly symmetrical in shape. Indeed the beauty of the human form, and also of
an animal such as the horse, is firmly based on their symmetrical proportions.
Thus artists such as Leonardo da Vinci and Albrecht Dürer were deeply concerned with the
proportions of the human form from a truly geometrical point of view.
Further,
sculpture and architecture have a strongly geometrical basis as is evident from
the wonderful work of the Italian Renaissance men Brunelleschi, Alberti and
Michelangelo.
The
geometry of perspective used by the Renaissance artists and subsequently by
many artists in later times has been subjected to detailed analysis by several
authors, for example in the book by Kemp1 devoted to optical themes
in western art.
The
repeating patterns and designs used in Islamic art are strongly based on the
geometry of the fundamental regular polygons such as the triangle, square,
pentagon, hexagon, octagon and so on, up to polygons with 24 sides, and also on
other beautiful geometrical shapes.
The
subject of astronomy has always been dependent to a great extent on geometry.
Johannes Kepler, the astronomer who discovered that the planets move round the
sun in ellipses, was greatly interested in the subject of geometry. He examined
how a plane could be completely filled with equal regular polygons such as the
equilateral triangle, the square and the hexagon with six equal sides, and also
studied the subject of polyhedra or convex solid bodies with flat faces.
In 1985
an International Congress on M. C. Escher was held in Rome. The graphic work of
Escher has been of profound interest to many mathematicians for a considerable
time and has led to many kinds of mathematical challenges, geometrical
investigations and research2. His work was much influenced by the
repeating patterns used by the Islamic artists of Granada in Spain.
Escher
was able to use the symmetrical shapes of living creatures such as birds,
fishes and insects of various kinds as well as flowers to create his wonderful
periodic patterns. In this context it is well worth recollecting that
scientifically minded people believe that the shapes of all living creatures
and vegetation, including plants and trees, have come about by the action of
Darwinian evolution arising from their successful adaptation to the ambient
environment in which they flourished over long periods of time often leading to
their beautiful symmetry and consequently their use as models for human art.
Thus we could say that art is founded, at least in part, on the past history of
life on Earth.
Music
and mathematics also have a close affinity as is readily apparent from the two
books of 24 Preludes and Fugues composed by Johann Sebastian Bach using the
equal-temperament scale, also known as the well-tempered Clavier. One remembers
that Pythagoras did fundamental work on the foundations of the musical scale
and on the vibrations of strings, and that the French mathematician Mersenne
was also a musician and was the originator of the equal-temperament scale.
Artists
have frequently shown a great affinity with music and their pictures often
include singers, musicians as well as musical instruments. The modern artist
Wassily Kandinsky originally wished to be a musician and another artist of the
same generation Paul Klee was a talented violinist throughout his life.
For all
these diverse reasons it is highly interesting as well as instructive and
revealing to investigate the relationship between art, music and mathematics in
its various aspects, and this is what I have tried to do in this article
employing just elementary geometry and algebra. It arose originally from a
lecture that I gave on Mathematics and Art at the British Association for the
Advancement of Science meeting held in Belfast in August 1987. More recently,
in 1999, I gave a lecture on the same subject in a somewhat modified form at a
meeting of the Belfast Branch of the Institute of Mathematics and its
Applications which was subsequently published in a considerably revised and
abbreviated form as an article in Mathematics Today, the Bulletin of the IMA3.
To
those who have a deeper interest in the mathematics involved in this article, I
have left the more advanced mathematical discussion to the Appendices.
I have
found many books very useful in the preparation of this article. In particular
I wish to acknowledge the importance of the book by Martin Kemp1 on
the Science of Art, the volume on the art of M. C. Escher edited by H. S. M.
Coxeter, M. Emmer, R. Penrose and M.L. Teuber2, the books by Dorothy
K. Washburn and Donald W. Crowe5 on the Symmetries of Culture, Sir
James Jeans9 on Science and Music, D'Arcy Thompson11 on
Growth and Form, Keith Critchlow12 on Islamic Patterns, Marilyn
Aronberg Lavin23 on Piero della Francesca, Caroline H. Macgillavry25
on the Symmetry Aspects of M. C. Escher's Periodic Drawings, and the article by
Martin Gardener27 on Roger Penrose's nonperiodic tessellations.
INTRODUCTION
In the
following I shall carry out an exploration of the nature of the relationship
between art, mathematics and music. Since true art as well as music has meaning
and significance, it must necessarily have pattern and structure and so one
would expect that it should therefore possess, at least to some extent, a
character based on symmetry, geometry, and even mathematics in certain
examples.
Although
the art of early cultures is often referred to as being primitive this by no
means implies that it was not highly significant and without great merit.
Indeed the cave artwork dating from many tens of thousands of years ago found
in Spain and France is very attractive and revealing as well as exhibiting a
close affinity with nature. It shows that primitive people were highly artistic
and not inferior to ourselves in mental capacity but simply limited by the
tools and pigments that were available for them to use in those days.
Stone
age artists carved interesting geometrical designs such as spirals on the large
stones in their passage-graves like that found at Newgrange in Ireland. The
significance of these spirals is not entirely clear but is very likely
associated with the seasonal decline and subsequent rise of the Sun since in
the case of Newgrange the Sun shines directly down the stone passage way
precisely at the winter solstice when the day is shortest and, for a few
minutes, lights up the grave chamber which is normally in utter darkness. Thus
spirals are associated with the Sun and its very regular periodic motion, and
may be a symbol for the life cycle as well as resurrection and a belief in the
possibility of immortality for the high ranking people buried in the tomb since
at the winter solstice the day is about to increase in length and heralds the
rebirth of the Sun and the coming of spring.
The
massive stones from which this passage grave was built were assembled so
accurately that it has survived for thousands of years without significant
movement and testifies to the mathematical understanding of the stresses and
strains in the megalithic construction by the Neolithic engineers who designed
it.
Similar
passage graves are found elsewhere, for example to the north of Scotland in the
Orkney Islands at Maeshowe a few miles away from Skarabrae on the Mainland of
Orkney, and at Midhowe on Rousay Island.
Spirals
were also used to decorate the beautiful ceramic pots made by the native
American people who lived about a 1000 years ago in the lost city known as
Cahokia in Missouri. This city also possessed many tumuli in which people of a
high status were buried. The modern day Choctaw Indians, who continue to use
spirals for decoration, may be connected with the people of Cahokia.
It is
well understood that some primitive art was strongly influenced by
considerations of symmetry4. Indeed Washburn and Crowe5
have pointed out that the various cultures spread around the world have such
marked preferences for particular types of symmetry that their works of art can
be identified by their choice of symmetry classes. The patterns used by the
ancient Egyptian artists as long ago as 1300 B.C. and earlier demonstrate that
they had a fair appreciation of the geometry involved in their use of symmetry.
Nevertheless their geometry was rather primitive as can be seen from the fact
that they calculated the area of a circle having diameter d using the
rule A = (8d/9)2 which is equivalent to taking p
to be 256/81 = 3.16... rather than the better approximation 22/7 = 3.14... used
by the ancient Greeks and even today for approximate work. Regular polygons
with 24 sides were used by the ancient Egyptians in the decoration of tombs
which may be associated with the fact that they divided the day into 24 hours
and moreover that the number 24 is exactly divisible by as many as six numbers,
namely the prime numbers 2, 3 as well as the composite numbers 4, 6, 8, 12.
Simple
geometrical figures such as spirals were used to decorate Mycenaean Greek jars
as early as the fifteenth century B.C. and later arcs and circles were used by
the Greeks to decorate amphora or storage vessels, and kraters or
two handled jars for mixing wine with water, in the tenth century B.C. known as
the Protogeometric period. Then more elaborate geometrical patterns such as
meanders, zigzags and triangles as well as animal and human figures were used
to decorate kraters in the ninth and eighth centuries B.C. in the full
Geometric period6.
The
nature and behaviour of the Sun and all the other heavenly bodies have always
produced wonderment in human beings. It is fascinating that the ancient Greek
Aristotelian interpretation of the motions of the Sun, the Moon, the wandering
planets, namely Mercury, Venus, Mars, Jupiter and Saturn, and the fixed stars,
by means of rotating crystalline spheres centred at the fixed Earth, was called
the music of the spheres by the Greek mathematician Pythagoras (c.
380-300 B.C.).
The
motions of the planets were given complicated but fairly accurate, although not
perfect, descriptions by the Greek mathematician and astronomer Ptolemy (c. 150
A.D.) making use of epicycles, that is paths traced out by points which
move on the circumferences of circles whose centres move on the circumferences
of other larger circles centred at the Earth (see Appendix B and Fig. 9). This
was done, in particular, to take account of the observed retrograde motions of
the planets in the night sky and thus avoid centering their motions around the
Sun although this had been known to simplify the interpretation of planetary
motions by Aristarchus (c. 281 B.C.)7. However it remained difficult
for most people at that time to believe that the Earth moved and was not the
centre of the universe.
Johannes
Kepler (1571-1630), the astronomer who discovered that the orbital paths of the
planets round the sun are ellipses, one of the conic sections which are curves
produced by cutting a cone with a plane, was deeply interested in spatial
relationships. He examined how a plane could be completely filled with
congruent regular polygons and also wrote on the subject of the convex solid
bodies with flat faces known as polyhedra. He first thought that the orbits of
the planets Saturn, Jupiter, Mars, the Earth and Venus were circles which could
be fitted between the regular polygons, namely an equilateral triangle, a
square, a regular pentagon, a regular hexagon, and so on. He found that this
simple geometrical structure failed but later on, in his treatise Mysterium
Cosmographicum, he attempted to place the orbits of the planets on spheres
which were fitted between the regular Platonic solids: that is the tetrahedron
with four faces, the cube with six faces, the octahedron with eight faces, the
dodecahedron with twelve faces and the icosahedron with twenty faces, which he
eventually realized was also not a feasible solution. He next decided that the
orbit of Mars, determined from the observations of the astronomer Tycho Brahe
(1546-1601), was an oval having an egg-like shape before eventually realizing,
as if he had been awakened from a sleep, that it was an ellipse as he explains
in his book Astronomia Nova, aptly named the New Astronomy. An
illuminating description of the way in which Johannes Kepler and those
astronomers who preceded him, like Nicolas Copernicus (1473-1543) and Tycho
Brahe, tried to understand the motions of the planets is given in the book The
Sleepwalkers by Arthur Koestler8.
The
Pythagoreans, believed that all natural phenomena, including the motions of the
planets already mentioned, are derived from harmony based on numbers. The
sounds produced by a vibrating string or a pipe were placed on a mathematical
basis by Pythagoras, who made the assumption that the notes should be separated
by well-defined regular intervals such as the frequency ratio 
corresponding to the dominant
note G which is a fifth above the first note C or tonic of an octave.
This idea was further developed by other mathematicians and musicians such as
the French philosopher and theologian Mersenne (1588-1648) who devised the
equal-temperament scale that was used by Johann Sebastian Bach (1685-1750) in
his two books of 24 Preludes and Fugues, das wohltemperierte Klavier or the
well-tempered Clavier, and by other composers of music9.
Obviously
there is a fundamental property that differentiates music from pictorial art.
Thus music is a form of art that is founded on the steady forward progression
of time, like the motions of the planets and the diurnal motion of the heavens,
whereas painting, drawing, sculpture and architecture are all static art forms.
However one recollects that Degas (1834-1917) once said that his chief interest
in dancers was in rendering movement, and the surrealist painter Paul Klee
(1879-1940) remarked that: 'Pictorial art springs from movement, is itself
fixed movement, and is perceived through movements'. It is instructive to
remember that Paul Klee was a talented amateur violinist and that another
modern artist Wassily Kandinsky (1866-1944) originally wished to be a musician.
Artists have frequently introduced musicians and musical instruments into their
artwork. For example Pablo Picasso (1881-1973) painted pictures having the
titles Man with a Guitar and The Three Musicians.29
The watercolour Moonlight Sonata by the artist Daniel Moiseiwitsch
(1919-1944) given in the frontispiece of this article also represents a picture
with a musical connection: it shows a grand piano seen through a window with
the music displayed of Beethoven’s Moonlight sonata in C sharp minor.
The
relationship between mathematics, science, music and art has been of great
interest for a long time. Indeed the mathematical sciences part of the medieval
liberal arts course known as the quadrivium comprised the four subjects
of arithmetic, geometry, astronomy and music. Further the great interest in
these subjects is made evident from the painting by Hans Holbein the Younger
(1497/8-1543) known as The Ambassadors, to be seen in the National
Gallery London, which depicts astronomical and musical instruments, a peculiar
mathematical distortion of a skull which can be corrected by viewing from the
far right or by using a glass cylinder placed on a photograph of the painting,
as well as the portraits of the two diplomats.
The
golden section ratio (1+Ö5)/2 = 1.61803... or its alternative reciprocal form (-1+Ö5)/2 =
2/(1+Ö5) = 0.61803... in the geometry of Pythagoras and Euclid which are
derived from isosceles triangles (see Fig. 2) with two base internal angles of
720 and a third internal angle of 360, adding together to
1800, is well established in art from ancient Greek to modern times.
Theodore Andrea Cook in his book The Curves of Life10 and
D'Arcy Thompson in his book On Growth and Form11 showed that
the golden section is important, for example, in the understanding of the
beautiful spiral shapes of various sea shells since it gives rise to an
equiangular or logarithmic spiral (see Appendices A and B). We shall see also
that the nonperiodic patterns produced by Roger Penrose26,27 using
just two tiles are based on the golden section.
Let us
reflect on the fact that nowadays biologists believe that the beautiful and
interesting structures of living creatures and plants arose from the process of
Darwinian evolution produced by their very successful adaptation to the
environment in which they lived. It is easy to understand, for example, that
birds require a pair of right and left symmetrically shaped wings which are
mirror images of each other, together with a symmetrically shaped lightweight
body, to be able to fly through the air, and that land creatures require two
pairs of right and left symmetrically shaped legs to walk on the ground under
gravity, although one pair of symmetrically shaped legs suffices for human beings
to walk upright. Furthermore, two symmetrically structured eyes are needed for
stereoscopic vision and two symmetrically formed ears are needed for
stereophonic hearing. Requirements such as these have led to the lovely
symmetry of the living creatures of our world and so to their use as models for
the artwork of humankind.
Spirals
occur in many different contexts, both in nature and in art. The convolvulus
plant climbs upwards in the form of a right-handed helix while the hop climbs
in the shape of a left-handed helix. A fine example taken from architecture,
cited by T. A. Cook, is the Palazzo Contarini del Bovolo in Venice with its
delightful right-handed spiral staircase Scala del Bovolo. It is intriguing to
discover that a feature of the Turbine Hall in the new millennium Tate Modern
art gallery in London is the three very high towers with spiral staircases
created by Louise Bourgeois.
Euclidean
geometry is fundamental to the art of Islam. Islamic art is strongly based on
various geometrical figures such as equilateral triangles, squares and many
different regular polygons with sides ranging in number from 5 to 24, as has
been beautifully illustrated in Keith Chritchlow's book on Islamic Patterns12.
Splendid examples of Islamic tessellations are to be found in Spain, for
example in the Alhambra in Granada, and elsewhere in the Muslim world19.
Much of
the pictorial art of Western Europe during the 12th to the 14th centuries
presents a somewhat naive, rather unnatural and doll-like appearance. It possesses
an almost symbolic form lacking in solid dimensionality that contrasts strongly
with the art of the Greeks and Romans who portrayed people very realistically
centuries before the birth of Christ. An exception to this was the marvellous
early 14th century Italian painter Giotto di Bondone (1267?-1337) who painted
people accurately and possessed a fine intuitive feeling for spatial
relationships. But, in general, this unnatural character remained until the art
of the Renaissance introduced some more reality into pictures in the late 15th
century onwards1,13 using the mathematics of perspective originated
by the sculptor and architect Filippo Brunelleschi (1379-1446) and also the
architect Battista Alberti (1404-72). Perspective was fully introduced into
painting by, for example, the Italian artists Paolo Uccello (1397-1475), Piero
della Francesca (1420-1492), Leonardo da Vinci (1452-1519) and Raphael
(1483-1520), as well as the German artist and engraver Albrecht Dürer (1471-1528). It is worth
emphasizing that Piero della Francesca was well known in his lifetime as a
mathematician in addition to his fame as an artist, and that Leonardo da Vinci
was greatly interested in mathematics as indeed was Dürer. They regarded a thorough
knowledge of mathematics as essential for an artist and indeed wrote books on
mathematical subjects such as plane and solid geometry.
Objective
art and the use of perspective then flourished in the painting of buildings and
landscapes until recent times. However a revolution occurred towards the end of
the 19th century and the beginning of the 20th century that resulted in a great
deal of modern art being rather abstract, mystical and non-objective. Although
this art certainly possesses structure and pattern, it employs a lot of symbolism
and does not show much direct connection with mathematics and moreover departs
from the use of the recognized rules of perspective. Indeed some modern art
often seems to deliberately avoid depicting the symmetry of nature and takes
delight in the distortion and metamorphosis of the appearance of people and
objects as was often done by Picasso and his contemporaries, albeit in an
interesting, colourful, very beautiful and imaginative way29.
However
there are some notable exceptions to the metaphysical, surrealist and
non-objective art of recent times, a particularly interesting example being the
graphic work of the Dutch artist Maurits Escher (1898-1972)2,14,25
much of which has a strong relationship to Islamic art. In fact Escher
developed some of his ideas for his graphic art from his visits to the Alhambra
in Granada, Spain where he drew pictures of the remarkable tiles that decorate
the walls in beautiful periodic patterns. Escher was strongly influenced by
mathematical ideas and produced, for example, interesting woodcuts illustrating
non-Euclidean geometry.
Periodic
patterns have also been used with great effect to produce beautiful textiles in
many parts of the world such as India, Pakistan, and Persia. An example from
the Victoria and Albert (V & A) Museum in London is a mid-17th century
carpet decorated with a design based on pots and flowers possessing horizontal
and vertical translation symmetries and vertical reflection symmetries. It is
made from wool pile with cotton warps and silk wefts, and probably comes from
Lahore.
In
England the textiles, tessellations and wallpapers designed by William Morris
(1834-96) illustrate various simple mathematical symmetries. An example, chosen
from the V & A, of his ornate tiling, depicts a complex collection of
flowers, stalks and leaves with a reflection symmetry. It was made in the
factory of William De Morgan in 1876-7 for the bathroom in Membland Hall in
Devon. Another example of the work of William Morris in the V & A is the
furnishing textile Tulip and Willow made of block printed and
indigo-discharged cotton. It possesses translation symmetries in both the
vertical and horizontal directions. Other beautiful examples of the work of
William Morris to be found in the Picture Library at the V & A that exhibit
translation and reflection symmetries are his Pimpernel Wallpaper (1876)
showing very large flower heads, which he hung in his dining room at Kelmscott
House, his Bird Curtains (1877-8) which he used in the drawing room of
Kelmscott House, and his Strawberry Thief pattern (1883) showing
thrushes about to eat strawberries.
We have
already noted that some modern art conspicuously possesses a non-objective,
spontaneous and rather discordant dynamic character, freed from motif,
which emanates largely from the unconscious mind and the emotional state of the
artist. Yet, in contrast, much primitive artwork is objective and exhibits a
surprisingly intimate connection with sophisticated geometrical symmetries and
thus with mathematics, although with a truly imaginative and creative flair,
and because of this it is to Primitive Art that we shall devote the first
section of this article.
1. PRIMITIVE ART
The
title of this section, Primitive Art, is the same as that used by previous
authors4,15 who were concerned with the art of the native peoples of
Africa, Australia and New Zealand, Asia, North and South America, and elsewhere
in the world. This art is often firmly and skillfully based on various patterns
derived from the fundamental one-dimensional and two-dimensional symmetries of
geometry. In no way does the use of the word 'primitive' imply a crude art
form: indeed even the earliest artwork of mankind usually has some mathematical
significance.
As long
ago as the fifteenth century B.C., Mycenaean pottery jars were decorated with
simple geometrical patterns such as the spirals on the jar found in a tomb in
Ialysos on the Island of Rhodes dating from 1450-1400 B.C. on display in the
British Museum. Somewhat later on in the tenth century B.C., known as the
Protogeometric period (1050-900 B.C.), the Greeks used various geometrical
patterns such as arcs and circles employing compasses and comb-like brushes to
decorate their vases or jars, sometimes placed over graves, called amphora.
Other simple geometrical patterns were also used, an example from Athens
(950-900 B.C.) displayed in the British Museum, has a chequer-board design on
the shoulder and three parallel wavy lines. In the full Geometric period,
occupying the ninth and eighth centuries B.C., more elaborate designs such as
continuous bands of meanders of various shapes sometimes with straight edges,
zigzags, triangles, and less often with circles and semi-circles, were used to
decorate two-handled vases and bowls or jars for mixing wine with water known
as kraters. In the early eighth century animals such as horses, deer,
goats and fowl were introduced into the decoration, and then in the latter part
of the eighth century human figures, chariots and warriors were used. Examples
are a krater from Athens c. 750 B.C., a late eighth century Athenian amphora
with deer, goats and geese, and a late eighth century vase decorated with
various geometrical designs, including a central flower-like symmetrical
pattern with eight petals, from the island of Thera. These are discussed in
rather more detail in the book on Greek Art written by John Boardman6.
Spirals
have been found in many kinds of primitive art. In particular they have been
found carved on massive stones in the Newgrange tumulus in Co. Meath, Ireland
dating from about 5200 years ago in the Neolithic age.
Of the
two spirals found connected together on the entrance stone of Newgrange
tumulus, one has a right-handed sense of rotation and the other has a
left-handed sense of rotation. A triple spiral is carved on one of the stones
of the central chamber of the Newgrange tumulus. These double spiral shapes and
triple spiral shapes are also found on Greek jars, an example being the
black–figured amphora depicting Achilles and Penthesileia painted by Exekias and
signed by him as potter (Athens 540-530 B.C.) which is on display in the
British Museum. It is interesting to note also that M. C. Escher used a dual
double spiral in a woodcut using three colours, namely Whirlpools, in which
infinite lines of fish proceed from one limiting point to the other limiting
point in both directions. This design has similarities to the Cornu double
spiral discussed in Appendix B.
Another modern intriguing double spiral composed of nine-sided polygons
has been constructed by Heinz Voderberg.
A
splendid example of African primitive art is a beaded mask from the Cameroons,
which is symmetrical about the vertical central axis and whose large circular
ears possess an inner sixfold symmetry and an outer 24-fold symmetry, shown in
the book on Primitive Art by Fraser15. We see that this primitive
art is very beautiful and has great merit.
Another
interesting and beautiful example of African primitive art given by Fraser15
is an ancestor screen from Ijaw, Nigeria having symmetry about a vertical
central axis.
A
further example this time from New Britain, is a warrior's shield, made of wood
and dyed cane, which is symmetrical with respect to reflections in two
perpendicular lines, apart from the small decorations at the top and bottom of
the shield15. Each reflection produces a colour reversal.
Fundamental Geometrical Symmetries of
Primitive Art
Much of
the beauty of primitive art arises from its expression of the various types of
the available geometrical symmetries. There are seven possible one-colour
one-dimensional patterns similar to friezes or bands, there are seventeen
possible two-colour one-dimensional patterns, seventeen possible one-colour
two-dimensional patterns similar to wallpaper patterns, and there are forty-six
possible two-colour two-dimensional patterns, many of which have been used by
artists in various primitive societies and have been beautifully put together
by Washburn and Crowe in their book Symmetries of Culture5.
The
seven one-colour one-dimensional patterns are based on symmetries that involve
translations, reflections, rotations, and glide-reflections, a glide-reflection
being a translation in a given direction followed by a reflection in a mirror
line parallel to the direction of the translation.
We
denote them by the two-symbol notation xy where, if there is a vertical
reflection x is m for mirror but otherwise 1; if there is
a horizontal reflection y is m, if there is a glide-reflection
but not a horizontal reflection y is g, if there is a half-turn
but not a reflection or glide-reflection y is 2, but otherwise y
is 1. These give rise to the seven cases 11, 1m, m1,
12, mm, mg, 1g. They are all repetitive patterns
and so they are invariant under a translation in the horizontal direction.
Ignoring their translational invariance, they possess the invariance properties
that are characterized by the symmetries that are generated as shown in Table
1.
Table
1. The seven one-colour one-dimensional patterns.
11: No transformation other than a
horizontal translation.
1m: Reflection in the horizontal
line.
m1: Reflection in the vertical line.
12: Half-turn or twofold rotation.
mm: Reflection in the vertical line
and reflection in the horizontal line.
mg: Reflection in the vertical line
and a glide reflection in the horizontal line.
1g: Glide reflection in the
horizontal line.
We have
referred to the two perpendicular directions as vertical and horizontal but
this designation is often just notional and is used here merely for
convenience.
The
seventeen two-colour one-dimensional patterns have been illustrated by Woods
(1935)16 and are also displayed in the book Symmetries of Culture
by Dorothy K. Washburn and Donald W. Crowe5. They can be represented
symbolically using the form G/G1 where difference of colour is
disregarded in G and colour is preserved in G1. This gives rise to a
four-symbol notation as follows: 11/11,
12/11, 12/12, 1m/11, 1m/1m, 1m/1g, 1g/11, m1/11, m1/m1, mm/12, mm/1m, mm/m1,
mm/mm, mm/mg, mg/1g, mg/m1, mg/12. They are generated as given in Table 2.
Table
2. The seventeen two-colour one-dimensional patterns.
11/11: Horizontal translation
through one step transforms one colour into the other colour producing colour
reversal.
12/11: Twofold rotation
transforms one colour into the other colour producing colour reversal.
12/12: Alternate twofold
centres where a twofold rotation produces colour reversal.
1m/11: Reflection in the
horizontal mirror axis produces colour reversal.
1m/1m: Reflection in the
horizontal mirror axis leaves colours unchanged but a one-step horizontal
translation produces colour reversal.
1m/1g: Reflection in the
horizontal mirror axis and a one step horizontal translation both produce
colour reversal.
1g/11: Glide reflection along
the horizontal mirror axis through one step produces colour reversal.
m1/11: Reflection in vertical
mirror lines produce colour reversal.
m1/m1: Reflection in alternate
vertical mirror lines produce colour reversal.
mm/12: Reflection in
horizontal and vertical mirror lines produce colour reversal.
mm/1m: Reflection in the
horizontal mirror axis preserves colours but reflection in all the vertical
mirror lines produces colour reversal.
mm/m1: Reflection in the
horizontal mirror axis produces colour reversal but horizontal translation leaves
colours unchanged.
mm/mm: Reflection in the
horizontal mirror axis leaves colours unchanged but reflection in alternate
vertical mirror lines produces colour reversal.
mm/mg: Reflection in the
horizontal mirror axis and reflection in alternate vertical mirror lines all
produce colour reversal. Also a glide reflection along the horizontal mirror
axis through one step preserves colours.
mg/1g: Reflection in vertical
mirror lines produce colour reversal but a glide reflection along the
horizontal mirror axis through one step preserves colours.
mg/m1: Reflection in vertical
mirror lines preserve colours but a glide reflection along the horizontal
mirror axis through one step interchanges colours.
mg/12: Reflection in vertical
mirror lines and a glide reflection along the horizontal mirror axis through
one step interchange colours.
The
seventeen one-colour two-dimensional patterns were first listed by the Russian
crystallographer Fedorov in 1891.
There
are five primitive cells for these two-dimensional patterns. They are:
A:
parallelogram.
B:
rectangle.
C:
rhombus.
D:
square.
E:
hexagon.
These
cells produce a lattice by means of translations.
Together
with their crystallographic symbols the patterns are generated as indicated in
Table 3 derived from those given by the geometer H. S. M. Coxeter in his
article on Coloured Symmetry in M. C. Escher: Art and Science2 and
in his book Introduction to Geometry17.
Table
3. The seventeen one-colour two-dimensional patterns.
p1: Two independent translations.
Parallelogram
lattice.
p2: Half-turns about three centres
which are not all in a straight line.
Parallelogram
lattice.
pg: Two parallel glide reflections.
Rectangular
lattice.
pm: Two parallel reflections and a
translation along the direction of the reflection lines.
Rectangular
lattice.
cm: A reflection and a parallel glide
reflection.
Rhombic
lattice.
pgg: Two perpendicular glide
reflections.
Rectangular
lattice.
pmg: A reflection and half-turns about
two points equidistant from the reflection axis or mirror line.
Rectangular
lattice.
pmm: Reflections in the four sides of
a rectangle.
Rectangular
lattice.
cmm: Two perpendicular reflections and
a half-turn about a centre that is not on either of the mirror lines.
Rhombic
lattice.
p4g: A reflection and a quarter-turn.
Square
lattice.
p4: Two quarter-turns.
Square
lattice.
p4m: Reflections in the three sides of
a right-angled isosceles triangle with two 450 angles.
Square
lattice.
p3m1: Reflections in the three sides of
an equilateral triangle.
Hexagonal
lattice.
p31m: A reflection and a threefold
rotation.
Hexagonal
lattice.
p3: Two rotations through 1200.
Hexagonal
lattice.
p6: One threefold and one sixfold
rotation.
Hexagonal
lattice.
p6m: Reflections in the three sides of
a bisected equilateral triangle with angles of 300, 600,
900.
Hexagonal
lattice.
In this
table the letter p denotes a primitive cell, the letter c denotes
a centred cell, the letter m denotes a reflection axis or mirror line,
and the letter g denotes a glide reflection.
The
centred cell only occurs in a rhombic lattice of points. It is a rectangle with
the mid-points of its sides at the vertices of the rhombus and has twice the
area of the rhombus.
The
forty-six two-colour two-dimensional patterns have been illustrated by Woods
(1936)16 and are also displayed in the book Symmetries of Culture
by Dorothy K. Washburn and Donald W. Crowe5. They may be represented
by the notation G/G1, where the difference between the two colours
is disregarded in G but retained in G1.
A simple
example of a two-colour two-dimensional pattern is a chess-board whose
chequer-board design occurs commonly in primitive art such as on the amphora
from Athens mentioned previously and on display in the British Museum. The
chess-board has the crystallographic pattern classification p4m/p4m.
In this pattern the mirror lines that pass along the two perpendicular sides of
the squares interchange the two colours. However the four mirror lines that
pass through the centre of a square in the two directions parallel to the sides
and along the two diagonals of the square leave the two colours invariant. A
rotation through 900 about a twofold center, where four squares
meet, interchanges the two colours.
We will
not classify the generators for the two-colour two-dimensional patterns in a
table although we shall introduce some examples of art works that exemplify
their symmetry characteristics both here and when we discuss the graphic work
of Escher in section 5 on modern art.
Primitive
art from the different cultures throughout the world, not necessarily from long
ago, such as are found in South America, North America and Africa take the form
of beautiful ceramics, bowls and jars, baskets, wooden vessels, mats, textiles
of various kinds such as clothes and blankets, shields, masks and mosaics. They
are usually of a practical nature although they are often used mainly for
decorative purposes.
In the
following we will give a few examples of primitive art works that exhibit some
of the various symmetry classes. They show that the artists who created them
had an excellent, if only intuitive, understanding of geometry.
Different
cultural groups favour different symmetries for their designs. The simplest
symmetry patterns tend to predominate. For example in the one-colour
one-dimensional, or band, designs of Inca pottery, Aschers (1981)5
found that 40 per cent were of the horizontal and vertical reflection symmetry
class mm, 20 per cent were of reflection symmetry class m1 while
11 per cent were of reflection and glide reflection symmetry class mg
and these preferences express their cultural identification.
With
the progression of time the symmetries used by a given cultural group may
change. Thus Zaslow and Dittert (1977)5 have found that the
red-on-buff band designs in the interior of the Hohokam ceramic bowls changed
from the one-colour one-dimensional translation symmetry class 11 to the
rotational symmetry class 12. In a later phase the designs changed to
include the one-colour two- dimensional glide reflection symmetry class pgg
although the rotational symmetry pattern was predominant to begin with.
An
example of a one-dimensional one-colour pattern is a twined basket from the
north west coast of California exhibiting an mm symmetry that belongs to
the collection of the California Academy of Sciences in Golden Gate Park in San
Francisco. It has been photographed by Christopher Thomas and is given in the
book by Washburn and Crowe5.
Examples
of one-dimensional two-colour patterns are a Cheyenne beaded pouch with a mm/mm
symmetry pattern, and a Germantown Navajo rug with a mm/mg
symmetry pattern in the central band, which however is not completely accurate.
They are to be found in the California Academy of Sciences in San Francisco,
and have again been photographed by Christopher Thomas5.
Next we
give some examples of two-dimensional patterns.
First
we consider some one-colour patterns:
A
Mohave ceramic scoop has the pmg symmetry pattern in which a reflection
in a mirror axis through the middle of any black triangle leaves the colours
invariant. There is a glide reflection in the perpendicular direction.
A
Huichol indian shoulder bag from the western Sierra Madre in Mexico possesses a
p4m symmetry pattern which is invariant with respect to rotations
through 900 and mirror reflections in perpendicular axes as well as
in axes along the diagonals.
An
Acoma Pueblo ceramic jar from New Mexico has a beautiful p2 symmetry
pattern that is invariant under twofold rotations through 1800.
A
Navajo indian woolen rug made about 1940 has a pgg symmetry pattern with
glide axes in two perpendicular directions. This is a one-colour pattern since
the white hooks can be superimposed on the white hooks and likewise the black
hooks can be superimposed on the black hooks but the white and the black hooks
cannot be superimposed upon each other.
Another
Navajo indian woolen rug made about 1930 has a pmm symmetry pattern
which is invariant under reflections in two perpendicular directions. This is
also a one-colour pattern since the designs in the white and dark diamond areas
are slightly different and cannot be superimposed on each other.
Next we
consider some examples of two-colour patterns.
A
pre-Columbian Peru bag from Nazca has a p2/p1 symmetry pattern
composed of black hooks and white hooks. There are rotation centres between the
black and white hooks that interchange colours but there are no rotations which
keep the two colours unaltered.
A
Woodlands indian, Great Lakes ribbon blanket has a pmm/pm
symmetry pattern with reflection axes in the vertical direction which keep the
colours unaltered but reflection axes in the horizontal direction which
interchange colours. All of the rotation centers lie on the reflection axes and
they all interchange colours.
A
basket from central Africa has a cmm/cm symmetry pattern in which
reflections in the axes that bisect the isosceles triangles do not change
colours but reflections in the perpendicular axes along the bases of the
triangles interchange the two colours. Also twofold rotations through 1800
interchange colours.
A
Middle Horizon, South Coast, Peru textile used as a burial wrapping has a pg/p1 symmetry pattern in which there are
glide axes going between the diamonds containing the small birds that face in
opposite directions. The colours of the diamonds are interchanged on performing
reflections in the axes through the centres of the diamonds parallel to the
translation axes.
An
Achomawi twined basket hat from California has a p2/p2 symmetry
pattern in which the colours are interchanged when a translation is performed
in the horizontal direction but unchanged when translated in the perpendicular
direction.
A
Navajo indian blanket or rug with a cmm/pmm symmetry two-colour
pattern has a design in which black and white colours are interchanged along
the diagonal glide lines but no colour interchanges occur about the reflection
lines.
Finally
we mention a plaited basket from Samoa with a cmm/pgg symmetry
pattern in which each rectangle contains four horizontal strips and five
vertical strips and so only a 1800 rotation can produce invariance.
A rotation through 1800 about the centres between the different
coloured steps of each cell interchanges the two colours. Further a reflection
in a horizontal or vertical mirror line interchanges colours also.
These
examples of works of art created by peoples belonging to the so-called
primitive cultures are evidently firmly based on a sophisticated understanding
of the complex symmetries of plane geometry.
They
are discussed fully in the book Symmetries of Culture by Washburn and
Crowe5 where many more cases of great interest are given.
2. MUSIC
It is
well understood that there is a strong connection between music and
mathematics. Indeed, as we shall see, both Johann Sebastian Bach and, perhaps
surprisingly, Chopin would surely have recognized such a relationship.
Pythagorean Scale and the Comma of
Pythagoras
The
first musical scale of 5 tones and 2 hemitones is often associated with the
ancient Greek mathematician Pythagoras. However the Pythagorean scale, that we
shall denote by the usual letters C, D, E, F, G, A, B, C forming an octave,
contains a difficulty known as the comma of Pythagoras. To demonstrate
this, consider the notes of an octave. According to the Pythagorean scale, the
ratios of the frequencies of the notes are given by the fractions listed in
Table 4.
Table
4. The Pythagorean musical scale and the equal-temperament scale.
|
|
Pythagorean scale |
|
Equal-temperament scale |
|
|
C |
1 |
|
1 |
|
|
|
|
tone |
|
tone |
|
D |
9/8=1.125 |
|
(12Ö2)2 =1.1224… |
|
|
|
|
tone |
|
tone |
|
E |
81/64=1.2656… |
|
(12Ö2)4 =1.2599… |
|
|
|
|
hemitone |
|
semitone |
|
F |
4/3=1.3333 |
|
(12Ö2)5 =1.3348… |
|
|
|
|
tone |
|
tone |
|
G |
3/2=1.5 |
|
(12Ö2)7 =1.4983… |
|
|
|
|
tone |
|
tone |
|
A |
27/16=1.6875 |
|
(12Ö2)9 =1.6817 |
|
|
|
|
tone |
|
Tone |
|
B |
243/128=1.8984… |
|
(12Ö2)11 =1.8877… |
|
|
|
|
hemitone |
|
semitone |
|
C |
2 |
|
2 |
|
The
upper C of the octave has twice the frequency of the lower C. Now there are
seven octaves from the lowest C to the highest C of a piano keyboard and so the
ratio of the frequency of the highest C to the lowest C is 27 = 128.
But if we go up the notes of a musical instrument tuned according to the
Pythagorean system rising by 12 intervals of a fifth or five notes of the
scale, each having a frequency ratio of 3/2 and corresponding to 3 tones and a
hemitone, as follows:
C ® G ® D ® A ® E ® B ®
G flat ® D
flat ® A flat ®E flat ® B flat ® F ® C
the
frequency ratio of the highest C to the lowest C becomes (3/2)12=129.746...
producing a discrepancy amounting to the factor of 1.0136... which is the comma
of Pythagoras.
Equal-temperament Scale and Key Signatures
To
overcome this problem the mathematician and musician Mersenne9,
noted in particular for his discussion of those prime numbers which have the
form 2n - 1 where n is also
a prime number, introduced the equal-temperament scale described in his Harmonie
Universelle (1636-37) which makes each of the 12 semitones of every octave
of a piano keyboard have the same frequency ratio 12Ö2=1.05946...
. This means that the 12 major scales, starting at the keys C, D flat, D, E
flat, E, F, G flat, G, A flat, A, B flat, B, will only differ in pitch so that
if an old-fashioned gramophone record of the note C, which is normally
standardized to a frequency of 262 vibrations per second, is played at an
appropriate higher speed of rotation the sound will be precisely the same as
the corresponding higher key. This observation similarly applies to the 12
minor keys.
Using
the equal-temperament scale composers were now able to write music for the
remote keys. This was not possible with the Pythagorean scale, or some of its
earlier modifications, which confined musicians to keys not far away from the
key of C major. However this meant that the notes of the equal-temperament
scale were never in precise harmony, for example in the scale of C major the
note G should have the frequency ratio of 3/2 = 1.5 referred to the note C for
exact harmony whereas in the equal-temperament scale it is (12Ö2)7
= 1.4983... , and the note A should have the frequency ratio 27/16 =
1.6875 for exact harmony whereas it is (12Ö2)9
= 1.68179... in the equal-temperament scale, as can be seen from Table 4.
Johann
Sebastian Bach wrote his two sets, Book I and Book II, of Preludes and Fugues
using the 24 keys of the equal-temperament scale in the straight-forward order
C major, C minor, D flat major, D flat minor, D major, D minor, E flat major, E
flat minor, E major, E minor, F major, F minor, G flat major, G flat minor, G
major, G minor, A flat major, A flat minor, A major, A minor, B flat major, B
flat minor, B major, B minor. Thus the two books have a total of 48 preludes
and fugues.
Chopin
also wrote a set of 24 preludes (but no fugues) in his Opus 28 although in the
order major to minor and of increasing fifths. Thus he chose the order C major,
A minor, G major, E minor, D major, B minor, A major, F sharp minor, E major, C
sharp minor, B major, G sharp minor, G flat major, E flat minor, D flat major,
B flat minor, A flat major, F minor, E flat major, C minor, B flat major, G
minor, F major, D minor, as displayed in the regular dodecagon or twelve-sided
polygon given in Fig. 1.
Fig. 1
In this
diagram the 12 major key notes C to F are given in clockwise order round the
outer rim of the dodecagon with the key of C major at the top. Within the
dodecagon the 12 minor key notes A to D are again given in clockwise order
round the polygon with the key of A minor at the top. Thus, bearing in mind
that C sharp is the same as D flat and G sharp is the same as A flat we see
that the minor key notes are displaced anticlockwise around the dodecagon by a
quarter-turn relative to the major key notes.
We
notice that the order in which the keys were used by Chopin in his 24 preludes
begins at the C major key at the top of the polygon, then goes to the
corresponding A minor key immediately within the polygon, then to the G major
key next clockwise along the outer rim of the polygon, then goes to the
corresponding E minor key within the polygon, and so on right round the polygon
to the keys of F major and D minor. There is a clear mathematical pattern to
this order of keys. As we go round the polygon in the clockwise sense both the
major and minor keys increase by fifths with the minor keys being thirds,
equivalent to a tone plus a semitone, below the major keys. The same order of
the keys was also used later on by the Russian composers Scriabin in his 24
Preludes Opus 11 and Shostakovich in his 24 Preludes Opus 34, but not by the
French composer Debussy in his Preludes.
The
numbers of sharps # or flats b in the
respective key signatures are as listed in Table 5. Note that F sharp major is
the same as G flat major and D sharp minor is the same as E flat minor.
Table
5. The numbers of sharps and flats in the key signatures of the 12 major and 12
minor scales.
|
|
|
|
sharps # |
|
C major |
A minor |
0 |
|
|
G major |
E minor |
1 |
F |
|
D major |
B minor |
2 |
FC |
|
A major |
F sharp minor |
3 |
FCG |
|
E major |
C sharp minor |
4 |
FCGD |
|
B major |
G sharp minor |
5 |
FCGDA |
|
F sharp major |
D sharp minor |
6 |
FCGDAE |
|
|
|
|
flats b |
|
G flat major |
E flat minor |
6 |
BEADGC |
|
D flat major |
B flat minor |
5 |
BEADG |
|
A flat major |
F minor |
4 |
BEAD |
|
E flat major |
C minor |
3 |
BEA |
|
B flat major |
G minor |
2 |
BE |
|
F major |
D minor |
1 |
B |
It is
worth remarking here that if the scale of Pythagoras is set out on a diagram
such as Fig. 1, it is endless since the note B sharp which we arrive at after
going round the figure once is no longer in the same position as the note C (to
which it is identical on the piano keyboard) due to the comma of Pythagoras.
The note G in the equal-temperament scale has the frequency ratio (12Ö2)7=1.4983...
relative to the note C whereas in the Pythagorean scale the note G has the
frequency ratio 1.5. Thus a diagram based on the dodecagon is not appropriate
for the Pythagorean scale and it is necessary to use a circular spiral figure
that is not closed.
At the
Edinburgh Festival in 1997 the excellent pianist Andras Schiff gave a recital
of Book I of the Preludes and Fugues by Johann Sebastian Bach, in the 24 keys
of the equal-temperament scale. Each of the Bach 24 pieces of music has a
tuneful preliminary part called a prelude followed by a complex contrapuntal
part known as a fugue. They involve complicated counterpoint based on the rules
of harmony and require great dexterity to play on the part of the performing
artist. In an interview on the wireless prior to the recital he said that he
thought of the keys as ranging over a spectrum of colours from the “white key”
of C major to the “black key” of B minor. Indeed it was Beethoven who first
referred to the key of B minor as a “schwarze tonart”. Both Bach and Chopin
wrote particularly soulful music in the key of B minor. Thus the Bach Prelude
No. 24 in B minor as well as his Mass in B minor, and the Chopin Prelude No. 6
in B minor, are characteristically rather dark pieces of music. Frederick
Niecks, in his Life of Chopin, quotes George Sand as saying that the Chopin
Prelude No. 6 was composed one evening when the rain was falling, and that it
“precipitates the soul into a frightful depression”. However it may be that
this composition was actually the so-called raindrop Prelude No. 15 in D flat
major which demonstrates the difficulty in ascribing a particular kind of
emotion to a piece of music.
It is
also interesting to note that the pianist Andras Schiff said that he regards
Bach as a sculptor and sculpture is just how Bertrand Russell described the
beauty of mathematics in the essay on The Study of Mathematics in his
collection entitled Mysticism and Logic18.
Time-signatures
Music
also has an important dependence on time that is another obvious connection
with mathematics. Each note has a given time duration denoted by, for example a
semibreve or whole-note, a minim or half-note, a crotchet or quarter-note and a
quaver or eighth-note. Each bar of music has a time-signature composed of two
numbers 
which expresses the
way in which the notes are played as time progresses forward. Here m denotes
the number of beats in a bar and n denotes the time duration of the beat, for
example 
denotes
four crotchet beats in a bar known as common time often indicated by C, 
denotes
three crotchet beats in a bar, 
denotes three minim
beats in a bar and 
denotes two quaver
beats in a bar. The rhythm of music is a clear indication of the mathematical
connection of music with time that is evident to the ear.
That
music is an art form that depends essentially on the forward progression of
time differentiates it from the static art forms of painting, drawing,
sculpture and architecture. It provides music with the ability to produce a
highly intense emotional experience that is rarely given by the static art
forms.
Characteristics
of Scales
As we
have already pointed out, in a piano accurately tuned to the equal-temperament
scale, every semi-tone is equal so that the 12 major scales and the 12 minor
scales separately only differ in pitch. Thus the feeling that the different
scales possess contrasting characteristics should be rather subjective although
nevertheless believed by some eminent people including the mathematician and
physicist Helmholtz (1821-1894)9. That the 24 scales are felt to
sound different to the ear may be due just to the fact that each scale begins
at a different key of the 12 keys comprising an octave and each scale has a
major form and a minor form in which the third note is a semitone lower than in
the major scale giving it a characteristic melancholic sound. Of course for a
pianist each scale has a different arrangement of white and black piano keys so
that each scale feels different to the touch of the fingers when played. For
example, the keys of B major and D flat major have all five black notes whereas
the key of C major has no black notes. Although the B major scale has five
black notes it is worth appreciating that Chopin recommended this scale for
beginners since the fingers of a hand fit naturally onto the keys of a piano in
this case.
However
it may be that musicians who can associate different colours with different
keys have a sense of absolute pitch which is relatively uncommon although we
know that Mozart had this sense of absolute pitch to an extraordinary high
degree even as a very young child.
It must
also be remembered that when any note is played on a piano or other musical
instrument it is accompanied by overtones or natural harmonics, that is by
tones which have frequencies which are pure multiples of the fundamental note,
and the combination of the fundamental note and its overtones produces a rich
musical sound.
Other
composers, as well as Beethoven, have also thought of music as being associated
with colours. For example the British composer Arthur Bliss wrote a Colour
Symphony. The first movement of this symphony is named Purple and was associated
by Arthur Bliss with the semi-precious stone amethyst and with pageantry,
royalty and death. The second movement is called Red and was associated
by the composer with the precious stone ruby and with wind, revelry, courage
and magic. The third movement is named Blue and was related to sapphire,
deep water, skies, loyalty and melancholy. The final movement is named Green
and was associated with the precious stone emerald and also with hope, joy,
youth, spring and victory. Bliss said that colour was very much in his mind
when he wrote this symphony prior to the time of its first performance in 1922.
Mathematical Nature of Bach's Music
It is
evident that the various associations with each colour in Bliss's symphony have
a lot to do with emotions and feelings although absolutely nothing to do with
mathematics. But music has structure and indeed possesses an abstract and
characteristically mathematical nature in the case of the Preludes and Fugues
by Bach.
Much of
Bach's music, because of its strict observance of the rules of harmony and the
precision of its rhythms and phrasing, gives the listener the impression of
being characterized by a mathematical discipline although with a great
emotional intensity. Again the pianist Andras Schiff, in a BBC television
broadcast on Bach, said that he found that playing Bach's Preludes and Fugues
had a cleansing effect on him and was pure to the spirit and the soul. This
suggests a purity that has been often associated with mathematics. Indeed the
subject is often called pure mathematics when applications are not
involved.
A
further example that can be provided in favour of the mathematical nature of
Bach's music is the series of complex pieces called the Musical Offering
that he composed in Leipzig in 1747 and were based upon a theme given by
Frederick the Great, the King of Prussia, to Bach on a visit he made to
Potsdam. The title-page contains the inscription Regis Iussu Cantio Et
Reliqua Canonica Arte Resoluta which roughly translated means “by the
King's Command the Theme and the Remainder Resolved by the Art of the Canon”.
The first letters of this inscription in Latin are RICERCAR which taken
from the Italian word ricercare implies a piece of research and is also
equivalent to the French word recherche. One of these pieces of music is a
canon in the key of C minor with a remarkable symmetry: it has two themes
performed together, the second of which is the original theme played backwards
in time. It is called a Canon Recte et Retro or sometimes a Crab-Canon.
Not
only does music have form and structure but so does painting, and art
generally, possess considerable structure or pattern. Without structure, music
would just be a discordant combination of sounds, and without pattern paintings
would just be composed of chaotic patches of colour, rather like one might
suppose a new-born child sees the world for the first time.
It is
well understood that it is pattern which is the essence of geometry and indeed,
in many respects, mathematics generally. Without pattern we have just
sensations and feelings, corresponding to the eye that peers inward. But the
eye also looks outwards and perceives structure in the universe and this is
what mathematics subjects to detailed analysis strictly based on logic. The
search for structure, for example in the motion of the sun, the moon, the
planets and the fixed stars in the sky, has been associated with religion
because religion presupposes the existence of a creator who made all things and
imposed harmony on the cosmos. And geometrical patterns are the basis of
Islamic art that I shall be discussing in the next section.
3. ISLAMIC ART
In the
art of Islam, in its purest form, only patterns and colour are permitted, with
the purpose of revealing and mirroring the underlying harmonic structure of the
universe as well as producing visual delight. No human figure or the shape of
any living creature may be present in the decoration of mosques or religious
buildings of any sort.
Arabic
art is strongly based on various notions such as displacement, reflection,
rotation, shape and symmetry that are all fundamental to the study of geometry,
and is closely related to the periodic structure of crystals. We recall that
the Russian crystallographer Fedorov first listed the seventeen one-colour
two-dimensional patterns in 1891.
This
means, of course, that Islamic art is concerned essentially with abstract
geometrical forms and is consequently very limiting and produces a great
challenge to the artist.
The
Arabs were great mathematicians who filled the role vacated by the ancient
Greek geometers until the Renaissance mathematicians of Western Europe began to
appear on the scene in the late 15th century.
Wonderful
examples of Islamic art can be found in Spain at the Alhambra in Granada and other
places throughout the Moslem world such as mosques in Persia, Pakistan, India,
Turkey and Egypt19.
Ceramic
tiles and mosaics arranged in various kinds of geometrical designs were used to
decorate the walls, ceilings and domes of mosques as well as all sorts of other
buildings.
These
Islamic designs have been classified and discussed in considerable detail by
Keith Critchlow in his book on Islamic Patterns with the subsidiary
title An analytical and Cosmological Approach12. They include
repeating patterns based on grids of regular polygons such as equilateral
triangles, squares, regular pentagons, hexagons, heptagons, octagons, and
regular polygons with 9, 10, 12, 14, 16 and 24 sides. Almost imperceptibly
irregular pentagons, hexagons and heptagons were also used. These grids were
employed by the Islamic mathematician-artists to construct beautiful and
intricate patterns composed of, for example, polygons, stars, petals and
arrowheads.
Basic Islamic Designs and Fundamental
Regular Polygons
The
complex structure of Islamic patterns has been discussed in considerable detail
in the book by Keith Critchlow12. Their composition can be best
understood by examining an illustration from this book of the fundamental
regular polygons having three to nine sides, namely the triangle, square,
pentagon, hexagon, heptagon, octagon and nonagon, their vertices on circles,
with some basic Islamic designs composed out of an interwoven tracery. There
are seven patterns in this illustration:
Pattern
A has a basic regular hexagon inside the equilateral triangular grid.
Pattern
B has a flower, in the form of a cross, with four petals inside the square
grid.
Pattern
C has a flower with five petals composed of slightly irregular hexagons
surrounding a pentagonal star within the regular pentagonal grid.
Pattern
D has a flower with six petals composed of basic regular hexagons surrounding a
hexagonal star within a regular hexagonal grid. It is interesting to see that a
star of David, composed of two equilateral triangles, is produced by connecting
the mid-point of the top side of the hexagon with the mid-points of its third
and fifth sides and connecting the mid-point of the bottom or fourth side of
the hexagon with the mid-points of its second and sixth sides.
Pattern
E has a flower with seven petals composed of slightly irregular hexagons
surrounding a seven-pointed star with seven arrowheads, all within a regular
heptagonal grid.
Pattern
F has a flower with eight petals composed of irregular hexagons surrounding a
eight-pointed star made from two superimposed squares together with eight
arrowheads.
Pattern
G has a flower with nine petals composed of irregular elongated hexagons
surrounding a nine-pointed star and nine arrowheads, all within a complicated
grid derived from a nonagon (also known as an enneagon).
These
patterns have been given a highly mystical and metaphysical interpretation
within the Muslim religion that is described in some detail in Critchlow's book
but which we will not discuss here.
Designs
based on a 24-fold symmetry have been used not only in Islamic art but also
elsewhere, for instance in the art of the Renaissance, an example being the
15-16th century paving within the 14th century Cathedral in Siena. In this
paving there is a 24-spoked wheel composed of columns and possessing rotational
symmetry inside a square pattern with an imperial eagle at its centre.
Finally
let us recall that the graphic work of Maurits Escher in this century was
strongly influenced by the remarkable Islamic tessellations he viewed in the
Alhambra in Granada.
4. RENAISSANCE ART
We
shall return to art based on repeating patterns later on in this article. But
for now we shall turn our attention to the pictorial art of Western Europe,
much of which, at its commencement, had a strong religious basis, as did the
art of Islam, although of an entirely different nature. In fact the early
Italian art of the 13th and 14th centuries was much devoted to a pictorial
description of various biblical personages and events. In many respects this
art was symbolic and no great attempt was made to produce pictures which gave a
true description of reality with the exception of the first great artist of the
Renaissance, the 14th century artist Giotto di Bondone (1267?-1337) who showed
a fine intuitive feeling for spatial relationships as can be seen from his
fresco Confirmation of the Rule of Saint Francis painted about 1325. In
his Lives of the Artists the artist and biographer Vasari20
writes that the use of foreshortening by Giotto was much praised and introduced
a new approach to pictorial art.
Perspective
In the
15th century a number of artists began to introduce rather more reality into
pictures by making use of the mathematics of perspective and in keeping with
the new scientific model of the universe that was being developed at that time.
These included the sculptor and architect Filippo Brunelleschi (1379-1446),
often thought of as the originator of perspective about or perhaps before 1413,
the architect and man of letters Leon Battista Alberti (1404-72) who wrote a
short treatise on painting Della Pittura (1436) which contained a
dedication to Brunelleschi in its preface and included the first published
description of one-point perspective, the artist Paolo Uccello (1397-1475), and
the universal artist, mathematician, scientist and engineer Leonardo da Vinci.
In addition we have the mathematician and artist Piero della Francesca (c.
1415/20-92) who wrote a book on perspective De Prospectica pingendi (c.
1474) and a book Libbellus de quinque corporibus regularibus (c. 1480)
on the five regular or Platonic solids: the tetrahedron with four congruent
equilateral triangular faces, the cube with six congruent square faces, the
octahedron with eight congruent equilateral triangular faces, the dodecahedron
with twelve congruent regular pentagonal faces, and the icosahedron with twenty
congruent equilateral triangular faces. In this book he also looks at five of
the thirteen Archimedian solids, namely the five truncated regular solids that
he is generally regarded as having rediscovered first. Piero also wrote a book
called Trattato d'abaco on arithmetic, algebra and geometry including a
discussion of two of the Archimedian solids, the truncated tetrahedron with
four equilateral triangular faces and four regular hexagonal faces and the
truncated cube with eight equilateral triangular faces and six regular
octagonal faces.
It is
interesting to note in this context that the discoverer of the three laws
governing the motions of the planets, Johannes Kepler (1571-1630), wrote a
treatise in 1619 entitled Harmonices mundi libri V or Five Books of the
Harmony of the World in which there is a discussion of convex polyhedra
including the thirteen Archimedean solids whose faces are not all identical.
Also,
the remarkable German artist, metal and wood engraver Albrecht Dürer made a set of four woodcuts
concerned with the use of perspective called the Designer Woodcuts. Dürer was very interested in
mathematics, like the Italian artists before him, and wrote a book on geometry Underweysung
der Messung (c. 1525) treating the subject of instruction on measurement
with compass and ruler as well as other geometrical topics. It is in four
parts: the first part is on plane curves such as conic sections and spirals;
the second part is on regular polygons; the third part is on pyramids,
cylinders and various instruments such as sundials; and the fourth part is on
polyhedra and perspective. Dürer
also wrote a book in four parts Vier Bucher von menschlicher proportion
(1523) on human proportion from a truly geometrical although characteristically
individualistic point of view. In addition he showed a considerable interest in
the symmetry of other living creatures and, for example, painted watercolours
of a young hare, the head of a roebuck, and a sea crab.
One of
the first painters to use perspective was the Florentine Renaissance artist
Paulo Uccello. An example of his work is The Profanation of the Host
which was originally an altar-piece composed of a sequence of six episodes. The
first two of these clearly illustrates his use of linear perspective in the
directions of the black and white floor tiles, arranged in a chequer-board
pattern, and the lines of the walls and ceiling. This set of pictures painted
about 1466 is to be found in the Galleria Nazionalle della Marche at Urbino.
A
beautiful illustration of the use of perspective can be found in the painting
by Raphael of the Marriage of the Virgin or 'Sposalizio' showing
a sixteen sided symmetrical building in which the vanishing point is clearly
discernible1,13. Another remarkable painting in which the vanishing
point can be readily inferred is the Annunciation by the 15th century
artist Carlo Crivelli (active 1457-d. 95) in the National Gallery in London,
perfectly illustrating what has come to be known as 'laser beam theology': a
pencil of rays of light descends from heaven, through a conveniently
constructed hole in a wall, passing through a hovering dove representing the
Holy Spirit and then onto the head of the Virgin Mary13.
As the
art of the Renaissance developed, perspective was increasingly used in the
composition of pictures, as can be well seen in the art of Dürer, for example his engraving of St.
Jerome in His Study21, the art of Sandro Botticelli (1446-1510),
Raphael, Leonardo da Vinci, and many others. Although Botticelli was not an
important figure in the development of perspective he made use of the method in
his art, for example in the Three Miracles of Saint Zenobius on display
in the National Gallery of London and which he painted about 1500. Although the
buildings in this picture are in perspective it depicts three events in the
life of Saint Zenobius that remarkably took place at different times.
The
vanishing point of linear perspective can be used to accentuate a particular
aspect of a painting such as occurs in the Last Supper by Leonardo da
Vinci where the vanishing point is at the head of Christ. Another beautiful
example is the Circumcision by Signorelli (1450-1523) who was trained by
Piero della Francesca and as a consequence became interested in geometry. In
this painting from the National Gallery London, the lines marked out by the
colourful paving stones point directly to the knife used by the mohel,
that is the man who is about to carry out the religious circumcision operation
on the baby Christ.
The
17th century Dutch painter Pieter de Hoogh (1629-84) was able to create a very
realistic feeling of depth in his pictures as can be seen in the painting of an
Interior with a Woman Drinking with Two Men and a Maidservant to be
found in the National Gallery London13.
Also
the Dutch artist Jan Vermeer (1632-75) painted many pictures that showed his
good understanding of perspective as in The Music Lesson. Philip
Steadman has constructed an interesting detailed model of the room and its
contents that are depicted from various points of view in a number of Vermeer's
paintings including The Music Lesson.
Symmetry
is an important aspect of both art and mathematics. It can be perceived clearly
in the art of Western Europe as well as in the art of Islam. But whereas
symmetry is precisely defined in Islamic art, even when it depicts living
creatures, such as in the 14th century Moorish silk textile in a Toulouse
collection19 depicting colourful imaginary birds and their mirror
images, the art of the Western Europe tends only to allude to symmetry as can
be seen in the drawing of a pair of horses and riders by Leonardo in a study
for the Adoration of the Kings (1481)22, and the painting of
the Three Graces by Raphael which also illustrates the fascination of
artists with the symmetrical beauty of the female human form. The Three
Graces may have been based on a Hellenistic sculpture group of which a copy
was found in Rome in the fifteenth century
(although it is now in Siena) and which also can be seen copied as a
wall painting in Pompei.
The
interest of the artist and mathematician Piero della Francesca in perspective
and symmetry is well exemplified in his painting Ideal Town where the
buildings with their arcades on either side of the square are very
symmetrically situated relative to the central cylindrical building and the
perspective is easily distinguished by the paving stones in the square. There
is perhaps a certain resemblance to the piazza in front of the octangular
Baptistry in Florence whose external walls were originally built of sandstone
but were subsequently covered in white Carrara marble and Green marble from
Prato.
The
German artist Hans Holbein the Younger (1497/8-1543), who spent some time in
England, exhibited the general interest in mathematics and astronomy, art and
music by his magnificent double portrait The Ambassadors which shows not
only the two diplomats Jean de Dinteville and Georges de Selve but also a
variety of scientific objects such as globes and astronomical instruments as
well as a six-stringed lute and a case of flutes. In this picture Holbein also
painted a peculiar diagonally shaped object that can be seen to be a distorted
skull, an indication of mortality, if viewed from an appropriate position on
the far right-hand side. Among the objects displayed on the top of the high
table is a celestial globe, a portable cylindrical sundial, a quadrant, and a
six-faced polyhedral sundial made of ivory with a magnetic compass set into its
upper face. Also there is a torquetum, together with its plumb-line, for
determining the positions of stars and planets.
Geometry of Piero della Francesca
Next, I
wish to draw your attention to a particularly remarkable painting also by Piero
della Francesca called The Baptism of Christ which can be found in the
National Gallery, London13. This picture has an interesting hidden
geometrical construction that is based on a circle centred at the fingertips of
Christ discovered by B. A. R. Carter23. This circle, representing
God, passes through the vertices of an equilateral triangle whose upper
horizontal side cuts through the hovering dove representing the Holy Spirit,
and whose vertices are placed at the upper corners of the rectangular main part
of the painting and the right foot of Christ. If a circular arc centered at the
foot of Christ is now drawn through his eyes, the length of its radius enables
a regular pentagon to be drawn with vertices on the original circle, whose five
angles of 72o each at the centre are related to the golden
section which is discussed in Appendix A and which will also be of interest
later on in the next section in connection with nonperiodic tiling.
To show
how Piero della Francesca could have constructed a regular pentagon using just
a straight edge and a compass, consider a right-angled triangle having the base
side of length a and the vertical side of length b = 2a.
The hypotenuse c of this triangle has length (Ö5)a
by the theorem of Pythagoras c2 = a2 + b2,
and can therefore be drawn using only a straight edge and a compass, from which
a length b = (Ö5 + 1)a/2 can then be marked out using a compass. Now the
isosceles triangle ABC having two equal sides AB and AC of length b and
a base BC of length a displayed in Fig. 2, where g = b/a is the
golden section ratio, can be constructed using a straight edge and compass.
Fig. 2
Taking
a circle of radius b centred at the fingertips of Christ situated at O,
ten lines of lengths a forming a regular decagon subtending angles of 360
at the centre can be marked out on the circle with only a compass and a
straight edge. Then, as required, five lines forming a regular pentagon and
subtending angles of 720 at the centre can be marked out on the
circle as depicted in Fig. 3.
Fig. 3
Symmetry
in its various forms is rather pervasive in most aspects of art. It can be
illustrated very well by reference to the work of John Ruskin24
(1819-1900) and in particular his remarkable set of three volumes entitled The
Stones of Venice published in 1851 that describes the architecture of
Venice and explains his deep interest in the philosophy of art. They contain
his beautiful drawing of a spandril decoration from the Ducal Palace possessing
reflection symmetries with colour reversal about the vertical and horizontal
axes as well as the two bisecting diagonals, and without colour reversal about
the other four alternate axes, and an eightfold rotation symmetry without
colour reversal and a sixteenfold rotation symmetry with colour reversal.
Spirals
We have
already pointed out that Dürer in the first volume of his treatise Underweysung
der Messung on geometry discussed the subject of spiral curves.
Spirals
occur in many forms in nature. Examples are the forms of sea-shells such as the
ammonite of the lower Jurassic period of the Mesozoic known only as fossils28,
the nautilus and the gastropods, and snail-shells, all of which do not change
their spiral shape as they grow in size but just increase the number of their
chambers, and the spiral horns of animals such as the rhinoceros, sheep and
goat. Also some climbing plants such as the well known convolvulus weed spiral
upwards, approximately in the form of a circular helix, along their supporting
plants. Some, such as the convolvulus, spiral in the right-hand sense while
others, such as the hop, spiral in the left-hand sense.
In
architecture spirals are used, for example, in the staircases of buildings, a
truly beautiful example being the Scala del Bovolo of the Palazzo Contarini in
Venice.
Another
interesting example is the open staircase in the Chateau de Blois close to
Cloux at Ambois near the river Loire where Leonardo da Vinci spent his last
days as an advisor to Francois I, King of France. An attractive example of a
modern spiral staircase in England can be seen in the De La Warr Pavilion, a
community centre as well as an arts centre having a theatre on the Bexhill sea
front in East Sussex, which was designed by Erich Mendelsohn and opened in
1935.
Appendix
B provides a short discussion of some aspects of the mathematics of spirals,
including the equiangular spiral, which shows a striking resemblance to the
form of the nautilus shell, the Cornu double spiral that is of importance in
connection with the theory of the diffraction of light, and the circular helix.
5. MODERN ART
Much of
the recent modern art of Western Europe is rather chaotic and seemingly
irrational at first sight, possessing a non-objective dream-like, surrealistic,
mystical and metaphysical quality with no obvious geometrical structure
although closer inspection often reveals an underlying pattern. This modern art
seems more concerned with communicating the inner unconscious feelings and
emotions of the artist rather than with an accurate description of the external
world and thus apparently little to do with mathematics. Occasionally even an
element of sheer chance has been deliberately introduced into the structure of
the artwork. Some of the renowned artists whose work displayed many of these
characteristics were Hans Arp (1887-1966), Max Ernst (1891-1976), Marc Chagall
(1887-1985), Joan Miro (1893-1983), Wassily Kandinsky (1866-1944, Salvador Dali
(1904-1989) and Paul Klee (1879-1940). In fact Kandinsky said that 'One thing
became clear to me: that objectiveness, the depiction of objects, needed no
place in my paintings, and was indeed harmful to them.' But he also wrote: 'The
final abstract expression of every art is number' and so he must have believed
in an underlying, perhaps intuitive, mathematical foundation for art.29
Influence of Cézanne and Picasso
Although
Wassily Kandinsky, Paul Klee and many of their contemporaries created art that
was non-objective and metaphysical, it is acknowledged that the French painter
Paul Cézanne (1839-1906) had a most important and fundamental influence on
modern art. His numerous paintings of La Montagne Sainte Victoire have
shown his fascination with the roughly pyramidal shape of the mountain
fashioned by erosion resulting from the action of the wind, rain and ice over
long epochs of geological time. Another painting of Cézanne's exhibiting his
interest in solid shapes is The Bathers in the National Gallery London
where he has painted the bodies and limbs of the female bathers with decidedly
emphasised cylindrical forms. In a famous letter to the painter Emile Bernard
he wrote: “traiter la nature par le cylindre, la sphere, le cone, le tout mis
en perspective...” which suggests that he thought that art should be based, at
least to some extent, on solid geometry. This may have led unintentionally to
the cubist movement as exemplified by the painting Portrait of Ambroise
Vollard (1909-10) by Pablo Picasso (1881-1973) and the painting Young
Girl with Guitar (1913) by George Braque (1882-1963) in which the subjects
are fragmented into straight-edged triangular- and rectangular-like pieces with
curved lines mostly excluded.
This
movement based on three-dimensional geometrical considerations did not survive
for any great length of time but it did lead subsequently to non-objective art.
Thus Picasso and those who followed him, deliberately introduced
non-symmetrical fantastic features into their paintings in order to emphasize
some of the irregularities of nature, to exhibit pathos and to shock the
viewer. Actually their clear feeling at that time was that the geometry of
linear perspective is very restrictive and in order to describe the
multi-dimensional character of a subject it was necessary to depart from
straightforward symmetry. This can be observed in several of the paintings by
Picasso such as the Weeping Woman or ‘Femme en Pleurs’ (1937),
the Seated Woman (1938), and the Woman in a Blue Dress (1941) in
which the faces of the women are seen from two different perspectives. However
even the artist Henri Matisse (1869-1954) who remarked that L'exactitude
n'est pas la verité and was a member of the group known as the Fauves or
wild beasts, used perspective in a roughly conventional way as found in the
painting Studio under the Eaves, where the vanishing point is clearly
situated in the sunlit open window of the dark studio. The use of perspective
is obviously necessary if any artist is to be able to depict the real objective
world in any kind of credible fashion. In addition Matisse possessed a
wonderful sense of the beauty of symmetry as can be seen from his lovely
colourful painting Lady in Blue.
It is
worth remembering here that in the painting Three Miracles of Saint Zenobius
by Botticelli the events depicted take place at different times, and this is
akin to avoiding single point perspective by introducing elements taken from
different viewpoints in three-dimensional space although this may appear very
peculiar and unrealistic as in the paintings of women by Picasso.
Picasso
was a highly versatile worker with a prodigious output who possessed great
ability as a graphic artist some of which he expressed in a neo-classical style
characterized by the choice of a quite realistic form into which he introduced
a non-rational and strange mystical quality. This was done, for example, by the
transposition, dislocation and metamorphosis of parts of the human body such as
the face and the eyes, and the use of symbolism to represent various ideas and
thus entering a world of fantasy where normal geometrical relationships are
absent.
An
important artist who was also strongly influenced by the work of Picasso, and
in particular the cubist movement, was Piet Mondrian (1872-1944) who developed
an abstract but objective art form sometimes based on coloured geometrical
shapes such as rectangles, an example being Composition with Red, Yellow and
Blue. He used the metaphysical phrase 'plastic mathematics' introduced by
the Dutch philosopher M. H. J. Schoenmaekers to describe his art. Schoenmaekers
also used the phrase 'positive mysticism' to characterize his philosophical
vision of Neo-plasticism which, like the strictly geometrical art of Islam,
attempts to picture the character and structure of the universe although in a
considerably less precise way.
Some
modern art gives the appearance of being really chaotic, an example being the
so-called 'action painting' carried out by the American artist Jackson Pollock
(1912-56) characterized by turbulence and great agitation and to which it is
rather difficult to ascribe any geometrical design.
There
are a number of interesting modern exceptions to the surrealistic approach of
Picasso and his followers including the group known as Op Artists doing
so-called optical art beginning with the work of the Hungarian artist Victor
Vasarely (b. 1908). An example of his art is Metagalaxy (1959) involving
a twofold rotation that produces colour reversal between black and white.
Another Op Artist is the American artist Richard Anuszkiewicz (b. 1930) whose Division
of Intensity (1964) has a symmetry which involves invariance with respect
to vertical and horizontal reflections.
Some of
the earlier work of the British artist Bridget Riley (b. 1931) gives the
impression of being founded on mathematical ideas since her early works done in
black and white are composed of lines and curves, an example being Twist.
Indeed a few of her recent paintings in colour retain the appearence of being
based on patterns with a flavour of geometrical symmetry. However there is also
some true computer art actually based on mathematical curves such as straight
lines, circles, and spirals. An interesting pattern possessing a reflection
symmetry about a vertical line through the centre, created by using
combinations of sines and cosines to shape curves, is Entwined Hearts
illustrated in Fig. 4.
Fig. 4. Entwined Hearts
Graphic Work of Escher
The
most remarkable example of modern art having a strong mathematical basis is the
entirely secular work of the Dutch artist Maurits Escher whose beautiful
graphic work2,14,17,25 has a very close connection with the
religious art of Islam which I was concerned with earlier in section 3. His art
takes the form of the regular close filling of a plane with repeating patterns
of geometrical figures and involves symmetries associated with various
mathematical transformations such as translations, reflections in a line
producing mirror images, and rotations about points which may be twofold,
fourfold, threefold or sixfold, producing lovely visual effects. For example,
his picture of starfishes, clams and shells has fourfold rotation points where
four starfishes and four snail shells meet and twofold symmetry points where
clams meet. In the standard symbolic notation of crystallography given in Table
3 of section 2 the fourfold symmetry pattern is denoted by p4.
Escher
also makes use of glide reflections, that is a translation or glide in a given
direction followed by a reflection in a line parallel to the direction of
translation, and often the symmetries he employs involve colour transformations.
In fact there are seventeen possible basic one-colour two-dimensional patterns
that can be used to close fill a plane by repetition as in a wallpaper or a
tessellation, and forty-six two-colour two-dimensional patterns.
The
one-colour two-dimensional patterns were first listed by the Russian
crystallographer Fedorov in 1891 and have already been given, together with
their crystallographic symbols, in Table 3 displayed in section 1 on Primitive
Art.
Whereas
the geometrical shapes used by the artists of Islam are mostly abstract, those
used by Escher include birds, fishes, lizards, bats, insects such as bees and
wasps, butterflies, flowers, shells, unicorns, horses as well as human figures14,25.
To help
understand his graphic work we shall discuss a number of his beautiful periodic
drawings and connect their designs with the crystallographic symbolic notation
associated with their characteristic symmetries.
We
begin by considering one-colour designs classified according to Table 3.
Firstly,
Escher’s drawing of light and dark geese is generated by two independent
translations and has the symmetry pattern p1. A careful inspection shows
that the shapes of the light geese and the dark geese are slightly different.
The drawing is composed of a lattice of unit cells that are congruent
parallelograms and enclose the basic pattern of the design. An example of a
unit cell in this design is the parallelogram formed from the straight lines
joining the tips of the beaks of four neighbouring light geese. Alternatively
we could use the tips of the beaks of four adjacent dark geese. The drawing is
then constructed from translations of this unit cell so as to entirely cover
the whole picture.
Next,
his drawing of birds and fishes has twofold rotation points where (i) heads of
birds meet, (ii) tails of birds meet, (iii) right wings or left wings of birds
meet. This has the symmetry pattern p2.
As an
example of the symmetry pattern pg we have his design composed of a
white man and a black man. There are rows of white men facing in opposite
directions separated by rows of black men facing in opposite directions so that
the unit cell is composed of two white men and two black men. By translating a
row of men in the vertical direction and carrying out a reflection in a
parallel mirror line it can be transformed into the row of men of the same
colour facing in the opposite direction. This is a glide reflection denoted by
the symbol g.
His
picture of flies, falcons, bats and butterflies, with two sets of perpendicular
mirror lines and twofold rotation points where the mirror lines intersect has
the symmetry pattern pmm.
Next
his picture of bees and wasps has a threefold symmetry pattern denoted by p31m.
There are threefold points where the abdomens of the insects meet and where the
legs of the insects meet. There are three sets of glide lines in this design.
His
picture of bats and two other creatures also has a threefold symmetry pattern
but now denoted by p3m1. The threefold rotation points occur where the
heads, or tails, of the creatures meet. Mirror lines pass through these
rotation points. Also there are three sets of glide lines in this pattern.
We next
consider two-colour symmetry patterns designed by Escher.
Our
first example is of black and white horsemen. Here black horsemen are converted
into white horsemen by a vertical colour glide transformation. A horizontal
translation converts white horsemen into white horsemen and black horsemen into
black horsemen. This picture of horsemen has the two-colour symmetry pg/p1.
Our
second example is of black and white beetles. Here vertical mirror lines
transform black beetles into black beetles and white beetles into white
beetles. Translations in the vertical direction also transform white beetles
into each other and black beetles into each other. Black beetles are
transformed into white beetles by a glide reflection, the colour glide line
being parallel to neighbouring mirror lines and halfway between them. This
picture of beetles has the two-colour symmetry cm/pm.
We next
consider some fascinating examples of polychromatic symmetry using three or
four colours.
His
picture of unicorns coloured in red, yellow and green has vertical glide
symmetry lines that transform unicorns facing in one direction into unicorns facing
in the opposite direction. Also vertical translations transform unicorns from
one colour into another colour. This picture of unicorns has a pg/pg
three-colour symmetry pattern.
Further
Escher’s picture of lizards involving a sixfold rotation symmetry with three
colours, namely white, red and black has seventeen twofold points. There are
twofold points between the tail ends of two lizards of the same colour. There
are threefold rotation points where the right rear legs of the lizards meet and
there are sixfold rotation points where the left front legs of the lizards meet
transforming their colours from white to red to black by rotations through 600.
This picture of lizards has a p6/p2 three-colour symmetry
pattern.
Also
his picture of red, black and grey tadpoles has mirror lines in the vertical
direction and along the lines making angles of 600 with the
vertical. Translations along these lines transform red into black into grey
tadpoles. There are threefold points where the heads of the tadpoles meet and
there are colour-threefold points where the legs of the tadpoles meet. In
addition there are glide lines present in this design. This picture of tadpoles
has a p31m/p3m1 three-colour symmetry pattern.
Escher
was fascinated by mathematics and indeed said that “although I lack any
training or knowledge in the exact sciences, I often feel closer to
mathematicians than to my fellow artists”. Some of his art is concerned with
going to infinity and involves representations of Lobachevskian or hyperbolic non-Euclidean
geometries inside a circle using, for example, alternating fishes along
circular arcs called Circle Limit III, and interlocking angels and devils
called Circle Limit IV.
In
Euclidean geometry, if a straight line L is chosen then, for any given point P
which is not on this line, there is just one straight line L’ passing through
P, in the plane of L and P, which does not meet L. The lines L’ and L are said
to be parallel. However in a hyperbolic geometry, there is more than one line
passing through P, in the plane of L and P, which does not meet the line L.
Lobachevsky (1793-1856), together with Gauss (1777-1855) and Bolyai
(1775-1856), were the first mathematicians to introduce non-Euclidean geometry.
Lobachevsky's work was published first in 1829-30 but was written in Russian
and was only noticed by a few people.
The
model of hyperbolic geometry that Escher used in his Circle Limit designs was
that introduced by the French mathematician Henri Poincare (1854-1912) in which
all the hyperbolic points lie inside a bounding circle. The hyperbolic lines
are arcs of circles that meet the bounding circle at right angles, with the
inclusion of the straight lines that are diameters.
Escher
had an interesting correspondence with the geometer H. S. M. Coxeter who
greatly admired his artwork and often wrote about it2,17.
Penrose Non-periodic Tessellations
Some
time ago the Oxford theoretical physicist and mathematician Roger Penrose
showed how two different tiles related by the golden section ratio g = 1.61803...
may be used to completely fill a plane with an approximate fivefold rotational
symmetry26,27. One pair of such tiles is two rhombuses, one of which
possesses internal angles of 720 and 1080 while the other
has internal angles of 360 and 1440, shown in Figs. 5a
and 5b.
Fig. 5a
Fig. 5b
Another
pair of such tiles is the kite and the dart, also having internal angles of 720
and 360 that are shown in Figs. 6a, 6b, 6c. Interesting arrangements
of these called the infinite sun pattern and the infinite star pattern can be
constructed27. These tiles have the angles 720 and 360
arising from the golden section discussed in Appendix A.
Fig. 6a
Fig. 6b Kite
Fig. 6c Dart
A plane
cannot be completely filled using a repeating pattern having a true fivefold
symmetry since it is mathematically impossible to do this using only regular
pentagons as can be done with equilateral triangles, or squares, or regular
hexagons, having common sides. An approximate fivefold symmetry analogous to
the Penrose mosaic pattern has been found in materials known as quasicrystals.
An example is shechtmanite, an alloy of magnesium, aluminium and zinc, which
produces a quasicrystal composed of icosahedrons with 20 triangular faces
having a fivefold symmetry.
Penrose
nonperiodic tiling has been used for decorating walls and floors. The decagons,
or polygons with ten sides, which are present throughout the Penrose mosaic
pattern all have the same orientation27.
Other
tilings exhibiting interesting patterns have been created. An intriguing spiral
tiling has been devised by Heinz Voderberg. It has two centres and is composed
of nine-sided polygons, two of which combine into an octagon. Another
interesting tessellation is composed of convex non-congruent heptagons.
Music
We have
seen that much recent art has a transcendental, abstract and non-objective
nature, unlike the tessellations we have considered above that have a decidedly
mathematical basis. This has been mirrored in the composition of music in
recent times. This may be said to have commenced with the music of Igor
Stravinsky (1882-1971) and in particular with his composition Le Sacre du
Printemps or The Rite of Spring dating from 1910-13, written even
before the first world war of 1914-18, the Russian revolution and the rise of
Nazism that destabilized European society to such a great extent. But although
there are vivid dissonances in this music, which was described by Stravinsky's
biographer Roman Vlad as brutal, savage, aggressive, and apparently chaotic, it
has very distinctive rhythmic beats characterized by detailed structure and
pattern and thus could be thought to have a kind of mathematical design. But
disharmony is nothing especially new in music since the so-called Dissonance
string quartet No. 19 in C major K.465 by Mozart begins with a discordant
harmony that was once a topic of some debate. Haydn made the comment that if
Mozart wrote the music in this way he must have had his own reasons for doing
so. It is simply analogous to a slightly broken symmetry in a geometrical
design.
6. EPILOGUE
Art is
a visual means of communication which employs patterns, shapes and colours to
convey emotions from within the artist and to describe the world without in a
characteristically beautiful manner which produces pleasure in the observer or
possibly shocks and perturbs him or her. It may be used to deliver a religious,
social or political message, or to describe people, places or events, and to
decorate buildings.
Art is
not simply an accurate picture of a person or place. This can be achieved by
just taking a straightforward photograph. It must always communicate something
else. However this cannot be carried out without some underlying structure that
is based on geometrical considerations or other mathematical principles
although this may be quite intuitive. The mathematical basis does not have to
be very explicit although in certain types of art it can be. For example
fractals, and in particular those based on Mandelbrot sets, briefly discussed
in Appendix C, can be used to create very weird and beautiful patterns composed
of the points c in the complex plane which lead to bounded sequences derived
from non-linear mappings such as z ® z2 + c
where z = x + iy is a complex number in which the square root of minus
one i = Ö{-1} is an imaginary number and x, y are real numbers.
However many would believe that, without some significant creative input from
the unconscious mind of the mathematician programming the computer, patterns
derived from Mandelbrot sets should not be regarded as true art.
It has
been often emphasized before that symmetry plays an important role in art. One
has only to examine the frescos painted by Michelangelo on the ceiling of the
Sistine Chapel in Rome illustrating various scenes from Genesis in the Old
Testament to see that they are arranged in a symmetrical, although cleverly
broken, pattern and revel in the symmetry of the naked human form. Moreover the
creation of Adam by God in the central panel exhibits a beautiful and revealing
symmetrical relationship between God and his human creation. The forefinger of
the right hand of God is pointing to the forefinger of the left hand
of Adam. No combination of physical translations and rotations can transform
Adam into God but a reflection in a mirror roughly parallel to their bodies
through the point where the finger tips of God and Adam meet can achieve this
in an approximate although, of course, not in an accurate way. Then Adam points
with his right hand whereas God now points with his left hand. This suggests
that Michelangelo is asserting that God created Adam not in his image but as a
mirror image of himself. Also the creation of Eve from Adam has an interesting
symmetrical pattern with Eve emerging along one diagonal and Adam lying approximately
along the other diagonal together with four naked youths at the corners of the
rectangular panel. The ancestors of Christ are symmetrically arranged in the
eight triangular spandrels around the Sistine Chapel ceiling, with seven
Prophets and five Sybils in between.
A
partial deviation from symmetry is often needed to adequately express an
artist's intentions or feelings. For example we may consider the painting by Dürer of Christ Among the Doctors
in which there is a beautiful indication of a certain rotational symmetry in
the hands of the young Christ and one of the wicked doctors who are in dispute21.
It should not be surprising to us that Dürer wrote a treatise in 1525 on geometry entitled Underweysung
der Messung or literally 'Instruction of Measurement'.
William
Blake, despite his mysticism, understood symmetry very well as is evident from
his poem “The Tyger” containing the words:
“What immortal hand and eye
Could frame thy fearful symmetry?”
and
from his watercolours30, for example 'The Ancient of Days' or 'God
Creating the World' which is the frontispiece for Europe and illustrates
A Prophecy:
“When he set a compass upon the
face of the deep”
suggesting
an intimate connection with geometry and the measurement of the universe. Here
Elohim or God is depicted as an old man with a white beard given the name
Urizen by Blake. Urizen, which may be a synonym for your reason and
represents the dark oppressive rational side of the creator, is leaning forward
with a compass from a radiant heavenly globe surrounded by clouds with his hair
blowing in a cosmic or solar wind. There seems to be a similarity to the figure
of Euclid in the painting named The School of Athens by Raphael in which
he is also leaning down to take a measurement with a compass13. This
painting has a strikingly symmetrical configuration with its vanishing point
between the heads of Plato and Aristotle in the centre of the picture. It is
well authenticated that William Blake held Raphael, as well as Dürer, in high esteem even as a youngster.
Indeed he wrote that “I am happy I cannot say that Rafael Ever was from my
Earliest Childhood hidden from Me”.
Although
William Blake had a distinct apathy towards science, and in particular the
supreme mathematician Isaac Newton who he also pictured looking downwards and
holding a compass like the demonic Urizen who symbolises a creator who uses
reason without imagination, many of his paintings display a great sense of
symmetrical design and thus some relationship to geometry.
In the
Palazzo Ducale in Urbino, Italy there is a very interesting and mysterious
painting The Flagellation of Christ by Piero della Francesca. In this
painting the perspective is so precise that it has been possible to construct
an accurate three-dimensional physical model of the buildings and placement of
the people in the picture, as has been shown by B. A. R. Carter23.
The
intricate design of the paving on the floor of the portico in this picture is
not at all clear from a cursory view. However an analysis carried out by T. C.
Czarnovski23, shows that Christ is standing on a large circular
black tile and that in the centre of the square foreground there is an
eight-pointed star composed of white rhombus-shaped tiles surrounded by eight
black square tiles. In addition there are four larger black square tiles in the
corners, and at the edges there are four black triangular tiles as well as
eight white irregular pentagonal shaped tiles. There is a similar design in the
background.
Carter's
model of The Flagellation of Christ shows a scene that may have a
religious and historical significance but nevertheless is entirely imaginary
and could never have occurred as depicted here.
On the
other hand the model built by Philip Steadman based on the group of paintings
by Jan Vermeer including The Music Lesson, is very likely a true
construct using perspective of an actual room and its contents and people.
Just as
a two-dimensional flat picture of a three-dimensional scene can lead to a
three-dimensional construction of the scene, as in the previous example, our
three-dimensional view of space, which necessarily involves a time dimension
due to the finite speed of light, can lead to a four-dimensional space-time
mathematical model of the universe. A major achievement of Einstein in his
theory of relativity was the reinterpretation of dynamics and gravity in terms
of a four-dimensional non-Euclidean space-time geometry as well as giving rise
to mathematical cosmology. Thus we understand that Geometry, and indeed
Mathematics generally, provides structure for both Science and Art.
Artists
and musicians, like mathematicians and scientists, think and feel deeply about
the structure of the world around them. Often they feel a mystical significance
towards their art as did the modern artist Piet Mondrian with his use of the
phrase 'plastic mathematics' to describe his art and its relation to the
universe.
Obviously much modern art is objective. As an illustration we show the
following indian ink drawing (Fig. 7) by the British war artist Daniel
Moiseiwitsch in which a soldier in the second world war of 1939-45 is riding in a
tram in the East-End of London. The tram rails, which take the role of the
perspective lines in the drawing, approximately converge to the distant
vanishing point, intimating the ultimate destiny of the soldier in the war.
Other pictures by Daniel Moiseiwitsch can be viewed at the V & A in London.
Fig. 7. Soldier riding in a Tram in
London
Earlier
in this book, I drew attention to the connection between music and mathematics,
in particular with regard to the Preludes and Fugues by Johann Sebastian Bach
using the equal-temperament scale. In the book entitled Mozart The Man and
his Works by W. J. Turner the author writes “One may speak often of a
movement of Mozart just as a mathematician might speak of a beautiful
proposition”. This is especially clear in the case of Mozart's last symphony
K.551 in the key of C major known as the Jupiter. The Allegro molto
fourth and last movement or finale, is written with incredible skill in a very
complicated contrapuntal manner. Mozart uses five themes, three from the
first-subject and two from the second-subject, that are made to combine
harmoniously together in a way which can only be described as a miraculous
mathematical skill, and which yet is intensely beautiful and moving as true art
should be.
But
there is an essential difference between music and pictorial art. Whereas music
is an art form that depends on the flow of time, pictorial art as well as
sculpture and architecture are transparently spatial in character and frozen in
time. However the music written down by the composer on sheets of paper
envisages the music as a single entity that can be read and relates progression
in time with position in space. In fact the performer of a piece of music, for
example for the piano, has to own the music, as has been said by the
renowned pianist Artur Rubinstein, that is the artist has to endeavour to
remember and also perform the music completely as a unified whole. It is
noteworthy that the theory of relativity combines space and time together into
a single geometrical space-time continuum and that this provides a remarkably
accurate way of describing the universe at the macroscopic level.
Finally,
it is noteworthy that so-called primitive art is strongly based on geometry and
mathematical symmetry, as indeed is the religious art of Islam. As our
knowledge of mathematics developed further during the early Renaissance and our
understanding of the physical world improved, the use of perspective and
geometry by the artists of that time in order to introduce more reality into
their pictures became increasingly evident, and subsequently was used very
successfully by many of the artists who followed them. It is therefore rather
interesting that towards the end of the 19th century and then in the
20th century, when the easy use of photography became widespread, a revolt
against the straightforward use of perspective came about and non-objective art
became fashionable. Nowadays it is still felt that depicting the real world
accurately, as was done in the past, is much less important than giving full
freedom to the expression of emotions and the unconscious mind. However there
now seems to be developing a turn away from this attitude and some interest is
being shown not only in the mathematical art of Escher but also in other modern
artists concerned with objectivity who are once again wishing to describe the
world in a reasonably accurate way based on considerations of geometry and
symmetry.
APPENDIX A. GOLDEN SECTION
If a
section of length a is chosen on a line of length b and we take
{b-a}/a = a/b
then a
is the golden section of the line of length b. It follows that
(b/a)2 - b/a – 1 = 0
and so,
setting g = b/a, we obtain the quadratic equation
g2 – g - 1 = 0
that
has the positive solution
g = {1+Ö5}/2 = 1.61803...
Now
consider the isosceles triangle ABC shown in Fig. 2 with equal sides AB and AC
of length b and base BC of length a having the angle 360
at the vertex A.
Then
the equal angles at B and C are 720. Next we construct an isosceles
triangle BCD, where D is on the line AC, with equal sides BC and BD of length a,
and angle 360 at the vertex B. Then the angle ABD must be 360
also and so ADB is an isosceles triangle with the length of the side AD being
equal to the length a of the side BD. Hence DC has length b-a and
so
(b-a)/(2a) = cos 720
giving
b/a = 1 + 2cos 720.
Also
b/a = 2cos 360
and
since, by the trigonometric formula cos 2q = 2cos2 q - 1,
we have
cos 720 = 2cos2 360
- 1,
we see
that b/a satisfies the quadratic equation derived above for the golden
section ratio g.
Thus
the golden section ratio is given by
g = 1/{g - 1} = {1 + Ö5}/2 = 1 + 2cos 720 =
2cos 360 = 1.61803...
Fibonacci Numbers
The
golden section is also related to the Fibonacci sequence of numbers
1,2,3,5,8,13,21,34,55,89,...
The nth
number an of the sequence is obtained by adding the previous
two
numbers an-1 and an-2 together giving:
an = an-1+an-2
which
is a formula first stated by the astronomer Kepler although it must have been
known to Leonardo Fibonacci (c.1170-1250), also called Leonardo of Pisa, who
originally introduced his sequence of numbers to describe the rate at which
rabbits would produce offspring if each pair gives birth to a new pair each
month, which produce at the end of two months, and the rabbits live for ever.
Then we
have
an/an-1 = 1 + an-2/an-1
and if
we set fn = an/an-1 we
get
fn = 1 + 1/fn-1.
It
follows from this result that fn > 1 and
fn+1 - fn = -(fn
- fn-1)/fnfn-1
so that
fn+1 - fn has the opposite sign to fn
- fn-1 and |fn+1 - fn| < |fn - fn-1|.
In fact
we have f2 = 2, f3 = 3/2 = 1.5, f4
= 5/3 = 1.66666..., f5 = 8/5 = 1.6, f6 = 13/8 = 1.625, f7 = 21/13
= 1.61538..., f8 = 34/21 = 1.61904..., f9 =
55/34 = 1.61764....
We now
observe that the limiting value f of fn as n ® ¥
satisfies the equation
f = 1 + 1/f
which
yields the quadratic equation f2 – f – 1 = 0 which is
the same equation as that satisfied by the golden section ratio g. Thus
the ratio fn of two successive Fibonacci numbers approaches
the golden section ratio as we proceed up the sequence. For example f10
= 89/55 = 1.61818... is only slightly larger and f11 = 144/89
= 1.61797... is only slightly smaller than
g = (1 + Ö5})/2 = 1.61803... .
APPENDIX B. SPIRAL CURVES
Gnomons and the Equiangular Spiral
A
certain class of spiral curves can be related to the golden section discussed
in Appendix A in the following way. Let us again consider the triangles shown
in Fig. 2. The triangle BCD is similar to the triangle ABC since the angles CBD
and BAC are equal and
AB/BC = BC/CD = CA/DB
that is
a cyclic formula obtained by going once round the corresponding sides of the
two triangles:
AB ® BC ® CA and BC ® CD ® DB.
The
triangle ADB is called the gnomon to the triangle BCD. The gnomon has a
very old history going back to Pythagoras, Euclid and Hero in ancient Greek
times and perhaps also to ancient Egypt. When a gnomon is added to a certain
geometrical shape such as a triangle or parallelogram the shape remains
unaltered. Thus in the case of the isosceles triangles in Fig. 2, when the
gnomon triangle ADB is added to the triangle BCD the similar triangle ABC is
obtained. If we now construct a series of gnomons as shown in Fig. 8 the
vertices A, B, C, D, E, F, G,... of the
successive gnomons lie on an equiangular or logarithmic spiral, that is a
spiral whose tangent at any of its points makes the same angle a with
the radius directed from the limiting point or centre of the spiral. If r
is the radial distance from the centre to a given point of the spiral and q
is the angle that the radius makes with a fixed line in the plane of the
spiral, its equation takes the form
r = aexp lq
so that
ln (r/a) = lq
and the
differential coefficient dr/dq is given by
dr/dq
= lr
where l =
cot a. Here ln denotes the natural
logarithm to base e = 2.71828... and cot denotes the cotangent or the
reciprocal of the tangent.
If we
again refer to Fig. 8, where we have shown an equiangular spiral passing
through the vertices of successive gnomons, it is interesting to note that the
centre of the spiral is situated where the dashed lines meet. These are the
straight line joining the vertex C to the mid-point of the opposite side AB of
the triangle ABC and the straight line joining the vertex D to the mid-point of
the opposite side BC of the similar triangle BCD and are called medians.
All the other corresponding medians, such as the one from the vertex E to the
mid-point of CD, also pass through the centre of the spiral. The appearance of
the equiangular or logarithmic spiral in Fig. 8 is strikingly similar to the
spiral shape of fossil nautilus shells28.
Fig. 8. Gnomons and Equiangular
Spiral
Cornu Double Spiral
The
Cornu double spiral, introduced by the French physicist Cornu in 1874 to
interpret the fringes produced by the diffraction of light by a rectangular
slit in a screen, is displayed in Fig. 9. It looks somewhat like the double
spirals found in the stone age art of 5000 years ago in the passage-grave
situated in Newgrange. The parametric equations of the Cornu spiral are
, 
where x,
y are the rectangular Cartesian coordinates of a point of the spiral curve
and the end points of the double spiral shown in the figure occur at the upper
limit value A = 5. One of the two limiting points of the double spiral
occurs when
=
, 
=
,
and the
other when x = y = -1/2. The integrals occurring here are known as
Fresnel's integrals.
Fig. 9
The
Cornu double spiral is invariant with respect to a twofold rotation about the
origin. Although the Escher woodcut in three colours, Whirlpools, appears at
first glance to be similar to the Cornu spiral, closer inspection shows that it
involves a dual double spiral which is invariant with respect to a twofold
rotation about the central point only if we ignore the colour reversal from red
to grey and grey to red.
Circular Helix
A
spiral staircase, such as the Scala del Bovolo, has the shape of a circular
helix that has the same form as a string tightly wrapped around a circular
cylinder in a spiral curve. It is given by the equations
x = acos q ,
y = asin q , z = bq
where x,
y, z are rectangular Cartesian coordinates with the z axis chosen
along the central axis of the cylinder, a is the radius of the cylinder,
q is the angle of rotation about the
axis and b is a constant which determines the rate of rotation. It can
also be expressed in the alternative form
x2 + y2 = a2,
y/x = tan(z/b), z = bq .
Epicycles
The
epicycles used by Ptolemy to interpret the motions of the planets in the night
sky correspond to the uniform motion of a point P on a circle C1
whose centre moves uniformly round a circle C2 centred at the Earth.
It has the flower-like shape displayed in Fig. 10 and is given by the Cartesian
coordinates
x = x1 + x2, y = y1
+ y2
where
x1 = a1cos(w1t + e1), x2 = a2cos(w2t + e2)
y1 = a1sin(w1t + e1), y2 = a2sin(w2t + e2)
Here t
is the time, w1 is the angular velocity, that is
the rate of rotation, of P round the circle C1 given by the equation
x12 + y12 = a12
and w2 is the angular velocity of the
centre of the circle C1 round the circle C2 given by the
equation x22 + y22 = a22.
Fig. 10 Epicycle
A closed
curve results if w1 is an integer multiple of w2, that is w1
= w2n where the integer n-1 is the
number of loops in the epicycle.
The
particular epicycle curve having 8 loops shown in Fig. 10 was obtained by
taking a1 = 0.2, a2 = 1, w1 = 9, w2
= 1, e1 = e2 = 0.
APPENDIX C. MANDELBROT SETS
Amazingly
beautiful and wonderfully intricate patterns can be produced by selecting those
points c for which a mapping such as z ® z2 + c produces
a bounded sequence. Such a set of points c is called a Mandelbrot set.
This
mapping is characteristicaly non-linear since it depends on the square of z.
Here z
and c are complex numbers. Thus we have
z = x + iy
where x
and y are ordinary real numbers and the imaginary number i
is the square root of -1, that is i = Ö{-1} so that i2 =
-1. The numbers z determine the position of points in the complex
plane. The initial number z0 is called the seed, and the
n+1 th number of the sequence zn+1 is given in terms of the n th
number zn by the formula
zn+1 = zn2 +
c
so that,
for example, z1 = z02 + c.
Then
the Mandelbrot set of numbers for this mapping are those complex numbers c for
which there are sequences of complex numbers zn which are
bounded, that is they are confined within a finite region.
If we
start the sequence with the number z0 = 0 we obtain
z1 = c, z2 = c2
+ c, z3 =(c2 + c)2
+ c, z4 = [(c2 + c)2
+ c]2 + c,
z5 = {[(c2 + c)2
+ c]2 + c}2 + c
and so
on. For example if we take c = i we get the bounded sequence 0, i,
-1 + i, -i, -1 + i, -i, ... but if we take c = 1 we
get the unbounded sequence 0, 1, 2, 5, 26, 677, ... and so i belongs to the
Mandelbrot set but 1 lies outside it.
However
even if c is a member of the Mandelbrot set not all values of the seed z0
may lead to a bounded sequence of numbers zn. The boundary
between those values of the seed z0 that produce bounded
sequences and unbounded sequences is called a Julia set and may have a
very complicated structure.
Mandelbrot
sets and Julia sets are examples of fractals which are characterized by
having the same repetitive structure as one examines the set in closer and
closer microscopic detail. This can be seen from a picture of a part of the
Mandelbrot set, coloured black, in the complex plane shown in Fig. 11.
Fig. 11.
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