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Is topology more than an abstract concept?
Loosely defined, the topology of an object is the property ‘that doesn't change when
you bend it or stretch it as long as you don't break anything1.’ Informal as this may
sound, this concept is at the very heart of every branch of topology. This feature gives
rise to the popular mathematical joke:
Q: What is a topologist? A: Someone who cannot distinguish between a doughnut and a coffee cup.2
In theory, a (remarkably malleable) coffee cup could be continuously transformed
into a doughnut shape without ‘breaking anything’ as both are objects with the same
topological structure: a single hole. The two objects are said to be homeomorphic;
this is a mathematical term for the idea that two objects are topologically equivalent.
In short, topologists are concerned about the way in which a part of an object is
connected to the other parts of that same object – features such as volume and surface
area are dispensable.
Point-set topology, or general topology, is the pure study of the topological
properties of various objects and the categorisation of the objects according to such
properties. The shift from the classical view of how geometry should be studied – that
is, a focus on the physical, real-life properties of tangible, real-life objects – to a more
abstract, ‘thinking-outside-the-box’ approach towards the end of the 19th century
prompted a new interest in topology, especially insofar as it related to exciting new
‘abstract spaces’ such as n-dimensional manifolds3. A multitude of branches of
topology now exist, ranging from the study of networks to the study of knots, and the
practical applications of the subject are many and varied. It would seem obvious that
such a subject was invaluable, but often the wider community underestimates the
importance of mathematics. Topology, given its association with unimaginable,
almost mystical shapes, can be perceived as fanciful, even useless, and yet the scope
of the subject speaks for itself and serves as an illustration of the achievements of
mathematical research.
It is hard to decisively pin down the origins of any branch of mathematics. In some
respects, the entirety of 21st Century mathematics is essentially founded in the work
of the first person to discover that 1 + 1 = 2. This is an extreme view, but that maths is
an evolutionary process is an incontrovertible fact. The work of many mathematicians
has accumulated and become interweaved over centuries to form the basis of the
topological concepts discussed in this essay. However, public recognition of the first
1.
2.
3.
Edward Witten, Viewpoints on String Theory,
http://www.pbs.org/wgbh/nova/elegant/view-witten.html, accessed 25/08/12
Paul Renteln & Alan Dundes, Foolproof: A
Sampling of Mathematical Folk Humour, http://www.ams.org/notices/200501/feadundes.pdf, accessed 25/08/12
Klaus Jänich, 1984. Topology. New York:
Springer-Verlag
topological problem to be treated as such arguably came in 1736 with Leonhard
Euler’s solution to the well-known Seven Bridges of Königsberg problem. The city
of Königsberg was effectively divided into four by the river that flowed through it,
and these four sections were connected by a total of seven bridges. Many a local had
spent an afternoon crossing and re-crossing the bridges in a failed attempt to walk
through the city crossing each bridge once and only once. Euler proved that there was
no solution by treating the walk through the city as a network, and consequently
focusing on the topological properties of the walk. His work is said to have laid the
foundations for Graph Theory, a branch of topology.
The map of Königsberg can be drastically simplified using topological principles
Euler noted that, with the exception of the start and finish points of the walk, when
you entered a land mass by a bridge you then had to leave it by a bridge. As each
bridge – edge in graph theory terms – can only be traversed once, this implied that
each land mass – node in graph theory terms – (other than the start and end points)
must have an even number of edges connecting it to other nodes, as half are used to
enter the nodes and half to leave. Each node was connected by an odd number of
edges to the other nodes, so it was impossible to walk across each bridge just once.
For this to have worked, there would have to have been either zero nodes with an odd
number of edges (of odd degree) or just two nodes of odd degree, where these two
nodes were the start and finish points4.
Topology was not, however, a clearly defined subject at this point. Euler continued to
reshape the mathematical community’s perception of geometry; he proved some
seemingly obvious properties of various shapes that had evaded discovery by other
mathematicians simply by shifting his focus to non-measurable properties. For
example, he proved that the formula for a polyhedron - a three dimensional
geometric solid with straight edges and flat faces, such as a square or a dodecahedron
– was
v – e + f = 2,
where v = vertices, e = edges and f = faces. Antoine-Jean Lhuilier expanded on this
formula in 1813, noting that it was inaccurate for polyhedra containing holes. He
showed that for a solid containing g holes,
v – e + f = 2 – 2g.
4. Simon Singh, 1997. Fermat’s Last Theorem. Great Britain: Fourth Estate
Lhuilier’s formula is the first known that considers a topological invariant, a
geometric property that is unaffected by any deformation that the shape undergoes.
However, it was not until 1847 that the word ‘topology’ worked its way into public
consciousness. Johann Listing, a German mathematician who was heavily influenced
by Carl Friedrich Gauss, wrote a relatively unimportant paper named entitled
‘Vorstudien zur Topologie.’ His 1861 paper was significant – he introduced the
concept of the Möbius strip, a famous and topologically interesting object. As the
name suggests, however, Listing was not
remembered for this achievement; rather,
it was August Ferdinand Möbius, who
published a description of the Möbius
strip in 1865, about which more will be
said later. Listing’s paper also discussed
the idea of connectivity, which
Bernhard Riemann had been studying
during the previous decade5.
Riemann, who ‘revolutionized virtually
Möbius strip
everything he touched6,’ was very much
a part of the movement that
was breaking away from standard, Euclidean geometry – that is to say,
geometry on what we think of as a ‘flat’ surface. He had studied the
geometry of curved surfaces and proved that the angles of a triangle did
not add up to 180° on a curved surface; he had shown that Pythagoras’
Theorem could not be used to measure the length of a line. He proposed in
his 1854 lecture that the shape of space may well not be Euclidean; now,
Euclidean space in two, three, four and even infinite dimensions was no
longer the only territory to explore. His work had implications in topology,
as ‘after Riemann, there could be no question that topological and
geometric ideas were essential to a deeper understanding of analysis 6.’
The angles of a triangle on a
Riemann’s discoveries also facilitated much of Henri Poincaré’s work:
sphere add up to 270°
the Poincaré conjecture, ultimately solved in 2003 by Russian
Mathematician Grigory Perelman, speculated about the structure of the
universe and was one of the most important questions in topology for over a century,
and Poincaré’s feud with German mathematician Felix Klein spurred on rapid
mathematical development. Born in 1854, he is viewed as the last great universalist,
meaning that he had a good grasp of all areas of mathematics – that this is now
virtually impossible is indicative of the spiralling growth of mathematics over the last
150 years.
Today, topology takes many forms. It is undoubtedly useful: when we use electricity,
we make use of network theory as it applies to electrical circuits; when we take the
London underground, we benefit from the simplified topological representation of all
the stops in the familiar tube map.
5.
6.
J.J. O’Connor and E.F. Robertson, A History of
Topology,
http://www-history.mcs.st-and.ac.uk/HistTopics/Topology_in_mathematics.html,
accessed 30/08/2012
Donal O’Shea, 2007. The Poincaré Conjecture. USA: Walker Publishing Company
Euler pre-empted simplifications of this type with his solution to the Königsberg problem
The pursuit of mathematical discovery is not, however, entirely driven by the search
for practical applications. The study of n-dimensional spaces cannot easily be applied
to the 3-dimensional world in which we live, nor does the search for the shape of the
universe yet have any purpose insofar as it relates to the advantage of the human race.
The four-colour theorem, for example, was only proved in 1976 – the first major
theorem to be proved using a computer – after over a century of attempts to solve the
topological problem. The four-colour theorem almost appears trivial and unworthy of
so much attention – it originates from Francis Guthrie’s conjecture that any map
could be coloured using just four colours. The mathematical historian Kenneth May
claims that ‘the four-colour conjecture cannot claim either origin or application in
cartography7,’ and yet mathematicians endeavored to find and still seek to improve
the existing solution. It is the mathematical equivalent of ‘art for art’s sake.’ This is
the driving force behind many mathematical discoveries; G.H. Hardy, an English
mathematician, asserted that ‘it may be fine to feel, when you have done your work,
that you have added to the happiness or alleviated the sufferings of others, but that
will not be why you did it8.’
Similarly, Henri Poincaré said that ‘the scientist doesn’t study mathematics because
it’s useful’ and the aforementioned Poincaré conjecture is a further example of this.
Poincaré is said to have had a formidable power of visualisation that enabled him to
consider shapes that could not be seen in three-dimensional space. The shape of space
is
primarily
a
topological
7.
8.
Mark Walters, It Appears That Four Colors Suffice: A Historical Overview of the Four-xx
Colour Theorem, http://home.adelphi.edu/~bradley/HOMSIGMAA/Walters.pdf,
accessed 05/09/12
G.H. Hardy, 1940. A Mathematician’s Apology. UK: Cambridge University Press
concern as nothing definite is known about the size of the universe in quantitative
terms; it is even possible, although unlikely, that it is infinitely large, in which case
traditional geometric measurements are of little use9. After centuries of striving to
determine the shape of the earth itself, the search for the shape of the universe was a
natural progression for mathematicians like Poincaré. To understand Poincaré’s
conjecture, it is necessary to first understand more about three-dimensional space and
two-dimensional manifolds. Three-dimensional objects have surfaces which are finite
two-manifolds – for instance, although the earth itself is the shape of a solid sphere,
its surface – if we were to remove it as though it
were packaging – would be flat and twodimensional. An infinitely extending twodimensional plane is an example of an infinite
manifold, but finite manifolds only are relevant
in this case. In this context, a two-sphere or a
torus (a bagel-shaped object) refers to the
hollow, ‘two-dimensional’ surface of those
objects. Nineteenth-century mathematicians
A torus
were interested in the number of forms a twomanifold could take, and surprisingly
discovered that any manifold was homeomorphic (meaning, as in the previous
example with the coffee cup and the doughnut, that one object can be deformed into
the other without being broken) to either the two-sphere or a torus with any number of
holes (a 2-torus has two holes, a 3-torus has three and so on). Having established this,
mathematicians then wanted to know how the shape of a two-manifold could be
established and defined, and they did so using the concept of loops. If a piece of rope
is looped around the two-sphere, it is always the case that as the rope is tightened it
will tighten to a point the top of the two-sphere and come free of it. However,
although a rope that is looped around the outer edge of a torus can also be tightened to
a point, a rope that is looped through the hole in the centre of the torus cannot be. As a
result, the sphere is defined as simply connected, but the torus is not.
Essentially, Poincaré took this idea up a dimension. If we think of two-manifolds as
modelling the possible shapes of the earth, then we can think of three-dimensional
manifolds as modelling the possible shapes of the universe, and Poincaré conjectured
that the only finite simply connected three-dimensional manifold is the threedimensional sphere. Three-dimensional manifolds are impossible to envisage, but
moving from the two- to the three-sphere is analogous to moving from a disc to the
two-sphere. The disc has a circular, one-dimensional manifold as its boundary, and
taking two discs and ‘gluing together’ the two boundaries creates the two-sphere.
Similarly, if we were to take two solid spheres and ‘glue together’ their boundaries –
that is to say, glue together the two-spheres that form the boundaries of the solid
spheres – this would create the three-sphere. It is
9.
In Our Time, 2006. The Poincaré Conjecture.
London: BBC Radio 4. February 11th [Audio Recording]
physically impossible to construct the three-sphere in three-dimensional space, but
theoretically possible that the three-sphere should serve as the boundary or surface of
a four-dimensional ‘hyper-sphere.’
To create the three-sphere, each corresponding square on the
spheres above must be ‘glued together’
Generally, three-manifolds are constructed ‘by joining opposite faces of regular solids
in different ways.’ The importance of the Poincaré conjecture lies in the fact that if it
were true, it could be proven that the shape of the universe is homeomorphic to the
three-sphere if every closed loop in the universe could be shrunk to a point. Is this
useful? It is certainly important, but cannot really be said to be directly useful today.
However, it has a deeper significance that transcends dimensions. Study of Poincaré’s
work and in particular this conjecture ‘gave topology wings.’ For instance, it sparked
an interest in topology that cultivated the growth of mathematics in America and in
particular the rise of the Princeton mathematics department, which went on to
overtake previously legendary departments like Göttingen University’s. Returning to
Euler’s ‘Seven Bridges’ problem, we see that although his proof appears to be merely
the solution to an entertaining puzzle, the knock-on effect of his work has led to
topology as we know it today. Furthermore, many important results can inadvertently
be arrived at in the search for a solution to a conjecture. Grigori Perelman’s final
paper, for example, included the proof of several other conjectures in order to arrive
at the implicit conclusion – that the solution to Poincaré’s 100-year-old problem, for
which the Clay Mathematics Institute had offered $1 million, had been found. His
work also created unforeseen links between the fields of geometry and topology, and
the significance of leaps of this kind in facilitating further mathematical development
cannot be understated6. This proof poses the question of whether any mathematics can
truly exist in abstraction from all other mathematics and by definition therefore from
applied mathematics.
Despite this, it is not difficult to understand why topology might seem somewhat
separated from reality upon first glance.
Two common objects with interesting and
seemingly fantastical topological properties
are the Möbius strip and the Klein bottle.
Far removed as mathematics may seem
from the arts, the Möbius strip is probably
known to many of us in one form
or another as the basis for a book, film,
video game, toy or artwork. M. C.
Escher, an artist whose work was greatly
influenced by mathematical concepts,
created ‘Möbius Strip II’ seen on the left. The interesting feature of the strip is that
although one might instinctively assume that it is a two-sided shape, it in fact only has
a single side.
Q: Why did the chicken cross the sssssssssssssMöbius strip? A: To get to the same side.2
If a Möbius strip is cut in half, yet another is created. Interestingly, there are actually
technical applications of the strip – conveyor belts, for
The ants move up and down the strip twice
example, are occasionally designed as Möbius strips as each
before coming back to the top
side is then only used half as much and takes twice as long to
wear out. Likewise, Möbius recording tapes have double the playing time.
Admittedly, such devices could potentially have been invented without the
intervention of mathematics, but the field of topology continues to find interesting
new objects like the Möbius strip that could well be used in a similar way in the
future10. The Klein bottle is even more abstract; it can be thought of as a square with
the
opposite
sides
connected together as shown
– an ant, to use Escher’s
idea, would leave the top left
corner at the side and
crawl back into the bottom
right corner, but on
leaving through the top edge
itself would re-enter at
the corresponding point on
the bottom edge rather
than on the opposite side.
Depicting this object in
3-D space can be done but it
is not entirely accurate;
the 3-D Klein bottle must
intersect itself, which in
theory it should not do. The
tube-like shape of the
bottle is due to the
correspondence of the
top and bottom edges of the
square, and the hole at
the top of the bottle is where
the left and right edges
meet. Admittedly, like the
Klein bottle, some of the
more entertaining concepts
and miscellany to arise
out of the study of topology
are not necessarily the
products of serious, gamechanging mathematics;
they are merely an accessible
manifestation of more
fundamental
ideas.
One
topological oddity is
torus chess, which involves
using a board on which
pieces can leave at the top and
re-enter,
The Klein bottle
10. Abram Teplitskiy, Student Corner: Marvel of the Möbius Strip,
ssssi http://www.triz- journal.com/archives/2007/01/07/, 25/08/12
seven squares down, at the corresponding place at the bottom of the board and vice
versa. Similar rules apply with regard to leaving the board at the left and at the right –
for example, a rook on the right-hand side of the board will, upon moving one square
to the right, end up seven squares to the left.
If this chessboard were a torus
chessboard, the White player would
have several ways of taking one of the
Black player’s pieces. They could move
the knight in g1 to h7 and take the black
pawn; alternatively they could move the
queen in e1 ‘backwards’ to e8 and take
the black queen. In a three-torus, which
can be thought of as a space in which
moving forward, backwards, right, left,
up and down all brings you back to your
starting point, you could chase yourself
endlessly, forever in sight but forever
eluding capture. Such scenarios capture
the imagination, and it is perhaps
mathematics such as this that can be
considered abstract, as it has no purpose other than to entertain.11
On the other hand, some branches of topology have some very clear commercial,
medical and practical applications. Knot
Theory is the study of mathematical
knots, which are defined as the
embedding of a circle in a 3-dimensional
Euclidean space, meaning that it differs
from what we think of as a knot in the
sense that it is a closed loop. At first
glance, this might seem to be an
interesting yet largely inapplicable branch
of mathematics; the creation and study of
intricate knots is usually seen as a pastime
reserved for Boy Scouts. However, being
able to understand the structure of knots
has proved vital in real-life situations – for
example, after Francis Crick and James
Watson discovered the double-helix shape
of DNA in 1953, it was clear that Knot
Theory
could
be
The classification of different mathematical knots
11.
Jeffrey R. Weeks, 1985. The Shape of Space. New York: Marcel Dekker
used to learn more about the way in which DNA becomes knotted and, in theory, how
to then untangle it. Knotted DNA causes problems as it prevents the DNA from
functioning as it should. Enzymes named topoisomerases which manipulate the
strands of DNA and wind or unwind them can be studied and classified according to
which knot they produce when allowed to act on a circular strand of DNA, and it is
therefore possible to know which topoisomerases counteract each other and we can
use this information to untangle DNA. Knot Theory also has applications in molecular
chemistry, statistical mechanics and particle physics12. The problem of how to
untangle DNA is a more recent one; as is sometimes the case, abstract mathematical
interest in a topic predated the real-life demand for research. Knot Theory began to
evolve in the late 18th Century, hundreds of years before the structure of DNA was
established. Alexandre-Théophile Vandermonde first developed a mathematical
theory of knots in 1771, and Carl Friedrich Gauss studied the subject further in the
19th Century. Lord Kelvin’s hypothesis in the 1880s, now known to be false, that the
universe was made from a substance called ether and matter could be explained as
knots in the ether, piqued interest in the subject. The emerging field of topology
eventually subsumed Knot Theory13.
Similarly, the topological field of Graph Theory has far-reaching uses. In particular,
a form of Graph Theory named Network Theory has proved an important tool.
Again, thinking back to Euler’s ‘Seven Bridges’ problem, we see an example of a
network in the simplified representation of the bridges and landmasses of Königsberg.
A network essentially consists of a set of vertices or nodes joined together by various
edges or links.
A
B
G
E
C
F
D
H
Network 1: The network above has nodes A, B, C… etc.
Th
The representation of social networks in this way has commercial uses: some
companies determine which node is the most influential in a network using various
measures of centrality and, for example, will subsequently decide that they are best
advised to target their advertisements at the website or person represented by this
node. The social sciences, as with so many fields of research across the board, have
been infiltrated by mathematical network analysts, with social scientist Linton
Freeman commenting in
12. Knot Theory: Applications of Knots. http://library.thinkquest.org/12295/, accessed
ddddi05/09/12
13. Erin Colberg: A Brief History of Knot Theory.
sssssshttp://www.math.ucla.edu/~radko/191.1.05w/erin.pdf, accessed 05/09/12
1984 that ‘there’s a whole lot of really high powered maths types running around in
the social networks arena.13’
The concept of centrality is best illustrated by
example, but generally speaking it refers to the
A
idea of which nodes are the most central and the
3
most important in a network. The crudest measure
B
3
of centrality is degree centrality, which simply
records the number of incident edges of each node.
C
2
The table on the left gives the degree centrality of
the nodes in Network 1, indicating that node D is
D
4
the most central. However, the influence of a node
E
3
goes deeper than this – it depends on how central
the nodes it is connected to are. For example,
F
3
although some node X is connected to four other
nodes of degree one, if some other node Y is
G
2
connected to two nodes of degree nine or ten, it
H
2
arguably has more influence than node X.
Closeness centrality is a measure that avoids this
issue by shifting the focus from the degree of a node to the ‘connectedness’ of a
node, so to speak. It records the sum of the shortest path lengths between a node and
all other nodes in the network. Nodes with low scores are well connected. By this
measure, as can be seen in the table to the right, node E is
Closeness
the most ‘central’ with node D coming a close second. Node
Centrality
Inspection of the network itself reveals the reason for this
A
16
– nodes E and F bridge the gap between the two separate
sides of the network, which also explains why node F has B
13
a low closeness centrality score. The two measures are
indubitably produce similar results, with nodes G and H C
17
judged the least central by both measures. There are
12
further subtleties to take into account, however – in using D
the measure of closeness, we are assuming that the E
11
traffic from one node to another takes the shortest path
possible. To put this in context, if the network F
13
represented the spread of illness through a network of
18
people, it is a reasonable assumption that the illness G
would travel along the shortest paths and not retrace its H
18
path. However, if the network represented the exchange
of money through a group of people, the movement of the
traffic from one node to another is unrestricted, as money could be returned or follow
an extremely indirect path from one node to an another14. More sophisticated
measures weigh up the importance of paths of different lengths and use, assigning a
lesser
importance
to
longer
paths
Node
Degree
Centrality
14. D.J. Higham, P. Grindrod, E. Estrada, 2011. People Who Read This Article Also
dddidRead...: Part I. SIAM News. 44 (1)
between nodes. One such measure is the measure of communicability, and an
explanation of this measure is provided for those more interested in the equations
underlying the concepts. It is calculated using the
1
adjacency matrix of a network, which again can
be best described through an example. In this
context, a matrix is essentially a table that shows
whether or not there is an edge between two
nodes. The matrix has as many columns and rows
3
4
2
C
D as the network has nodes, and each column and
B
each row represents one of the nodes. Where two
nodes have a path between them of length one, as
in Network 2 1&2, 1&3, 2&3, 3&4 and 3&5 do,
the intersecting column and row of these two
nodes will have a value of one. All other values in
5
Network 2
the matrix will be 0. An adjacency matrix
representing all paths of length one for network 2 can be seen on the right. The
important point is that the nature of matrix multiplication means that the square of the
matrix,
represents, in a similar way, how many paths there are of
length two between any chosen pair of nodes. M3 tells us
about paths of length three, and more generally MN
Adjacency matrix (M) of Network 2;
tells us about paths of length n. The communicability
column 1 and row 2, for example, has
value 1 as there is an edge between
score works on the premise that longer paths tend to be
M
nodes 1 and 2.
less suggestive of centrality; it uses the expansion of e
which diminishes the importance of longer paths between
the nodes. The result is a weighted adjacency matrix,
and to calculate the communicability score for a
particular node it is necessary to find the sum of its row
or column (it does not matter which because, as can be
seen in the adjacency matrices of Network 2, the matrices
are symmetrical15.)
Using these tools, interesting analysis can be done on a
Adjacency matrix (M2) of Network 2
wide variety of networks. It is a very current
mathematical topic: the network on the following page,
for
example,
shows
the
connections
between
various
figures
15. Ernesto Estrada, 2012. The Structure of Complex Networks: Theory and Applications.
xxxxxNew York: Oxford University Press
16
15. Ernesto Estrada, 2012. The Structure of Complex Networks: Theory and Applications.
xxxxxNew York: Oxford University Press
The tables are an example of what mathematics can reveal about subjects that it seems to be
unrelated to
in the recent phone hacking scandal, and the analysis of centrality has given a
mathematical estimate of who were the key players in the affair. Indubitably there is
correlation between the degree rank and communicability rank, but whereas the
number of links connecting Andy Coulson to others would suggest that he is the most
important member of the network, the communicability score indicates that Rebekah
Brooks has a more fundamentally influential position15. It would have been possible
to weight the edges of the network according to what type of connection existed
between the two nodes – social, work or social/work – but this can be complicated as
it is impossible to put an exact value on the strength of such a connection, just as it
would be impossible to definitively rate the relationship between two people as some
score between 0 and 1 in real life. Hence a general problem arises with applying
mathematics to real life situations – it can be argued that the precision of a
mathematical model alienates it from the unpredictability of reality. However, the
purpose of a model is not to provide certainties but to give indications: Newton’s
laws of motion, for example, are subject to external forces and variation and yet they
are applied successfully on a daily basis; statistical models like the normal
distribution weigh up the options with extreme accuracy, and yet they cannot see
into the future. In the same way, network theory can be invaluable in analysis despite
the fact that it generalises the complexities of real networks, and the scope of the field
is extremely broad: Frank Harary, a pioneer of modern graph theory, authored and
co-authored over 700 papers in areas from music to linguistics to geography.
Topology, and by extension Network Theory, which strips mathematics down to the
fundamentals, is arguably as close to perfect for use in applied mathematics as can be.
Of course, the ideas presented here are only a summary of some of the most
fascinating aspects of topology; the mathematics behind the subject runs far deeper
and is the product of decades of concerted effort by the mathematical community.
However, time and again the importance of topology, and more generally
mathematics, in every day life comes up – in some cases the applications of certain
areas of research are not glaringly obvious, but there are always some to be found –
perhaps many that are yet to be discovered. In every area there seems to be potential
for inventions and techniques that are beneficial to the human race, whether maths is
facilitating or driving these leaps forward. When Isaac Newton and Gottfried
Leibniz invented calculus – to decide who would necessarily be the topic of an
entirely different essay – they presumably did not expect that credit card companies
would some day be using their ideas. Topology is only ever superficially abstract; it is
more that it is inaccessible, to an extent, as it is rarely taught in schools. Consideration
of the subject shows that we can relate it to our lives in more ways than we might
expect.
Bibliography: (Listed by footnote number)
1.
Edward Witten, Viewpoints on String
Theory, http://www.pbs.org/wgbh/nova/elegant/view-witten.html, accessed
25/08/12
2.
Paul Renteln & Alan Dundes, Foolproof:
A Sampling of Mathematical Folk Humour,
http://www.ams.org/notices/200501/fea-dundes.pdf, accessed 25/08/12
16. BBC News: Phone Hacking Scandal: Who’s linked to who?
http://www.bbc.co.uk/news/uk-14846456, accessed 23/10/12
3.
Klaus Jänich, 1984. Topology. New
York: Springer-Verlag
Simon Singh, 1997. Fermat’s Last Theorem. Great
4.
Britain: Fourth Estate
J.J. O’Connor and E.F. Robertson, A
History of Topology,
http://www-history.mcs.stand.ac.uk/HistTopics/Topology_in_mathematics.html, accessed 30/08/2012
6.
Donal O’Shea, 2007. The Poincaré
Conjecture. USA: Walker Publishing Company
7. Mark Walters, It Appears That Four Colors Suffice: A Historical Overview of
the Four-Colour Theorem,
http://home.adelphi.edu/~bradley/HOMSIGMAA/Walters.pdf, accessed
05/09/12
8. G.H. Hardy, 1940. A Mathematician’s Apology. UK: Cambridge University
Press
9.
In Our Time, 2006. The Poincaré
Conjecture. London: BBC Radio 4. February 11th [Audio Recording]
10.
Abram Teplitskiy, Marvel of the Mobius
Strip,
http://www.trizjournal.com/archives/2007/01/07/, accessed 25/08/12
11.
Jeffrey R. Weeks, 1985. The Shape of
Space. New York: Marcel Dekker
12.
Knot Theory: Applications of Knots.
http://library.thinkquest.org/12295/, accessed 05/09/12
13.
Erin Colberg: A Brief History of Knot
Theory. http://www.math.ucla.edu/~radko/191.1.05w/erin.pdf, accessed
05/09/12
14.
D.J. Higham, P. Grindrod, E. Estrada,
2011. People Who Read This Article Also Read...: Part I. SIAM News. 44 (1)
15.
Ernesto Estrada, 2012. The Structure of
Complex Networks: Theory and Applications. New York: Oxford University
Press
16.
16. BBC News: Phone Hacking
Scandal: Who’s linked to who? http://www.bbc.co.uk/news/uk-14846456,
accessed 23/10/12
5.
Other Resources:

S. P. Borgatti, Centrality and Network
Flow. http://www.analytictech.com/borgatti/papers/centflow.pdf, accessed
23/10/12

M.E.J. Newman, 2010. Networks: An
Introduction. New York: Oxford University Press
Image Sources: (all found through Google Images. Listed in order of appearance
throughout the document)

Transformation of a coffee cup into a doughnut:
http://youngmathwizards.com/lessons/images/Topology/TransformDonutToC
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offeeCup.gif
Map of Königsberg:
http://upload.wikimedia.org/wikipedia/commons/5/5d/Konigsberg_bridges.pn
g
Network Diagram of Königsberg:
http://www.learner.org/courses/mathilluminated/images/units/11/3095.png
Möbius Strip: http://paulbourke.net/geometry/mobius/mobius3.gif
Triangle on a sphere: http://www.math.cornell.edu/~mec/tripleright.jpg
Satellite tube map: http://i.telegraph.co.uk/multimedia/archive/01487/mapphys_1487192c.jpg
Tube map: http://d2mns3z2df8ldk.cloudfront.net/images/explorermap/tubemap-2012-01.png
Torus: http://upload.wikimedia.org/wikipedia/commons/1/17/Torus.png
Sphere:
http://upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wirefra
me_10deg_6r.svg/220px-Sphere_wireframe_10deg_6r.svg.png
Escher’s ants:
http://cache2.allpostersimages.com/p/LRG/7/725/XSSA000Z/posters/escherm-c-ants.jpg
Klein bottle:
http://upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Klein_bottle.svg/
240px-Klein_bottle.svg.png
Chessboard:
http://upload.wikimedia.org/wikipedia/commons/a/a7/Chess_board_blank.svg
Knots: http://accuracyandaesthetics.com/wpcontent/uploads/2008/09/knot_tables.jpg
Source Evaluation:
As my essay was largely factual in content, my sources were not generally clouded by
any question of bias. G.H. Hardy’s book ‘A Mathematician’s Apology,’ which I did
not read with the intention of including in my EPQ but found relevant to one of the
central ideas in my essay about the purpose of mathematics, was however quite onesided in that his argument relentlessly stressed the importance of studying
mathematics for its own sake. Although he was well placed to comment on the issue,
being a mathematician himself, his views are not necessarily representative of the
entire mathematical community; however, I found similar opinions expressed
elsewhere, for example by Poincaré. My main difficulty in finding appropriate
sources was that many subject-related books were at undergraduate level, and so I had
to use the Internet to supplement my reading or could only use several pages of the
book. Both ‘the Shape of Space’ and ‘the Poincaré Conjecture’ were extremely useful
in this respect as they were aimed at those that did not necessarily have a
comprehensive understanding or indeed much knowledge at all of the subject. I
generally found that although the Internet can at times be a dubious source of
information, as facts are less likely to be checked, I tried to use reputable articles as
much as possible to ensure that this was not a problem. Furthermore, I ensured that
the information I found on the Internet was corroborated by my general knowledge of
the topic. Another point is that my knowledge of graph theory was largely shaped by
my work experience at Strathclyde University over the summer and whilst it is based
on the ideas contained in the book ‘The Structure of Complex Networks: Theory and
Applications,’ my understanding would have been lesser had it not been for the
explanations provided to me whilst at the University. Overall I feel that whilst it was
difficult to find accessible material, I succeeded in finding a number of useful
resources ranging from textbooks to podcasts, and I consider them to have been
largely reliable.