TITLE
The Circle Limit Drawings of M.C. Escher
SOURCE
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
AREA
Geometry, symmetry
KEYS
Geometry, group theory, symmetry
LEVEL
final year
LENGTH
1/8 final year. 30-40 pages
PREREQ
a basic knowledge of group theory.
HISTORY
used once
DESCRIPTION
In [1] Coxeter describes how he sent a reprint
to Escher which inspired his circle limit drawings.
In his reply Escher said: "If you could give me a simple
explanation of how to construct the following circles,
whose centres approach gradually from the outside till they
reach a limit, I should be immensely pleased and very
thankful to you".
The general aim of this project will be to supply
such an explanation with particular reference to
the Circle Limit 2,3,4 drawings of Escher.
In [1] Coxeter indicates briefly that these patterns
arise from symmetry groups in hyperbolic geometry
and describes the type of group which arises. It will
thus be necessary to find out about:
(i) hyperbolic geometry, (ii) how hyperbolic geometry
is modelled by the unit disc with straight lines
(geodesics) becoming circles and straight lines
orthogonal to the unit circle, (iii) the isometries
of the hyperbolic plane, (iv) the symmetry groups
and their relationship to hyperbolic triangles.
Every detail of the steps involved here need not be
gone into but you should aim to have a clear view
of the ideas involved to give a mathematical answer to
Escher's question.
Most of the necessary ideas involved in (i)-(iv) are
contained in [2]. Further references will be found in
[1],[2],[3]
References:
[1] Coxeter, H.S.M "Angels and Devils" in
`The Mathematical Gardner' ed. Klarner.
[2] Magnus, W. (1974) "Non-Euclidean tesselations and their
groups" Academic Press
[3] "M.C. Escher --- His life and complete Graphic work"
ed. J.L. Locher (review Math.Intell. 7(1985) 1).
[4] Coxeter, H.S.M & Moser, W.O.J (1972) "Generators and
relations for discrete groups" Springer
[5] Ford L.R. (1951) "Automorphic Functions" Chelsea
[6] Greenberg M.J. "Euclidean and non-Euclidean geometries;
Development and History" Freeman
TITLE
Wallpaper Patterns
SOURCE
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
AREA
Geometry, symmetry
KEYS
Geometry, group theory, symmetry
LEVEL
final year
LENGTH
1/8 final year. 30-40 pages
PREREQ
a basic knowledge of group theory.
HISTORY
used once
DESCRIPTION
A full discussion of the group of isometries of 2
dimensional Euclidean space is required. It should
lead to the discovery and classification of the
17 possible wallpaper patterns.
In detail:
a) Classify the types of isometry in E2 - translation,
rotation etc. Also develop some general results about
the isometry group, eg that it can be generated by
reflections.
b) Discuss the idea of conjugacy in a group and find
simple forms for representatives of the conjugacy
classes for this isometry group.
c) Discuss translation subgroups and point groups.
d) Frieze groups
e) Wallpaper patterns
The work can be extended towards similarity groups in E2,
similar work for E3, tilings and their symmetries.
References:
[1] M.A. Armstrong "Groups and Symmetry"
[2] R.C. Lyndon "Groups and Geometry"
[3] Grunbaum & Shephard "Tilings and Patterns"
[4] R.L.E. Schwartzenberger `The 17 Plane Symmetry Groups'
Math Gazette 58 (1974)
[5] H. Weyl `Symmetry' Princeton
TITLE
Perspective and Projective Geometry
SOURCE
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
AREA
Geometry, Algebra
KEYS
perspective, projective geometry
LEVEL
final year
LENGTH
1/8 final year. 30-40 pages
PREREQ
HISTORY
used
DESCRIPTION
European painting took a big step forward in the
early 15th century with the discovery of a proper
theory of perspective --- how to give a 2-dimensional
painting the illusion of depth. The theory was discovered
by professional artists, but it is a mathematical theory
and the artists concerned were interested in both the
practical and the theoretical aspects of what they
discovered.
Later the mathematical side was taken further by men who
were mathematicians first and other things second.
In their hands it turned into a new type of geometry,
called Projective Geometry.
The project is for you to write an essay explaining the
mathematical theory of perspective and how it led to
projective geometry.
References:
As starting points:
[1] Dan Pedoe : "Geometry and the Liberal Arts"
[2] John Stillwell : "Mathematics and its History"
Then chase on from there.
TITLE
Torus knots
SOURCE
T.P.McDonough, Department of Mathematics, University of Wales,
Aberystwyth. e-mail:tpd@aber.ac.uk
AREA
Geometry
KEYS
Geometry, curves, surface, periodicity.
LEVEL
elementary undergraduate.
LENGTH
One of three investigations undertaken by students in a group
of about four. The course is of one semester duration and is
1/12 of a student's degree commitment for the year.
Each student presents a report on each of the topics
investigated by the group. This report is the basis of the
student's assessment.
PREREQ
A-level Mathematics.
HISTORY
The project has actually been used in this form.
DESCRIPTION
The torus knots form a family of 3-dimensional curves which
may be described in parametric form by the equations
\halign{\tabskip=20pt#&${#}$\hfill\cr
&x(t)&= (3 + \cos at)\cos bt\cr
&y(t)&= (3 + \cos at)\sin bt\cr
&z(t)&= \sin at\cr
}
where $a$ and $b$ are constants.
Find the orthogonal projections of these curves onto a
selection of planes and for various values of the parameters
$a$ and $b$.
What can you say about the plane cross-sections of these
curves?
Describe the curves, indicating how they depend on the
parameters $a$ and $b$ and making use of the plane
projections you have found as an aid to your description.
References:
[1] "The torus and an associated coordinate system";
F.Chorlton; Math.Gaz.vol.65;434
TITLE
Saturn's Co-Orbital Moons
SOURCE
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
AREA
Mechanics
KEYS
Mechanics, numerical ode
LEVEL
final year
LENGTH
1/8 final year. 30-40 pages
PREREQ
elementary mechanics of planetary orbits, numerical solution
of ODE.
HISTORY
used once
DESCRIPTION
Saturn has two small moons S-10 and S-11 with a rather
interesting property. S-10 has an essentially circular
orbit of radius 151400 km. S-11 has an essentially circular
orbit, in the same plane, of radius 151450 km. Both moons
have radii greater than 50 km. This obviously presents a
problem.
The visual evidence suggests that the two moons have been
moving happily in these orbits for a long time.
Why do they not collide?
The aim of the project is to study the rather interesting
interaction between the two moons when they come very close
to each other, in particular the phenomenon of orbit-switching.
Set up differential equations for the motions of the two
moons on the simple (and reasonable) assumption that they
are moving only under the gravitational attraction of Saturn
itself and their mutual gravitational interaction.
This mutual interaction is clearly trivial unless the moons
are very close, so some analysis of the equations is
necessary to isolate the significant problem of this
interaction (there are wildly different time scales at work).
When the problem has been properly cooked the last phase
will be to perform numerical calculations to find the precise
behaviour during the interaction.
References:
[1] W. M. Smart `Celestial Mechanics'
[2] W. M. Smart `Spherical Astronomy'
[3] `Science' magazine April 1981
[4] `Astronomy' magazine February 1981
TITLE
Non-Newtonian Fluids.
SOURCE
Dr R.M. Thomas UMIST, PO box 88, Manchester M60 1QD
tel 0161 200 3657
AREA
fluid mechanics
KEYS
fluid mechanics, non-Newtonian fluids, constitutive equations
LEVEL
Final Year project
LENGTH
1/6 of final year. About 40 pages (12000 words)
PREREQ
HISTORY
used
DESCRIPTION
Review the mechanics of non-Newtonian fluids and characterise
their behaviour both in shear and elongational flow,
including steady-state and time-dependent effects.
Formulate the main types of constitutive equation and see how
well they are able to describe the physical properties of
certain materials.
The starting point for the work should be chapters 3 and 4
of [1].
References
[1] R.I. Tanner "Engineering Rheology " OUP
[2] W.R. Showalter "Mechanics of non-Newtonian fluids"
Pergamon
[3] Bird, Hassager et al. "Dynamics of Polymeric Liquids"
Wiley
TITLE
Musical Tones
SOURCE
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
AREA
acoustics
KEYS
music, acoustics, wave equation
LEVEL
final year
LENGTH
1/8 final year. 30-40 pages
PREREQ
familiarity with the 1-dimensional wave equation. Separation
of variables. Some musical knowledge.
HISTORY
used once
DESCRIPTION
Why are discords discordant? What is the real mathematical
and physical basis for the classical musical theory of
concord and discord?
Helmholtz developed a theory based on the actual physics of
hearing and the behaviour of the ear drum when excited into
vibration. This is elaborated in his big book [1].
Report on the relevant work in this book. In particular, on
the theory of combination tones and the non-linear behaviour
of acoustic systems.
It would probably be sensible to preface this by a description
of the standard acoustic theory, based on the one-dimensional
wave equation and Fourier series.
References:
[1] H.L. Helmholtz "On the Sensations of Tone" Dover
TITLE
Population models
SOURCE
T.P.McDonough, Department of Mathematics, University of Wales,
Aberystwyth.
e-mail:tpd@aber.ac.uk
AREA
Differential equations
KEYS
Modelling, differential equations.
LEVEL
elementary undergraduate.
LENGTH
One of three investigations undertaken by students in a group
of about four. The course is of one semester duration and is
1/12 of a student's degree commitment for the year.
Each student presents a report on each of the topics
investigated by the group. This report is the basis of the
student's assessment.
PREREQ
A-level Mathematics.
HISTORY
The project has actually been used in this form.
DESCRIPTION
The population sizes $x$ and $y$ of two species which occupy
a bounded environment and compete for a fixed food resource
are functions of time $t$; i.e. $x = x(t))$ and $y = y(t)$.
The changes in these populations in time may be modelled by
a system of differential equations
\halign{\tabskip=20pt#&$#$\hfill\cr
&x'(t) = px + qy\cr
&y'(t) = rx + sy\cr
}
Describe what you would consider to be reasonable constraints
on the parameters $p$, $q$, $r$ and $s$, noting their effects.
Examine the system graphically when
\halign{\tabskip=20pt#&\hfill#&\vbox{\hsize=4.5in#}\hfill\cr
&(i)&$p$, $q$, $r$ and $s$ are constants.\cr
&(ii)&$q = q'x$, $r=r'y$ and $p$, $q'$, $r'$ and $s$ are constants.\cr
&(iii)&$p=p'+p''x$, $q = q'x$, $r=r'y$, $s=s'+s''y$
and $p'$, $p''$, $q'$, $r'$, $s'$ and $s''$ are constants.\cr
}
In each of the cases (i), (ii) and (iii), make a few choices
of the relevant constants.
Where possible, interpret your solutions as statements about
the situations being modelled.
How would your analysis change if only one of the populations
used the food resource while the second preyed on the first
for its food.
References:
"Some nonlinear differential equations"; A.Sofo;
Math.Gaz.vol.65; no.432
"An Evolution game for a Prey-Predator Ecology";
J.M.T.Thompson; Bull.I.M.A.; vol 15;7
"Quantitive Population Ecology: Elegant Models or
Simplistic Biology";
D.W.Morris; Bull.I.M.A.; vol 21; no.11/12.
TITLE
Determination of Parabolic Orbits
SOURCE
Dr John Pulham, Department of Mathematical Sciences,
University of Aberdeen.
e-mail pulham@maths.abdn.ac.uk
AREA
Classical Dynamics.
KEYS
Dynamics, Orbits, Gravitation.
LEVEL
Final Year Project
LENGTH
1/8 of final year work. 30-40 pages
PREREQ
Elementary mechanics
HISTORY
Used once.
DESCRIPTION
This looks at one aspect of the classical problem of
determining the gravitational orbit of an object (planet,
asteroid, comet) from visual observations.
Leaving aside the whole theory of perturbations, the motion
of bodies in the solar system is essentially described by
the Keplerian laws, which are derivable from Newton's
dynamical theory and his law of gravitation.
Deducing the orbit of a given object from observational
evidence is much more difficult. Before we went into
space the only real observations available were measurements
of directions at given times (you point the telescope and
look). No distances were available as primary data and the
angular data was given, in the first instance, relative to
axes fixed to the Earth, which is itself an object orbiting
the sun. Complicated.
This project looks at one fairly simple case of the problem:
that of determining the elements of the orbit of a comet
(the orbit being assumed to be parabolic) from three
positional observations.
The student will need to learn about the details of parabolic
orbits including the time dependence, the essential ideas of
orbital elements, the basic coordinate systems (helio- and
geo-centric) that need to be used, and some numerical method
(like conjugate gradient) for minimising a function of several
variables.
References
[1] A very old book which does cover the material is
Russell T. Crawford: "Determination of Orbits of Comets
and Asteroids" McGraw-Hill 1930
[2] Most of the required background is in, for example
W.M. Smart "Textbook on Spherical Astronomy" CUP 1977
Bate, Mueller and White: "Fundamentals of Astrodynamics"
Dover 1971
TITLE
Newton's Early Work
SOURCE
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
AREA
History \& Calculus
KEYS
Newton, calculus, history
LEVEL
final year (can be used much earlier)
LENGTH
1/8 final year. 30-40 pages
PREREQ
elementary calculus
HISTORY
used twice
DESCRIPTION
Newton's notebooks have been published, in wonderful form, by
Whiteside. The early parts deal with Newton's progress towards
the calculus around 1666. The aim of the project is to study
and present Newton's development of the binomial series and
his technique of formal interpolation. The student should try
to assess the mathematical climate that produced such arguments
(did Newton think he was `proving' anything?). A short biography
of Newton's early life should be included together with a brief
discussion of the state of mathematics at that time.
References:
[1] D.T. Whiteside `The mathematical papers of Isaac Newton'
Vol 1. CUP 1967
[2] R.S. Westfall `Never at Rest -- a biography of Isaac
Newton' CUP 1980
[3] C.B. Boyer `The History of Calculus ...' Dover 1959
[4] E.T. Bell `Men of Mathematics' Vol 1
TITLE
The Greek Theory of Ratio
SOURCE
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
AREA
Analysis, history
KEYS
Real Numbers, analysis, Euclid, Eudoxus, history
LEVEL
final year
LENGTH
1/8 final year. 30-40 pages
PREREQ
Would help if the student had already met something like
Dedekind Cuts.
HISTORY
used twice
DESCRIPTION
Dedekind and Weierstrass are usually credited with the
formal construction of the real number system from the
rational numbers. Dedekind's work was published in about
1860 [1]. This is rather odd since an almost equivalent
theory is available in book 5 of Euclid's Elements [2]
`published' in about 300 BC.
The Greeks had discovered irrational numbers and were sharp
enough to realise that they created problems for any simple
theory of arithmetic and, in particular, ratio based on
`fractions'. The work in Euclid's Book 5 is a solution to
this problem (the work is usually credited to Eudoxus).
Find out about the state of Greek mathematics before this
point and something about the discovery of irrationals.
Present the theory given in Book 5 of the Elements.
Compare and contrast it with Dedekind's work as presented
in his famous paper. Is it really fair to say that the
ancient Greeks anticipated modern analysis by over 2000 years?
References:
[1] Richard Dedekind `Essays on the theory of numbers'
Dover 1963
[2] T. L. Heath `The 13 books of Euclid's Elements'
Cambridge 1908, Dover 1956
[3] C. B. Boyer `A History of Mathematics' Wiley 1968
and other sources mentioned in the above
TITLE
Charges on a Sphere
SOURCE
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
AREA
theoretical physics, computing
KEYS
electrostatics, optimisation
LEVEL
final year
LENGTH
1/8 final year. 30-40 pages
PREREQ
basic electrostatics, some experience of optimisation and
computing
HISTORY
used once
DESCRIPTION
If $n$ equally charged particles are free to move on a
circle then an equilibrium position is one in which all
the particles are equally spaced round the circle. That
is easy.
Now put $n$ equally charged particles on a sphere.
Common sense suggests that their equilibrium position should
be some kind of symmetrical distribution on the sphere ---
regular solid rather than regular polygon. Unfortunately,
there are only a few regular solids available and none has,
for example, 5 vertices. So what does happen? What are the
equilibrium positions? (To further increase the confusion,
the regular solids do not always give the equilibrium
configurations for the appropriate number of particles.)
The problem should be studied and the references examined.
The current knowledge of the problem should be summarised.
The student should also attempt to write a computer program
that will allow him to conduct a numerical investigation of
the problem, at least for small values of $n$. The
calculation is really an optimisation problem where you are
trying to find the minimum energy configurations.
References:
[1] Edmundson JR Acta Cryst. A48 1992 60-69
[2] Erber T and Hockney GM J. Phys A 24 L1369-76 1991
[3] Webb S Chem Phys Letters 129 310-14 1986
[4] Calkin MG, Kiang D and Tindall DA Am J. Phys. 55 1987
157-58
[5] Berezin AA Chem Phys Letters 123 1986 62-64