Exploring University Mathematics. Lectures Given at Bedford College, London, Volume 3

126 Pages·1969·5.64 MB·English

by  Mary Bradburn, G. T. Kneebone, M. R. Hoare and N. J. Hardiman (Auth.)

EXPLORING UNIVERSITY
MATHEMATICS 3
LECTURES GIVEN AT BEDFORD COLLEGE,
LONDON
by
MARY BRADBURN G. T. KNEEBONE
M. R. HOARE C. A. ROGERS
P. C. KENDALL A. GRAHAM
H. G. EGGLESTON
Edited by
N. J. HARDIMAN
PERGAMON PRESS
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First edition 1969
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EDITORIAL
THE seven lectures comprising the chapters of this book formed
the programme for the 1967 Easter Conference in Mathematics
at Bedford College, London. These conferences are given
annually and the 1965 and 1966 lectures have been published
in Exploring University Mathematics, vols. 1 and 2 (Pergamon
Press). In the Foreword to vol. 1, Professor Eggleston says:
'The lectures are primarily designed for students about to
embark on a degree course of which mathematics is a major
part. Although those attending are drawn from schools in all
parts of the country, the number involved each year is, un-
fortunately, very limited The organizers of the conferences
felt that these lectures, given by professional mathematicians
on subjects of current mathematical interest and yet assuming
little mathematical background, would be of interest to a wider
public. It was therefore decided to publish them in a book and
so increase the 'audience' many times."
The scope of the lectures is fairly wide and is divided between
pure mathematics and applied mathematics, with a natural
bias towards the former at this level. Each lecture is quite
independent, so that getting "lost" in one lecture does not
mean that a subsequent lecture is unintelligible. This, of course,
is less important in the book, as the reader has time to take
each lecture as slowly as necessary for complete comprehen-
sion. Wherever possible, a list of suggestions for further reading
is given.
Vll
Vlll EDITORIAL
Two of the lectures were given by members of the Mathe-
matics Department at Bedford College, one by a member of
the Physics Department, whilst for the other four lectures we
were very pleased to welcome Professor Rogers, F.R.S. (Uni-
versity College, London), Dr. Mary Bradburn and Professor
Eggleston (Royal Holloway College, London), and, once
again, Professor Kendall (University of Sheffield). Each
lecturer chose a subject in which he is an expert, either as
a teacher or as a research worker.
Dr. Bradburn chose as the subject for her lecture "Symmetry
of Pyramids and Prisms". This is in some ways an extension
to three dimensions of the chapter in vol. 2 on "Wallpaper
Patterns". The lecture was illustrated by a large number of
three-dimensional models which cannot be easily reproduced
as diagrams in this book, but the reader might well profit by
making some for himself. (Hint: Dr. Bradburn used brightly
coloured paper as a code for quick recognition of the type of
symmetry involved in the model.)
Dr. Hoare is a member of the Physics Department, but his
own field of research, statistical mechanics, is on the borderline
between mathematics and physics and this is reflected in the
topic he chose for his lecture. He gives a thoughtful and closely
reasoned account of the close relationship between the two
subjects, best described in his own words: "I must give you a
large-scale picture of the part played by mathematics in modern
science, explain why it is indispensable, at least to the physical
scientist, how it not only makes possible his calculations but
shapes his whole attitude to Nature, and the sort of language
he talks."
Although the lecture on "Cushion Craft" was delivered by
Professor Kendall, it was written jointly by P. C. Kendall,
R. R. McLone and P. Polak, and so all three names appear in
the heading for Chapter 3 and I would like to extend my thanks
to McLone and Polak for their share in the preparation of
this lecture. The lecture falls into two parts: the first part
EDITORIAL IX
describes the isometric transformations involved when a sur-
face formed by an inextensible membrane is deformed, and
the second part develops the mechanics involved in the problem
and gives the theory of a hovercraft skirt in two dimensions.
Each year we try to include at least one lecture on the
lecturer's own research work, although the topics which can
be presented at this level are, of course, very limited. Dr.
Kneebone is Reader in Foundations of Mathematics at Bed-
ford College and his lecture on "Logic" is a clear and concise
introduction to the work in which he is interested. He has
written an authoritative book Mathematical Logic and the
Foundations of Mathematics, and Bedford College is a centre
of research work in Logic in the University of London.
Professor Rogers' lecture started with a simple illustration of
a three-dimensional house, but quickly and almost impercep-
tibly he led his audience with him to the mathematician's
imaginary, yet familiar, /i-dimensional space. I think all those
privileged to be present at the lecture would agree that in
the hands of an expert «-dimensional space was made to sound
deceptively simple.
Calculating machines and electronic computers have de-
veloped so rapidly since the war that they are now everyday
equipment both in research establishments and in industry.
Mr. Graham is a lecturer jointly at Bedford College and the
Institute of Computer Science, and in his lecture he talked
about some of the mathematics which makes it possible to
use these machines — although they operate so very quickly,
all calculations must first be expressed in terms of four basic
operations: addition, subtraction, multiplication and division.
Hence behind every computer there are mathematicians.
Professor Eggleston's lecture on "Some Irrational Numbers"
was given to an invited audience of the more advanced pupils
(i.e. those who had already passed A-level), the school teachers
and the university lecturers. Professor Eggleston was Head
of the Mathematics Department at Bedford College from 1958
X EDITORIAL
to 1966, and it was largely due to his enthusiasm and support
that these conferences have become an annual event, and also
that the 1965-7 conferences are being published in book form.
It was, therefore, very fitting that he should be invited to give
this "special" lecture and we were delighted when he agreed to
give it.
I should like to thank all those who have taken part in the
writing and proof-reading of this book, especially Dr. Sargent
and Mr. E. L. Knight, and the Pergamon Press for the care
which they have given to the production of the book.
Bedford College, London N. J. HARDIMAN
July 1967
CHAPTER 1
SYMMETRY OF PYRAMIDS
AND PRISMS
MARY BRADBURN
Mappings
An idea which plays a very important part in university
mathematics is the idea of a map. A map is a familiar thing in
many walks of life. There are maps in an atlas in which a
portion of the surface of the earth is mapped onto the page
of a book. A street map of a town, road maps for the motorist
and representational diagrams of the underground system are
other geographical kinds of maps. There are, however, other
uses of mappings; architect's plans in which a building is
represented formally by ground plans and elevations and the
blue prints used by engineers to represent a machine, or a
structure, or an electric circuit are also maps. Photographs,
sketches and paintings are mappings of a three-dimensional
subject on to a plane. Representations need not be limited to
those on a plane, three-dimensional models such as model
ships, aeroplanes, trains and cars are maps of the originals
with the dimensions scaled down.
In mathematics a mapping provides a correspondence
between elements of one set with elements of another. An
element P of one set is mapped onto an element P' of the
i
2 EXPLORING UNIVERSITY MATHEMATICS 3
second set and P' is uniquely determined for a particular
element P and for a given mapping. If the sets are sets of points
the mapping has a geometrical significance similar to the
examples described. Though it is an essential and common
feature of all mappings that the point P' on the map is uniquely
determined for a chosen point P of the original, there are
mappings in which several points P P ,... ,P on the original
u 2 W
set map onto the same point Pf. This happens when a solid
is represented on a sheet of paper. If a wire tetrahedron made
up of six wire edges is represented on a sheet of paper, the
map will be the six lines formed by the four sides of a quadri-
lateral and its two diagonals, shown in Fig. 1.1. The point
FIG. 1.1
of intersection P of the diagonals AC and OB on the diagram
will be derived from two distinct points P and P on opposite
x 2
edges of the tetrahedron.
Isometric mappings
Maps in which each point P' on the map corresponds to one
and only one point P on the original structure are a special
SYMMETRY OF PYRAMIDS AND PRISMS 3
type and are called for obvious reasons one-to-one mappings,
written (1,1). The mapping of part of the surface of the earth
onto the page of an atlas is (1,1) but in addition to this property
these maps can be designed to preserve various features of
the original. Some maps preserve angles without preserving
relative size and others preserve relative size without preser-
ving shape, but all maps of the surface of a sphere onto
a plane produce some kind of distortion. Most maps designed
to show the major land masses of the earth distort the relative
size or shape of the polar regions and other regions round the
edge of the map. All geographical maps are on a scale much
reduced from the original.
Maps which take surfaces into surfaces and solids into solids
can be designed to be (1,1) and also to preserve some geo-
metrical features. A mapping which preserves shape is called
an isomorphism. A mapping which preserves shape and also
preserves size is called isometric. Such a mapping produces
an exact replica of the original or a point for point representa-
tion with distances between pairs of points preserved exactly,
and the map can be recognized immediately as closely related
to the original. Isometric mappings are the class of mappings
needed in the study of symmetry.
A design on a sheet of paper can be mapped isometrically
on to another sheet by taking a tracing and the tracing can be
looked at from above or below. A solid can be mapped by
making an exact replica. One cube can be mapped onto an
equal cube. The second cube can be turned around and moved
about at will and whatever its position it will be possible to
define a mapping which takes the original cube into the model
and the mapping will preserve size and shape. This collection
of mappings is too haphazard for the study of symmetry and
some kind of order must be produced from all the arbitrary
positions of the model.
A convenient way to describe the positions of a set of points
is to choose three mutually perpendicular axes fixed in space
4 EXPLORING UNIVERSITY MATHEMATICS 3
through a point O and label the three lines Ox, <9y, Oz. The
position of a point is determined by its three coordinates
(x, y, z). This is just an extension of the point in two-dimen-
sional coordinate geometry to include the third dimension
by the use of z. In Fig. 1.2 the x-axis and y-axis are intended to
r(P) ZKP)
XI (PI
f 111 .1 11■ f P
1
1 y
'
1 1 Λ
11 s * 1 x /
ΚΡ)χ^[ 1 X(P)
-1
I
z
X I
/
/
Z(P) YKP)
FIG. 1.2
lie in the familiar position in the plane of the paper and the
z-axis is perpendicular to the plane directed towards the reader.
The three distances x, y and z giving the coordinates of the
point P are shown.
A mapping can be denoted by a symbol M and M(P) would
be the symbolic way of writing "the point into which P is
mapped by the mapping M". If this is a point P\ it is possible
to write M(P) =P', and the coordinates (V, y\ z') of P'
referred to the fixed axes will depend on the mapping M and
the coordinates (JC, y, z).