Dan - Geometry and the Imagination.pdf - Topologia i Geometria - renvox

Geometry and the Imagination

Xiong Dan

An academic exercise presented in partial fulfillment for the degree of

Bachelor of Science with Honours in Applied Mathematics.

Supervisor: Associate Professor Helmer Aslaksen

Department of Mathematics

National University of Singapore

2003/2004

Acknowledgements

I would like to thank my supervisor, A/P Aslaksen, for conducting this wonderful

project. I am grateful to him for his time and his patience with me. I benefit greatly from

discussions I have had with him on this project, and I hereby express my appreciation

for his guidance full of inspirations. This project has been a meaningful and pleasant

experience with excitements and joys of discoveries, which I will always remember.

I would like to dedicate the regular octahedron shown below to my lovely girlfriend,

Goose. To me, she is as nice and perfect as a regular octahedron.

I would like to dedicate the space hexagon highlighted in bold on the regular octahedron

to all the people who have been kind to me. I would like to share my joys with you all.

I would like to dedicate the hyperboloid of one sheet that contains all six sides of the

space hexagon to my parents. I would like to express my gratitude to them for their love

to me over the years. Something is behind the scene, silent and invisible, yet you know.

CONTENTS

Acknowedemens i

Summy v

Author’s Contributions i

Inoduon 1

Chapter I: The Simplest Curves and Surfaces 5

1.1 Properties of second order plane curves 5

1.2 s f ltin 1

1.3 Rld s 6

Chapter II: Strain Transformations 20

2.1 Hilbert’s dilatations 0

2.2 Dilatations, shears, strains and linear transformations 23

.3 tis f tis 2

Chapter III: Geometry and the Imagination 45

3.1 From skew lines to a hyperboloid of one sheet 45

3.2 The closest regular packing of spheres 58

Bibaphy 2

iii

Summary

This thesis begins with discussions of properties of second order plane curves and

quadratic surfaces in Chapter I. In the study of quadratic surfaces, the concepts of

“surfaces of revolution” and “ruled surfaces” are introduced. As a particular example,

the surface of the hyperboloid of revolution of one sheet is closely examined by the

author. After showing that a hyperboloid of revolution of one sheet can be obtained by

rotating a straight line in space, the author goes on to examine the properties of the

straight lines lying in the surface. Following that the author extends these properties to

the hyperboloid of one sheet of the most general type. This process gives rise to a new

concept, called a “strain”. It is a transformation in space which can deform surfaces of

revolution to general type surfaces.

In Chapter II, the author examines the concept of “strain” and explores the properties of

this kind of transformation in space. When exploring the properties, the author notices

that strain transformations preserve collinearity, concurrency as well as tangency. From

this, the author points out the connection between the idea of perspectives in projective

geometry and the nature of strain transformations. As a demonstration, the author

concludes that if we can prove Brianchon’s theorem in the circle case, we will get the

ellipse case “for free”, by using an argument based purely on the properties of strain

transformation.

In Chapter III, two interesting problems are discussed. In the first problem, the author

presents a proof for the fact that given any 3 skew straight lines in space which are not

parallel to a common plane, there always exists a hyperboloid of one sheet containing

these three lines. The basic idea of the author’s proof is to find some strain

transformations that can transform the given 3 lines into positions such that a known

hyperboloid of one sheet contains all 3 of them. Due to the arbitrariness and ambiguity

of the positions of the three given straight lines, it proved difficult to determine such

strain transformations. In the author’s method of finding the desired transformations, a

very special space hexagon is constructed from the three given straight lines, and from

this, the author finishes the rest of the proof using properties of strain transformations.

In the proof of the first problem in chapter III, the author has a close look at the

structure of the regular octahedron, and discovers that the structure of the regular

octahedron can be used to visualize the connections between the face-centered cubic

lattice packing and the face-centered hexagonal lattice packing when constructing the

closest regular packing of spheres in space. This interpretation of the author’s is

explained in the second problem which concludes this thesis. □

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