Geometry at Work: Papers in Applied Geometry available in Paperback
Geometry at Work: Papers in Applied Geometry
- ISBN-10:
- 0883851644
- ISBN-13:
- 9780883851647
- Pub. Date:
- 01/28/2000
- Publisher:
- Mathematical Association of America
- ISBN-10:
- 0883851644
- ISBN-13:
- 9780883851647
- Pub. Date:
- 01/28/2000
- Publisher:
- Mathematical Association of America
Geometry at Work: Papers in Applied Geometry
Paperback
Buy New
$28.95Buy Used
$20.37-
SHIP THIS ITEM— Temporarily Out of Stock Online
-
PICK UP IN STORE
Your local store may have stock of this item.
Available within 2 business hours
Temporarily Out of Stock Online
-
SHIP THIS ITEM
Temporarily Out of Stock Online
Please check back later for updated availability.
Overview
Product Details
| ISBN-13: | 9780883851647 |
|---|---|
| Publisher: | Mathematical Association of America |
| Publication date: | 01/28/2000 |
| Series: | Notes Series |
| Edition description: | CAS & BOOK |
| Pages: | 208 |
| Product dimensions: | 8.50(w) x 10.90(h) x 0.40(d) |
Table of Contents
1. Geometry at work in art and architecture K. Williams, P. Calter and J. Kappraff; 2. Geometry at work in Vedic civilization D. W. Henderson and J. F. Price; 3. Geometry at work in the classroom M. Ascher and M. V. Pokrovskaya; 4. Geometry at work in engineering J. Casey, R. Shahidi, B. Servatius, G. Nagy and B. Mishra; 5. Geometry at work in decision-making processes D. G. Saari, K. P. Bennett and E. J. Bredensteiner; 6. Geometry at work in mathematics and science A. Boerkoel, C. Radin, L. H. Kauffman, T. Pisanski, M. Randic, T. Burger and P. Gritzmann.Introduction
There is no branch of mathematics, however abstract, that will not eventually be applied to the phenomena of the real world.
-N.I. Lobachevsky
Geometry is the study of shape and form, with origins in surveying land, designing buildings, and measuring volumes. More importantly, geometry is a point of view that sees the shapes and forms that are intrinsic to any mathematical concept or relationship. Indeed, almost every area of mathematics incorporates geometric concepts and the geometric viewpoint in a fundamental way. For example, Descartes used analytic geometry to display the lines and curves associated with algebraic equations. A differential equation has inherent within its formulation a geometric phase space. When confronted with a new group, and algebraist might view it in terms of the symmetries of a higher-dimensional polyhedron to better understand its structure. Binary relations can be understood in terms of trees, and even probability has its foundation in the properties of measure spaces.
Today, geometry encompasses many different approaches, techniques, and theories, including Euclidean and non-Euclidean geometries, projective geometry, finite geometries, transformational geometry, computational geometry, differential geometry, discrete geometry, tilings, and knot theory. The papers in this collection show that the geometric point of view, in all its many different varieties, has many diverse applications going far beyond its origins in measuring distances, areas, and volumes.
Euclidean geometry, which has dominated the development of Western geometry, uses compass and straightedge to study the flat plane or three-dimensional space. We see the influence of Euclidean geometry in many of the papers in this collection. It is used in the field of art in papers by Kim Williams and Jay Kappraff and we see it used by the engineer as a vital tool for design and communication in the paper by Marina Pokrovskaya. Both Ramin Shahidi and Paul Calter show that measuring distances with Euclidean geometry continues to find new applications.
Non-Euclidean geometry takes the view that the intrinsic shape of space need not be flat but may be curved. Many of the shapes in our environment are curved, and Jim casey shows how the engineer can make use of Riemannian geometry to understand these curved shapes.
Analytic geometry integrates algebra and geometry, and it has allowed the geometer to make use of the powerful tools of algebra. Papers by Thomas Burger and Peter Gritzmann, Kristin Bennett and Erin Bredensteiner, Ton Boerkoel, Bud Mishra, and Don Saari demonstrate the effectiveness of integrating algebraic formulations with the study of geometric information.
Recent research in geometry has focused on invariants and other properties subtler than those of measurement. This work includes tilings, knot theory discrete geometry, computational geometry, and graph theory. Applications of these approaches can be found in the papers by Lou Kauffman, Charles Radin, George Nagy, Brigitte Servatius, and Tomaz Pisanski and Milan Randic.
Finally, cultures different from our own have seen geometry quite differently. In papers by Marcia Ascher, Jay Kappraff, John Price, and David Henderson we see that our own understanding of geometry can be used to analyze the work of other cultures and to bridge what gaps there may be in time or space between these cultures and ourselves.
That papers in this collection, written by pioneers and leading experts in their fields, are a valuable resource for geometry teacher and student alike. All of the papers are accessible to anyone having a college-level course in geometry and it is hoped that they will provide a broad vision of applied geometry-geometry at work.
1 The nature of Applications of Knowledge
Some have the attitude that
knowledge exists for its own sake, that applications need not be of concern to
the mathematician, and in fact may not even be possible for some areas of
mathematics. G.H. Hardy espoused the view that number theory, which he
considered to be the most beautiful and profound area of mathematics had not the
"slightest 'practical' importance." The comment by N.I. Lobachevsky, one of the
founders of non-Euclidean geometry, given at the beginning of this introduction,
takes quite a different stance. History has proved Hardy wrong-today the very
mathematics he cited as an example of the unproductive is the heart of the
widely used RSA cryptosystem. Lobachevsky, on the other hand, displayed
remarkable prescience, since the revolutionary geometry that he discovered,
opposed to the contemporary view of space and originally viewed as unnatural and
invalid, turned out to be precisely the viewpoint needed by Albert Einstein in
his general theory of relativity. Similarly, many new ideas and discoveries in
mathematics and science have eventually proven to have more useful application
despite unpromising beginnings. For this reason, it is worthwhile to understand
how the theoretical knowledge of mathematics can have real-world applications.
Although in all disciplines the general processes of gaining and
applying knowledge are similar, there are significant differences between
mathematics and the sciences which can serve to highlight the unique role of
mathematics in applying knowledge generally and in the types of applications we
see here. In the sciences, physical phenomena are the objects of study, while in
mathematics, the objects of study are purely intellectual concepts and
constructs. The real number line, for example is purely conceptual, having no
physical existence itself. In the sciences, observations of physical phenomena
are made by conducting experiments and recording measurements. For the
mathematician, computations, examples, counterexamples, special cases, and
diagrams replace microscopes and telescopes as a way of observing and measuring
the structure and behavior of mathematical objects. From observations,
principles of knowledge are derived by the scientist or mathematician which
describe pattern of behavior common to a significant class of examples. In
science, these principles are verified by experimentation and further
measurement; in mathematics, principles are verified intellectually by
mathematical proof.
Applying knowledge involves extending these
general principles to guide progress in some specific area of life. By its very
nature, knoeldge will be relevant to that area of life from which it is derived
and can serve as the basis for applications in that area. However, it is our
experience that mathematics has applications far beyond the boundaries of the
area in which it was first developed. Indeed, mathematics is striking in that
the concepts, principles, and techniques developed for the understanding of
purely non-physical mathematical constructs provide the essential tools that
scientists in all areas use to understand their observations and measurements of
the physical world. Eugene Wigner, in discussing the success of mathematics in
physics, says,
It is important to point out that the mathematical
formulation of the physicist's often crude experience leads in an uncanny number
of cases to an amazingly accurate description of a large class
phenomena.
Moreover, once the scientist's observations have been given a
mathematical formulation, the methodology and computations of mathematics can
extend these formulations to predictions about behavior that has not yet been
observed. This is of enormous importance to the scientist because, as Richard
Hamming points out,
Constantly what we predict from the manipulations of
mathematical symbols is realized in the real world. ? For glamour, I can cite
transistor research, space flight, and computer design, but almost all of
science and engineering has used extensive mathematical manipulations with
remarkable success.
The fact that abstract mathematical concepts,
which are often only suggested by observations of the physical world and depend
mostly on the imagination of applications is indicative of a common source for
both mathematics and the physical world.
As a subjective discipline,
mathematics depends on the creativity and aesthetic sensibilities, as well as
the intelligence, of the mathematician. Mathematical progress is always in the
direction of locating deeper and more abstract concepts, structures, and
relationships, every further removed from the physical world. Yet when we look
at mathematics in general, or at geometry in particular, we see that the deeper
and more abstract the concepts, the greater is the range of application in the
real world. In other words, the greater the subjective component of a
mathematical theory, the more effective is that theory in its objective role of
applications. William Thurston sees this as a natural phenomenon:
My
experience as a mathematician has convinced me that the aesthetic goals and the
utilitarian goals for mathematics turn out, in the end, to be quite close. Our
aesthetic instincts draw us to mathematics of a certain depth and connectivity.
The very depth and beauty of patterns make them likely to be manifested, in
unexpected ways, in other parts of mathematics, science, and the world.
Mathematical formulations of abstract patterns and relationships appear to
be in many cases our deepest understanding of principles that exist
throughout nature. Indeed, we can make the case that the wide applicability of
mathematics suggests the interconnectedness of all spheres of life, from the
abstract to the concrete. The beauty, orderliness, and universality that we see
in all areas of mathematics are reflected in the beauty and orderliness that
scientists find in their physical world.
2 The Role of Applications in the Study of Geometry
The full range of
geometry is from the most theoretical and abstract theorems based on axioms and
undefined terms to varied applications in science and technology, as well as in
other areas of mathematics. In recent years, applications of geometry have taken
on a more prominent and exciting role, both within and outside of mathematics.
For example, computers have spurred the development of many new areas, including
computational geometry, image processing, visualization, robotics, and dynamic
geometry.
Today more than ever, the study of applications is an
important component in the study of geometry, a subject traditionally valued for
its practicality. We gain a broader and richer understanding of geometrical
concepts when we see the unexpected applied contexts in which they appear. In
applied settings, geometrical concepts can take on new and quite different
interpretations; for example, a finite geometry can become a graph or a knot can
become a description of a quantum mechanical operator. Finally, there is charm
in seeing familiar theorems and principles showing their value in many different
roles. Without studying applications, a student will never see the complete
character of geometry.
This collection is an abundant resource for
those wishing to include applications in their study of geometry.
This
collection is an abundant resource for those wishing to include applications in
their study of geometry. We see here geometry used to describe and understand
the shapes that we see in the world around us and geometry used to design the
shapes that we construct to enrich our environment. We also see how geometry is
used in other branches of mathematics and in science to give shape and form to
mathematical data or concepts that are not inherently or intrinsically
geometric.
3 Geometry Used to Understand the Environment
Many papers in this
collection show ways of applying the tools of geometry to the analysis of shapes
that exist in our environment. An important such use is to transform
measurements that one is capable of obtaining into the kind of information that
one can really use. An old example of this is, of course, surveying, but there
are very modern examples as well.
Ramin Shahidi in "Geometry to the
Aid of Surgeons" and Paul Calter in "Fa?ade Measurement by Trigonometry" both
use similar geometric techniques. In the first case, measurements of a patient's
anatomy made by medical imaging machinery are converted into a potential
surgical trajectory. In the second case, measurements made of the fa?ade of a
building by surveying instruments are converted into distances between specific
points on the fa?ade.
George Nagy has an analogous problem, that of
converting the geographical measurements in a Geographical Information System
into a useful format. In a GIS, however, there is so much data that the role of
geometry is to synthesize the data into a format that can readily be
interpreted by the researcher. For example, elevation data can be transformed
into visibility graphs that can then be used for the optimal placement of fire
towers or radio transmitters.
Of a subtler nature is the question of
the arrangement of atoms in a quasicrystal, a kind of material having a new and
surprising symmetrical structure as measured by X-ray diffraction. Studying the
possible arrangements of atoms in a quasicrystal, Charles Radin has developed
the concepts of statistical symmetry, a way of measuring regularities in an
arrangement of shapes that is not, strictly speaking, symmetrical.
The
traces left behind by other cultures are sometimes difficult to understand and
they can easily be misinterpreted according to the learned fashions of the day.
When cultural legacies have shape and form, a geometrical analysis can give us a
quite reliable basis from which to make an interpretation of what has been left
to us. Marcia Ascher and Jay Kappraff demonstrate how to undertake a geometrical
analysis of our cultural legacies. They lead us to a broad-minded appreciation
of the possible interpretations that can legitimately be given to what we see in
other cultures.
In her paper on spirals, Kim Williams explains
techniques for constructing spirals, volutes, and rosettes so that we can
understand the challenges facing architects who have incorporated these
beautiful geometric shapes into their work. With this, we gain insight into the
intentions of the architects and deeper appreciation for their work.
David Henderson and John Price both look at writings left by the Vedic
civilization and, through close geometric analysis, give insight into the
possible reasoning, computations, and motivations behind the geometric
constructions given in the Sulba Sutras. As with the examples given by Ascher,
Kappraff, and Williams, we see that this expands our view of the achievements of
the past.
4 Geometry Used to Build the Environment
Geometry is essential for
designing the shapes that we build to mold our environment. Descriptive, or
projective, geometry has been the main tool used by the engineer to develop,
record, and communicate plans and designs. Marina Pokrovskaya connects fur us
the theoretical aspects of descriptive geometry to simple but realistic examples
of the practical applications of descriptive geometry. She shows how these
examples can be used in the classroom to train those who will be using
descriptive geometry in their work.
In her paper on rigidity of
frameworks, Brigitte Servatius examines the structure of rectangular grids to
determine whether or not they are rigid. She shows how the determination whether
or not they are rigid. She shows how determination of the rigidity of a
structure begins with the side-side-side theorem of Euclidean geometry, and then
goes on to show that many other geometric ideas can be applied to the analysis
of frameworks. In particular, graph theory is very effective in this area since
a framework can be viewed as a collection of rods (edges) connected at joints
(vertices).
In addition to flat surfaces, the designer or engineer
needs curves and curved surfaces. Jim Casey develops our understanding of
curvature and Riemannian geometry through measurement and experimentation and
then introduces us to the analysis of the structural properties of curved
surfaces.
A recent engineering creation is the robot. How should a
robot hand be designed so that it can securely grasp any shape? Using the
properties of convex sets, Bud Mishra is able to determine the exact number of
fingers a robot hand must have, and further, he shows how to determine the
placement of those fingers to grasp any given object.
Kim Williams and
Jay Kappraff have used geometrical analysis to deduce the intentions of artists
from geometrical evidence in their work. In Paul Calter's paper "Sun Disk, Moon
Disk," the artist himself explains his experience of integrating his aesthetic
purposes with his mathematical thinking in the design and construction of a
massive sculpture. This paper shows firsthand the nature of the mathematical
thinking that can go into creating a work of art.