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As the study of shape and form, geometry is able to model the space around us and the forms inhabiting this space. For this reason, geometry has always been highly regarded for its practical value, and geometry and its applications have long been a central part of the study of mathematics. Centuries ago, in prescribing geometry to be a part of the standard educational program for youth, Plato recognized that "for the better apprehension of any branch of knowledge, it makes all the difference whether a man has a grasp of geometry or not."
While there are many textbooks presenting a pure or theoretical approach to geometry and many monographs investigating a single aspect of applied geometry, it is difficult to find a wide-angle view of applied geometry. The purpose of this collection is to give as broad a picture as possible of the applications of geometry.
This collection will be a rich resource for the geometry instructor, whether as a supplement to standard textbook material, as reference material for student reports and projects, or as the starting point for a research program. The papers vary in difficulty, but are accessible to anyone having a college-level acquaintance with geometry.
Part 1: Art and Architecture
Spirals and the Rosette in Architectural Ornament, Kim Williams
Sun Disk, Moon Disk, Paul Calter
Façade Measurement by Trigonometry, Paul Calter
A Secret of Ancient Geometry, Jay Kappraff
Part 2: Vedic Civilization
Square Roots in the Sulba Sutras, David W. Henderson
Applied Geometry of the Sulba Sutras, John F. Price
Part 3: The Classroom
Ethnomathematics for the Geometry Curriculum, Marcia Ascher
Education with Fascination: Teaching Descriptive Geometry with Applications, Marina V. Pokrovskaya
Part 4: Engineering
Making Measurements on Curved Surfaces, James Casey
Mathematics to the Aid of Surgeons, Ramin Shahidi
The Geometry of Frameworks: Rigidity, Mechanisms and CAD, Brigitte Servatius
Geometry and Geographical Information Systems, George Nagy
On the Other Hands: Geometric Ideas in Robotics, Bud Mishra
Part 5: Decision-Making Processes
Decisions through Triangles, Donald G. Saari
Geometry in Learning, Kristen P. Bennett and Erin J. Bredensteiner
Part 6: Mathematics and Science
The Geometry of Numbers, Antonie Boerkoel
Statistical Symmetry, Charles Roadin
Three-Dimensional Topology and Quantum Physics, Lour H. Kauffman
Bridges between Geometry and Graph Theory, Tomaz Pisanksi and Milan Randic
Polytopes in Combinatorial Optimization, Thomas Burger and Peter Gritzmann
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 "Façade 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 façade of a building by surveying instruments are converted into distances between specific points on the façade.
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.