A Mathematical Space Odyssey

A Mathematical Space Odyssey: Solid Geometry in the 21st Century

Claudi Alsina
Roger B. Nelsen
Volume: 50
Copyright Date: 2015
Edition: 1
Pages: 288
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  • Book Info
    A Mathematical Space Odyssey
    Book Description:

    Solid geometry is the traditional name for what we call today the geometry of three-dimensional Euclidean space. Courses in solid geometry have largely disappeared from American high schools and colleges. The authors are convinced that a mathematical exploration of three-dimensional geometry merits some attention in today’s curriculum. A Mathematical Space Odyssey: Solid Geometry in the 21st Century is devoted to presenting techniques for proving a variety of mathematical results in three-dimensional space, techniques that may improve one’s ability to think visually. Special attention is given to the classical icons of solid geometry (prisms, pyramids, platonic solids, cones, cylinders, and spheres) and many new and classical results: Cavalieri’s principle, Commandino’s theorem, de Gua’s theorem, Prince Rupert’s cube, the Menger sponge, the Schwarz lantern, Euler’s rotation theorem, the Loomis-Whitney inequality, Pythagorean theorems in three dimensions, etc. The authors devote a chapter to each of the following basic techniques for exploring space and proving theorems: enumeration, representation, dissection, plane sections, intersection, iteration, motion, projection, and folding and unfolding. In addition to many figures illustrating theorems and their proofs, a selection of photographs of three-dimensional works of art and architecture are included. Each chapter includes a selection of Challenges for the reader to explore further properties and applications. The book concludes with solutions to all the Challenges in the book, references, and a complete index. Readers should be familiar with high school algebra, plane and analytic geometry, and trigonometry. While brief appearances of calculus do occur, no knowledge of calculus is necessary to enjoy this book. The authors hope that both secondary school and college and university teachers will use portions of it as an introduction to solid geometry, as a supplement in problem solving sessions, as enrichment material in a course on proofs and mathematical reasoning, or in a mathematics course for liberal arts students.

    eISBN: 978-1-61444-216-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Preface
    (pp. ix-x)
    Claudi Alsina and Roger B. Nelsen
  3. Table of Contents
    (pp. xi-xiv)
  4. CHAPTER 1 Introduction
    (pp. 1-26)

    Welcome to amathematical space odyssey! In 1968 the celebrated film2001: A Space Odysseypremiered, directed by Stanley Kubrick, along with the classic novel of the same title by Arthur C. Clarke (who was co-writer with Kubrick of the movie’s screenplay). In this story a geometric object called amonolithplayed an important symbolic role. The monolith was a black rectangular slab measuring precisely 1 X 4 X 9 units. There are many other very interesting geometric objects in “space” associated with mathematics, mathematics that is often surprising and beautiful. In this introduction we present some examples of the material...

  5. CHAPTER 2 Enumeration
    (pp. 27-44)

    Mathematics is often said to be the study of patterns. Enumerative combinatorics is a branch of mathematics that counts the number of ways certain patterns can be formed. In combinatorial problems it is frequently advantageous to represent the patterns geometrically, as configurations of solids such as spheres, cubes, and so on. In many cases we have a sequence of configurations or patterns, and we seek to count the number of objects in each pattern in the sequence. We also exhibit examples where such a two-dimensional counting problem has a three-dimensional solution, and examples where a three-dimensional counting problem has a...

  6. CHAPTER 3 Representation
    (pp. 45-64)

    A visual method for establishing properties of positive numbers is to represent the numbers by volumes of solids. For example, we can illustrate identities for positive real numbers by using geometric transformations of solids that preserve the volume of the solid representing the number. Similarly we can illustrate certain inequalities by showing that the solid representing one number is a subset of another, so that the volume of the first is less than or equal to the volume of the second. We begin with some elementary algebra.

    For any positive real numbera, the numbercan, of course, be...

  7. CHAPTER 4 Dissection
    (pp. 65-82)

    Dissection is a remarkably effective technique for the study of solids, especially polyhedra. In this chapter we use dissection to establish the volume formulas for a variety of polyhedra. We begin with parallelepipeds, the parallelograms of space, and use them to derive the volume formulas for related solids including prisms, pyramids, frustums of pyramids, and four of the Platonic solids. We also employ dissections to study the isosceles tetrahedron and the rhombic dodecahedron. We conclude the chapter with various dissections of cubes into smaller cubes.

    Aparallelepipedis a solid whose six faces lie on three pairs of parallel planes,...

  8. CHAPTER 5 Plane sections
    (pp. 83-116)

    Slicing a solid with a plane to reveal a plane section (or plane cross-section) of the solid is a well-known technique for studying geometric properties of both the solid itself and the geometric configuration in the plane section. After a brief visit with the hexagonal section of a cube, we examine two important procedures, based on plane sections of a solid, for computing volumes of certain polyhedra—theprismoidal formulaandCavalieri’s principle. Next we examine deGua’s theoremfor the right tetrahedron, a three-dimensional analog of the Pythagorean theorem for right triangles. By examining plane sections of cones, we...

  9. CHAPTER 6 Intersection
    (pp. 117-132)

    In Example 1.5 on page 5 and Challenge 5.5 on page 111 we saw how the intersection of two simple objects, such as a pair of cylinders, can produce a more sophisticated object, the bicylinder. In this chapter we continue our examination of intersections in space. We begin with skew lines in space and concurrent lines in the plane. Next we examine the tricylinder, a solid resulting from the intersection of three cylinders. We study the angles in tetrahedra, both the dihedral angles formed by the pairs of planes containing adjacent faces, and the trihedral angles formed by triples of...

  10. CHAPTER 7 Iteration
    (pp. 133-146)

    In mathematics,iterationmeansrepetition, in the sense of repeating an operation or process with the goal of reaching a desired result, often requiring infinitely many repetitions. Each repetition builds upon the result of the previous repetition, and each of the repetitions in the process is also called an iteration.

    In this chapter we use iteration to derive a number of results in space, some of which may be surprising since they are profoundly different from the corresponding results in the plane. We show that there is no analog of the four color theorem for three-dimensional maps, and that it...

  11. CHAPTER 8 Motion
    (pp. 147-164)

    Isometriesin space are linear transformations that preserve distances. They include translations, rotations, and reflections. We visualize such isometries as motion in space, and usemotionto prove Viviani’s theorem for regular tetrahedra, to examine relationships among pairs of polyhedra with equal volumes using hinged dissections, and to solve a popular puzzle requiring moving cubes into a desired configuration.

    In later chapters we consider motions in space that are not isometries, such as projection of points (in Chapter 9) and folding and unfolding of surfaces (in Chapter 10).

    Points in space are the building blocks of all geometric figures. Simple...

  12. CHAPTER 9 Projection
    (pp. 165-192)

    Perhaps a child first encounters the representation of a three dimensional object in a two dimensional plane when he or she sees his or her shadow on the ground. So, in a sense, the concept of a mathematical projection is an ancient one. Projections serve many purposes, from the creation of perspective in painting, the use of multiple projections in architectural drawings, and the many types of projections of the earth’s surface to produce maps. In fact,nearly every illustrationin this book is a projection of a three-dimensional object onto a two-dimensional page!

    In this chapter we explore a...

  13. CHAPTER 10 Folding and Unfolding
    (pp. 193-226)

    You may have encountered folding and unfolding in solid geometry when you constructed paper or cardboard models of polyhedra, such as those illustrated in Figure 8.6.1 on page 155. These techniques can also be applied to study some of the properties of polyhedra. Using the polyhedral nets pioneered by the artist Albrecht Dürer we study the collection of polyhedra known as deltahedra. By simply folding paper we can construct a model of the regular pentagon, and also “solve” the ancient problem of duplicating the cube. Extending “unfolding” to include unrolling cylinders and cones leads us to a method for finding...

  14. Solutions to the Challenges
    (pp. 227-258)

    Many of the Challenges have multiple solutions. Here we give but one solution to each Challenge, and encourage readers to search for others.

    1.1. (a) four is the number of vertices and of faces in the tetrahedron.

    (b) six is the number of vertices in the octahedron and the number of edges in the tetrahedron.

    (c) six is the number of faces in the cube and the number of vertices in the octahedron, eight is the number of vertices in a cube and the number faces in the octahedron, and twelve is the number of edges in both the cube...

  15. References
    (pp. 259-264)
  16. Index
    (pp. 265-271)
  17. About the Authors
    (pp. 272-272)