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Math Made Visual

Math Made Visual: Creating Images for Understanding Mathematics

Claudi Alsina
Roger B. Nelsen
Copyright Date: 2006
Edition: 1
Pages: 190
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  • Book Info
    Math Made Visual
    Book Description:

    Is it possible to make mathematical drawings that help to understand mathematical ideas, proofs and arguments? The authors of this book are convinced that the answer is yes and the objective of this book is to show how some visualization techniques may be employed to produce pictures that have both mathematical and pedagogical interest. Mathematical drawings related to proofs have been produced since antiquity in China, Arabia, Greece and India but only in the last thirty years has there been a growing interest in so-called "proofs without words". Hundreds of these have been publised in Mathematics Magazine and The College Mathematics Journal, as well as in other journals, books, and on the Internet. Often times, a person encountering a "proof without words" may have the feeling that the pictures involved are the result of a serendipitous discovery or the consequence of an exceptional ingenuity on the part of the picture's creator. In this book the authors show that behind most of the pictures "proving" mathematical relations are some well-understood methods. As the reader shall see, a given mathematical idea or relation may have many different images that justify it, so that depending on the teaching level or the objectives for producing the pictures, one can choose the best alternative.

    eISBN: 978-1-61444-100-7
    Subjects: Mathematics

Table of Contents

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  1. Front Matter
    (pp. i-viii)
  2. Introduction
    (pp. ix-x)
    Claudi Alsina and Roger B. Nelsen

    Is it possible to create mathematical drawings that help students understand mathematical ideas, proofs and arguments? We are convinced that the answer is yes and our objective in this book is to show how some visualization techniques may be employed to produce pictures that have both mathematical and pedagogical interest.

    Mathematical drawings related to proofs have been produced since antiquity in China, Arabia, Greece and India but only in the last thirty years has there been a growing interest in so-called “proofs without words.” Hundreds of these have been published inMathematics MagazineandThe College Mathematics Journal, as well...

  3. Table of Contents
    (pp. xi-xvi)
  4. Part I: Visualizing Mathematics by Creating Pictures

    • 1 Representing Numbers by Graphical Elements
      (pp. 3-6)

      In many problems concerning the natural numbers (1, 2, …), insight can be gained by representing the numbers by sets of objects. Since the particular choice of object is unimportant, we will usually use dots, squares, spheres, cubes, and other common easily drawn objects.

      When one is faced with the task of verifying a statement concerning natural numbers (for example, showing that the sum of the firstnodd numbers isn²), a common approach is to use mathematical induction. However, such an analytical or algebraic approach rarely sheds light onwhythe statement is true. A geometric approach, wherein...

    • 2 Representing Numbers by Lengths of Segments
      (pp. 7-12)

      A very natural way to represent a positive numberais to construct a line segment of lengtha. In this way many relationships between positive numbers may be illustrated with figures, and relationships among lengths of line segments in those figures.

      Given two segments of lengthsa,b> 0 and a segment of unit length, we describe representations of some basic quantities associated withaandbin Figure 2.1.

      The best known and most common way to “average” two numbersaandbis thearithmetic mean(a+b)/2, which always lies betweenaandb. But...

    • 3 Representing Numbers by Areas of Plane Figures
      (pp. 13-18)

      Another very natural way to represent a (positive) number is by the area of a region in the plane. The simplest such regions are squares and rectangles, and calculus can be used to represent a number as the area under the graph of a function. Counting problems now become area computations, and inequalities between numbers can be established by showing that one region has a larger or smaller area than another.

      In Chapter 1 we encountered several visual representations for thenth triangular number${{T}_{n}}=1+2+\cdots +n$. If we use a square of unit area to represent the number 1, two...

    • 4 Representing Numbers by Volumes of Objects
      (pp. 19-22)

      In this chapter we represent a positive number by the volume of an object. In the simplest cases we can represent a product of three positive integers by the volume of a rectangular solid. We also can represent an integer by a collection of unit cubes, and establish identities by computing the volume. In many instances we may need to alter or rearrange the parts of an object before computing the volume.

      The reader is probably well acquainted with the following area representation of the familiar formula for factoring the difference of two squares:\[{{a}^{2}}-{{b}^{2}}=(a-b)(a+b).\]

      Using volumes, we have an...

    • 5 Identifying Key Elements
      (pp. 23-26)

      Mathematical pictures go beyond usual artistic representations because they often contain a great deal of information. They are sophisticated creatures that contain symbols as well as lines, angles, projections, measures, etc. In this chapter we illustrate the technique of introducing special “marks” in the pictures to identify relevant parts: equality of segments, equality of angles, repetitions, similar or congruent subsets, etc. In many cases appropriate identification of key elements readily yields a proof of the desired result. This is also the case in Euclidean geometry where, using straightedge and compass, one must construct figures using a collection of related elements...

    • 6 Employing Isometry
      (pp. 27-30)

      Isometries in the plane are linear transformations that preserve distances. The basic isometries are rotations, translations, and reflections. Preservation of Euclidean distance yields the fact that isometries preserve angles and areas, so shapes of figures are invariant. Two geometric figures related by an isometry are congruent.

      Perhaps the simplest (and most elegant) proof of the Pythagorean theorem is the following one from theChou pei suan ching, a Chinese document dating to approximately 200 b.c. It uses only translations of triangles within a square:

      The total area of the white portions inside the large square remains unchanged as three of...

    • 7 Employing Similarity
      (pp. 31-34)

      In this chapter we explore applications of similarity of geometric figures—primarily triangles—as a tool for illustrating and proving theorems in geometry. We’ve employed this technique in earlier demonstrations, for example, in Section 2.1, where we used similar triangles in Figure 2.2(a) to establish the fact that the altitude to the hypotenuse in a right triangle is the geometric mean of the segments it determines on the hypotenuse.

      The following theorem—and proof, using only similar triangles—is due to Ptolemy of Alexandria (circa 150 a.d.).

      Theorem.In a quadrilateral inscribed in a circle, the product of the lengths...

    • 8 Area-preserving Transformations
      (pp. 35-42)

      The basic length-preserving transformations in the plane are the isometries, which we examined in Chapter 6. Since isometries preserve lengths, they necessarily preserve angles, areas, volumes, etc. We now consider transformations in the plane which may not preserve lengths and angles, but do preserve areas.

      First consider triangles and parallelograms. If two triangles have a common base and if their vertices lie on a line parallel to the base, they must have equal areas, as illustrated in Figure 8.1(a). Similarly, two parallelograms with a common base and equal heights also have the same area, as illustrated in Figure 8.1(b). We...

    • 9 Escaping from the Plane
      (pp. 43-46)

      We have a natural tendency to solve planar problems in the plane and spatial questions in space. But restricting our arguments to the given context (plane or space) may be too limiting! There are many examples of planar problems which are easier to solve if we look at them “from a spatial point of view.” Conversely, one can also find spatial problems that may be more easily solved when reduced to planar problems.

      Our message in this chapter is to keep in mind the possibility of combining planar and spatial techniques.

      Let’s consider a beautiful result in the plane, known...

    • 10 Overlaying Tiles
      (pp. 47-54)

      Atilingof the plane is a countable family of closed sets (thetiles) that cover the plane without gaps or overlaps [Grünbaum and Shepard, 1986]. In Figure 10.1 we see portions of two examples, composed of tiles that are squares of two different sizes in (a) and rectangles and squares in (b).

      Indeed, tilings such as those above have been used for many centuries in homes, churches, palaces, etc. [Eves, 1976]. If we overlay a second grid of “transparent” tiles, we can construct visual demonstrations of a variety of mathematical theorems. We begin with the tilings in Figure 10.1,...

    • 11 Playing with Several Copies
      (pp. 55-58)

      In Chapters 3 and 4 we saw several instances where making several copies of the figure representing a mathematical expression made it easier to evaluate that expression. For example, we used two copies of an area representation of the sum$1+2+\cdots +n$in Figure 3.2 to show that the sum was one-half the area of a rectangle, and we used two copies of a volume representation for the double sum$\sum\nolimits_{i=1}^{n}{\sum\nolimits_{j=1}^{n}{(i+j-1)}}$in Figure 4.6 to show that the value of the double sum was one-half the volume of a rectangular box.

      In this chapter we present additional examples of this technique,...

    • 12 Sequential Frames
      (pp. 59-62)

      In many instances a sequence of pictures can be employed to illustrate a particular idea in mathematics. We used this idea in several sections in Chapters 4, 8, and 11, for example. One can think of a sequence of pictures as stills or “frames” from a motion picture, or for a Java-driven illustration on the web. In this chapter we provide a few more examples of this technique.

      Did you know that in any parallelogram, the sum of the squares on the diagonals is equal to the sum of the squares on the sides? See Figure 12.1:

      In the next...

    • 13 Geometric Dissections
      (pp. 63-68)

      Geometric dissections have long been popular in recreational mathematics. The basic task in these recreational puzzles is to cut up a geometric figure into pieces and reassemble the pieces to form a different figure. Sam Loyd (1841–1911) created a great many dissection puzzles, such as the “sedan chair” in Figure 13.1, where one was asked to “cut the sedan chair into the fewest possible pieces, which will fit together and form a perfect square, so the men will appear to be carrying a closed box” (see Challenge 13.1 at the end of this chapter). The “tangram” puzzles in Part...

    • 14 Moving Frames
      (pp. 69-72)

      Drawing an approximate graph of a function has been, for decades, a common exercise on visual representation in mathematics classes. While hand-made graphs on paper or a blackboard have always been a rather tedious procedure, today’s calculators and computers help us to draw sophisticated graphs (and to work with them) rapidly and efficiently.

      We assume the reader is already familiar with the use of such devices for graphing functions. So in this chapter we focus our attention on showing how some important functional properties may be better understood with visualization.

      The sumf+gand the productf·...

    • 15 Iterative Procedures
      (pp. 73-78)

      Closely related to the idea of using multiple copies of a picture (see Chapter 11) is the following procedure wherein we employ multiple (indeed, sometimes infinitely many) copies of a picture, but in such a way so that part of the picture is a scaled version of the entire picture. For example, in Figure 15.1(a) the northeast quarter of the square is a scaled version of the entire square. If we now let the side length of the largest square be 1 and label the interior rectangles and squares with their respective areas, we have in Figure 15.1(b) a visual...

    • 16 Introducing Colors
      (pp. 79-82)

      We often introduce color in mathematical pictures for aesthetic reasons or to distinguish various parts of the picture. In this chapter we illustrate the use of color (or just black and white, or various shades of gray or patterns) to enhance an argument. This idea is especially useful when working with tilings (tilings were introduced in Chapter 10).

      Given a standard 8 × 8 checkerboard (see Figure 16.1(a)), it is a simple matter to “tile” the board with 321 × 2 dominoes of the appropriate size (that is, place the dominoes on the board so that they don’t overlap and...

    • 17 Visualization by Inclusion
      (pp. 83-86)

      This technique is especially powerful for proving numerical inequalities between positive numbers. Its secret consists of the basic fact that whenever a setAis included in another setB, necessarily any measure ofA(cardinality, length, area, volume, …) will be less than (or equal to) the corresponding measure ofB.

      Given any triplea,b,cof positive numbers, there exists a triangle with sides of lengthsa,b, andcif and only ifa+b>c,b+c>a, andc+a>b. Without loss of generality, assumeabc. Then only the...

    • 18 Ingenuity in 3D
      (pp. 87-96)

      The aim of this chapter is to use concrete examples to show how some geometrical problems can be readily solved in three dimensions by means of ingenious “hands-on” strategies, while it would be almost impossible (or extremely tedious) to address the same problems in the traditional way.

      We start with a collection of problems that motivate the development of three-dimensional strategies.

      Problem 1Suppose we are sitting at a table, on which there is a box of unknown dimensions and a tape measure. What is the easiest way to determine the length of the diagonal of the box?

      The key...

    • 19 Using 3D Models
      (pp. 97-108)

      Many mathematical properties of three-dimensional objects may be seen easily by making appropriate three-dimensional models. A large collection of physical resources for the construction of models is presented in Part II.

      There are precisely five Platonic solids—polyhedra whose faces are congruent regular polygons and where the same number of faces meet at each vertex. They are illustrated in Figure 19.1, and their names and descriptions are: thetetrahedron(four triangular faces); thecube(six square faces); theoctahedron(eight triangular faces); thedodecahedron(twelve pentagonal faces); and theicosahedron(twenty triangular faces). To be consistent, the cube should be...

    • 20 Combining Techniques
      (pp. 109-116)

      In solving mathematical problems, it is often advantageous to combine various problem-solving techniques. The same is true for creating visual proofs of mathematical theorems. In this chapter we present a variety of examples combining many of the techniques found in earlier chapters.

      Heron’s remarkable formula$K=\sqrt{s(s-a)(s-b)(s-c)}$for the areaKof a triangle with side lengthsa,b, andc, andsemiperimeter s= (a+b+c)/2 can be proven by a variety of methods (see [Nelsen, 2001] for references). In this section we present visual proofs of two lemmas that reduce the proof of Heron’s formula to...

  5. Part II: Visualization in the Classroom

    • Mathematical drawings: a short historical perspective
      (pp. 119-121)

      Since its very beginning, the discipline of mathematics has combined three different resources for its development: anatural language(hieroglyphic Egyptian, Greek, Latin, English,…), asymbolic language(with signs +, −, ×, =,… and symbolsx,y,z,f,…) andfigures. There were two key reasons for introducing images in mathematical texts: to substitute appropriate drawings for long linguistic descriptions; and to facilitate mental reasoning based upon graphical intuition.

      In this photograph of a portion of the Moscow papyrus (Egypt, circa 1850 bc) we can see text, symbols and … the image of a trapezoid.

      In the beginning, most mathematical...

    • On visual thinking
      (pp. 121-123)

      The quotations above represent two extreme positions concerning the use of figures. For Dieudonné, as well as for many other mathematicians, the only correct way to present mathematics is by means of formal discourse based on formal languages. For Pólya, mathematical problem solving is often best done beginning with a visual representation.

      In addition to the classical arguments of mathematicians tominimize the role of intuition in order to save rigor, some other objections have been raised in terms of “perception” problems: images may give a false perception of realities. For example, a recent announcement of the FlatronTM company, which manufactures...

    • Visualization in the classroom
      (pp. 123-124)

      Among the standards presented by the National Council of Teachers of Mathematics in 2000 [NCTM, 2000] we find problem solving, reasoning, proof, communication, connections and representation. All of these require the student to think visually or may be achieved by means which incorporate visualization.

      Historically, visualization in the classroom has occurred with pencil on paper or with chalk on the blackboard. While this practice may be changing with more and more students having access to computers and graphing calculators, the traditional methods may never completely disappear.

      Technology opens new possibilities to visual experiences, from the modest superposition of transparencies on...

    • On the role of hands-on materials
      (pp. 124-127)

      In Chapters 1–17 we focused our attention on graphical images and in Chapters 18, 19 and 20 we presented a small sample of materials that facilitate an intuitive approach to mathematics, or to develop spatial intuition. What is the value of experimentation in mathematics? Following [De Villiers, 2003] we can consider experimentation as comprising non-deductive methods including intuitive, inductive or analogical reasoning. Its important aspects are:

      conjecturing(looking for an inductive pattern, generalization,…;

      verification(obtaining with certainty the truth or validity of a statement or conjecture);

      global refutation(disproving a false statement by generating a counterexample);

      heuristic refutation(reformulating,...

    • Everyday life objects as resources
      (pp. 127-132)

      To study the geometric characteristics of a building you could make a scale model of it … but there is a better alternative: use the real building! In this section we invite you to use materials readily available as objects with mathematical interest.

      Most of the man-made shapes that we see around us are the result of design: houses, streets, cars, beds, bells, pencils, etc. In this designed reality there is a strong mathematical component, from sizes to shapes. Most of these objects have been designed to satisfy some desirable purpose or function. In the classical dialogue between form and...

    • Making models of polyhedra
      (pp. 133-135)

      Polyhedra are among the best known three-dimensional geometrical objects. Families of polyhedra have been studied throughout the history of mathematics and models of them have become the standard way of visualizing them. One can construct accurate models from cardboard or plastic, or purchase sophisticated models with metal or plastic edges, hinged faces, etc. There is a large market for kits of polyhedra, and polyhedra may well be the geometrical topics that receive the most attention on the Internet.

      Our aim here is not to present a grand tour of the realm of polyhedra, but to present some examples and some...

    • Using soap bubbles
      (pp. 135-136)

      Experiments with soap solutions are both entertaining and convincing. The key observation here is the physical fact that soap films will form a minimal surface whenever they are limited by a frame.

      Every child knows that when a wire ending with a circular loop is immersed into a soap solution, then blowing into the soap circle creates a soap bubble which floats in the air for a short while.

      To experiment with soap films in the classroom you need to prepare your own soap solution: mix together one gallon of water, one cup of liquid detergent and one tablespoon of...

    • Lighting results
      (pp. 136-137)

      Light plays a crucial role in physics. Its maximum speedcappears in Einstein’s theory of relativity (E=mc²) and its behavior when reflected in a mirror (the angle of incidence is equal to the angle of reflection) or when crossing boundaries between liquids are well-known effects seen in the real world.

      Let us start with a very practical problem using light. Suppose you have a table which appears to be flat. How can you be sure that it is really flat? Take an object such that you are confident about its flatness on one face (a book, a...

    • Mirror images
      (pp. 138-140)

      Virtual images date back to the time when prehistoric men and women first gazed into a puddle of water and saw their reflections. High quality metals provided the first technological alterative to water. These were followed by mirrors made from silvered glass. Mirrors allow us to see images obtained by reflections and provide some rather intriguing views, since reflections are isometries that change orientation.

      For mirror experiments, it is possible to buy inexpensive plastic mirrors and mirrored craft paper or silvered Mylar film that can be easily cut with scissors.

      With a single mirror we see reflected images in which...

    • Towards creativity
      (pp. 140-142)

      The Catalan architect Antoni Gaud (1852–1926) was a very creative architect and investigator of new geometrical forms to be employed in architecture. He regularly used hands-on materials and produced scale models in expressing his remarkable three-dimensional creativity. In Figure II.24 we have photographs of his workshop in Barcelona.

      His most important project was the Temple of the Holy Family (Sagrada Familia) which is still under construction in Barcelona. The two basic surfaces in this project are the hyperbolic paraboloid and the hyperboloid of one sheet. Since both surfaces are ruled surfaces, i.e., formed by straight lines, they are easy...

  6. Part III: Hints and Solutions to the Challenges

  7. References
    (pp. 161-168)
  8. Index
    (pp. 169-172)
  9. About the Authors
    (pp. 173-173)