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CHAPTER 3
Experiencing Meanings in Geometry
David W. Henderson and Daina Taimina
What geometrician or arithmetician could fail to take pleasure in the symmetries, correspondences, and principles of order observed in visible things? Consider, even, the case of pictures: those seeing by the bodily sense the products of the art of painting do not see the one thing in the one only way; they are deeply stirred by recog- nizing in the objects depicted to the eyes the presentation of what lies in the idea, and so are called to recollection of the truth – the very experience out of which Love rises. (Plotinus, The Enneads, II.9.16; 1991, p. 129)
In mathematics, as in any scientific research, we find two tenden- cies present. On the one hand, the tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations. As to geometry, in particular, the abstract tendency has here led to the magnificent systematic theories of Algebraic Geometry, of Riemannian Geometry, and of Topology; these theories make extensive use of abstract reasoning and symbolic calculation in the sense of algebra. Notwithstanding this, it is still as true today as it ever was that intuitive understanding plays a major role in geom- etry. And such concrete intuition is of great value not only for the research worker, but also for anyone who wishes to study and appreciate the results of research in geometry. (David Hilbert, in Hilbert and Cohn-Vossen, 1932/1983, p. iii; italics in original )
It’s a thing that non-mathematicians don’t realize. Mathematics is actually an aesthetic subject almost entirely. (John Conway, in Spencer, 2001, p. 165)
The artist and scientist both live within and play active roles in constructing human mental and physical landscapes. That they should share structural intuitions is less surprising than inevitable. What is surprising and wonderful is how these intuitions have manifested themselves in the works of innovative artists and scien- tists in culturally apposite ways. (Kemp, 2000, p. 7)
The authors quoted above all stress the importance of the deep experience of meanings. It is these experiences in geometry (and indeed in all of math- ematics, as well as in art and engineering) that we believe deserve to be called aesthetic experiences. Mathematics is a natural and deep part of human experience and experiences of meaning in mathematics should be accessible to everyone. Much of mathematics is not accessible through formal approaches except to those with specialized learning. However, through the use of non-formal experience and geometric imagery, many levels of meaning in mathematics can be opened up in a way that most people can experience and find intellectually challenging and stimulating. A formal proof, as we normally conceive of it, is not the goal of math- ematics—it is a tool, a means to an end. The goal is to understand meanings. Without understanding, we will never be satisfied—with understanding, we want to expand the meanings and to communicate them to others (see also Thurston, 1994). Many formal aspects of mathematics have now been mecha- nized and this mechanization is widely available on personal computers or even on hand-held calculators, but the experience of meaning in mathematics is still a human enterprise. Experiencing meanings is vital for anyone who wishes to understand mathematics or anyone wanting to understand some- thing in their experience by means of the vehicle of mathematics. We observe in ourselves and in our students that such experiencing of meaning is, at its core, an aesthetic experience. In this chapter, we recount some stories of our experience of meanings in geometry and art. David’s story starts with art and ends with geometry, while Daina’s story starts with geometry and ends with art. However, the bulk of what follows we both share.
David’s Story: from Art to Mathematics
I have always loved geometry and have been thinking about geometric kinds of things ever since I was very young, as evidenced by a drawing I made when I was six years old (see Figure 1). The drawing is of a cat drawing a picture of a cat (who is presumably drawing a picture of a cat …). Notice the perspective from the point of view of the cat—for example, the drawing shows the underside of the table. I was already experiencing geometric meanings. But I did not realize then that the geometry that I experienced was mathematics or even that it was called ‘geometry’. I did not call it ‘geometry’ —I called it ‘drawing’ or ‘design’ or perhaps failed to call it anything at all and just did it. I did not like mathematics in school, because it seemed very dead to me—just memorizing techniques for computing things and I was not very good at memorizing. I especially did not like my high-school geometry course, with its formal, two-column proofs. However, I kept on doing geometry in various forms: in art classes, in carpentry, by woodcarving, when out exploring nature or by becoming
Chapter 3 – Experiencing Meanings in Geometry
except for analytic geometry and linear algebra, which only lightly touched on anything geometric. So, it was not until my fourth and final year at the university that I switched into mathematics and I only did so then because I was finally convinced that the geometry that I loved really was a part of mathematics. Since high school, I have never taken a course in geometry, because there were no geometry courses offered at the two universities I attended. Now I am a professional geometer and I started teaching an undergraduate Euclidean geometry course in the mid-1970s. My concern that both my students and I should experience meaning in the geometry quickly led me into conflict with traditional, formal approaches.
Daina’s Story: from Mathematics to Art
I took a lot of geometry, both in grade school and at the university. But I only had a very few art lessons in school. From them, I developed the impression that I could not draw and that I had little artistic talent. But I liked geometry precisely for its aesthetic values. My mathematics teachers always paid a lot of attention to how we drew geometric diagrams; they encouraged Euclidean constructions with compass and straight-edge, but also supported the free-hand drawing of geometric figures, while insisting on accurate shapes and proportions. At university, besides other traditional geometry courses, I also took a course in descriptive geometry, as well as a short course on how to draw three-dimensional geometric diagrams—both of these latter courses contained a lot about perspective. I always enjoyed and excelled at the drawing aspects of geometry, but I did not think it had anything to do with art or aesthetic sensibilities. When teaching the history of mathematics, I was particularly interested in the history of geometry and, because of my interest in art appreciation and art history, was happy to find so many connections between geometry and art. I was fascinated with the golden ratio, with the story of projective geometry arising from painters’ perspective prior to it becoming a pure mathematical subject and with the considerable impact of mathematics on art in the twentieth century (for example, in cubism and, later, in the work of M. C. Escher). I was also teaching a university course on ‘the psychology of mathematical thinking’, which led me to wonder about all creative thinking. I have had many students in my mathematics classes tell me that they were taking my class just to fulfill a distribution requirement. But they would also assert that they were no good at mathematics, because they are artists (poets, musicians, actors, painters) and their thinking is different. This made me wonder: is creative thinking really different in its very essence? So I decided as an experiment to take a watercolor class, knowing that I had never been any good at art. I wanted to get a glimpse of the emotions one goes through as a student in a subject for which one has no talent. I started the watercolor class not really understanding what techniques I should use
Chapter 3 – Experiencing Meanings in Geometry
Mathematics and the Aesthetic
for my brush, how to mix colors and other such technical details. But then I realized it was only the techniques I did not know. I found that my aesthetic experiences with drawing in geometry gave me a feel for how to use my skill at geometric drawing in painting. Ideas of composition and perspective in painting are all so geometrical. I enjoyed reading books about composition and perspective, as well as finding out how much I already knew from my earlier geometry studies. Proportions (the golden ratio, particularly) and shapes are directly related to composi- tion, but I had to learn about the use of colors. For perspective drawings, I already knew from three-dimensional geometric drawing how to draw in linear perspective, but I had to learn how to create an atmospheric perspec- tive. It was crucial for me to find out that I had had similar experiences already—albeit ones obtained in different ways and for different purposes. Below in Figure 2 is the painting I did after attending only eight water- color classes. I started it in class and later the same day finished it at home because I could not stop. When it was dry, I looked at it and could not believe I had painted it.
Experiencing ‘Undefined’ Terms
In geometry, ‘point’ and ‘straight line’ are usually referred to as “undefined terms”. In a formal sense, something has to be undefined, because it is impossible to define everything without being circular. However, if we want to pay attention to meanings in geometry, then we must still ask what is the
Figure 2: Daina’s watercolor painting (“Sunset on Oregon Coast”, 27.5 ” x 17.5 ” , photograph by Daina Taimina)
Mathematics and the Aesthetic
This symmetry meaning is in line with Heath’s (1926/1956) translation of Euclid’s definition of straight line as “a line that lies evenly with the points on itself” (p. 153), which Heath then attempts to clarify in a footnote:
we can safely say that the sort of idea which Euclid wished to express was that of a line […] without any irregular or unsymmetrical feature distinguishing one part or side of it from another. (p. 167)
Using these experientially-based meanings of straightness, we can ask what are straight lines on the surface of a sphere. If we look at this question from a point of view outside of the sphere, then clearly the answer is that there are no straight lines on a sphere. This is the extrinsic point of view. On the other hand, there is an intrinsic point of view. Imagine yourself to be a bug crawling on a sphere. The bug’s universe is just the spherical surface. What paths on the sphere would the bug experience as straight? After some exploration, we can convince ourselves that the great circles on the sphere are the curves that have the same symmetries (with respect to the sphere) that ordinary straight lines have with respect to the plane. We thus say that the great circles are intrinsically straight. A much more usual approach in texts is simply to define straight lines on the sphere to be the great circles—but, again, this blocks contact with the meaning (and, thus, the potential for aesthetic experience). So, again, why and in what way are these two meanings (“shortest” and “symmetric”) related? On the sphere, we can see that (Figure 4), for two nearby points of the equator (a particular great circle), the shortest distance is along the equator. However, there is another straight path (in the sense of “symmetric”) between the same two points that traverses the equator in the opposite direction (going the long way round). Thus, the “symmetric” meaning is not always the “shortest” meaning. In addition, there are surfaces with corners (see Figure 5) for which the shortest path is not symmetric.
Figure 4: ‘Intrinsically straight’ on a sphere
A simple question that may seem intuitively straightforward at first glance, namely “what is the meaning of ‘straight’?”, reveals some deeper intuitions about symmetry and shortest distance, which may only become meaningful when explored in different geometrical contexts.
Proofs as Convincing Communications
that Answer the Question Why?
Much of our own view of the nature of mathematics is intertwined with our notion of what a proof is. This is particularly true with geometry, which has traditionally been taught in high school in the context of ‘two-column’ proofs (see Herbst, 2002). Instead, we propose a different view of proof as “a convincing communication that answers a why -question”. The book entitled Proofs Without Words (Nelsen, 1993) contains numerous examples of visual proofs that provide an experience of why something is true—a experience that is, in most cases, difficult to obtain from the usual formal proofs. For example, Nelsen writes about the following result, which is usually attributed to Galileo (1615) – see Drake (1970).
We can easily check that this is true by simply adding the numbers.
These are the cases n = 2 and n = 3 of the more general equality.
Chapter 3 – Experiencing Meanings in Geometry
Figure 5: “Shortest” is not the same meaning as “symmetric”
=^1
1 + 3 + ... + (2 n – 1)
(2 n + 1) + (2 n + 3) + ... + (4 n – 1)
=^1
Chapter 3 – Experiencing Meanings in Geometry
Sometimes we have legitimate why- questions even with respect to state- ments traditionally accepted as axioms. One is Side-Angle-Side (or SAS):
If two triangles have two sides and the included angle of one of them that are congruent to two sides and the included angle of the other, then the triangles themselves are congruent.
SAS is listed in some geometry textbooks as an axiom to be assumed; in others, it is listed as a theorem to be proved and in others still as a definition of the congruence of two triangles. But clearly one can ask: why is SAS true on the plane? This is especially true because SAS is false for (geodesic) triangles on the sphere. So naturally one can then ask: why is SAS true on the plane, but not on the sphere? Here is another example – the vertical-angle theorem: If l and l′ are straight lines, then angle α is congruent to angle β.
The traditional proof of this in high-school geometry is to label the upper angle between α and β as γ, and then assert α + γ = 180˚ and γ + β = 180˚. The usual proof then concludes that α is congruent to β because they are both equal to 180˚ – γ. This proof seems fine until one worries about whether the rules of arithmetic apply in this way to angles and their meas- ures. The traditional solution in high school is to use several ‘ruler and pro- tractor’ axioms to assert the properties needed. We do not know of anyone for whom this proof with the attendant axioms has aesthetic qualities (though it may be convincing). We do not usually perceive a proof as aes- thetically pleasing when it is mostly repeating a list of axioms in a way that the meaning does not come through clearly. This proof seems to be an unnecessarily complicated answer to the question: why are vertical angles congruent to one another? For about ten years of teaching this theorem in his geometry course, David was satisfied with the idea of this proof, though he managed to simplify and make more geometric the necessary assumptions contained in the ‘ruler and protractor’ axioms. But then one student suggested that the vertical angles were congruent because both lines had half-turn symmetry about their point of intersection, P. David’s first reaction was that her argu- ment could not possibly be a proof—it was too simple and did not involve
Figure 7: The vertical-angle theorem
Mathematics and the Aesthetic
everything in the standard proof. But she persisted patiently for several days and David’s meanings deepened. Now her proof is much more convincing to him than the standard one, because it directly clarifies why the theorem is true. Even more importantly, the meaning of the student’s ‘half-turn’ proof is closer to the meaning in the statement of the theorem. To see this, look at the situation depicted in Figure 8.
Here, there is no symmetry: yet, the standard proof seems to apply and gives a misleading result. By means of either zooming in on the point of intersection until the curves are indistinguishable from straight-line segments (or by means of defining this angle to be the angle between the lines tan- gent to the curves at the intersection), symmetry arguments can be shown to apply and, hence, it is possible to argue that the angles α and β are con- gruent. However, the standard proof does not provide a way to discuss this, except by means of a discussion of when the ‘ruler and protractor’ axioms are valid. One could ask:
But, at least in plane geometry, isn’t an angle an angle? Don’t we all agree on what an angle is? To which a reply could be: Well, yes and no.
Consider the acute angle depicted in Figure 9.
The angle is somehow at the corner, yet it is difficult to express this formally (note that the zooming meaning of ‘point’ seems to be involved here). As evidence of this difficulty, we have looked in all the plane geometry books in Cornell University’s mathematics library for their definitions for ‘angle’. We found nine different definitions. Each expressed a different meaning or aspect of ‘angle’ and, thus, each could potentially lead to a different proof for any theorem that crucially involves the meaning of ‘angle’.
Figure 9: Where is the angle?
Figure 8: Are the opposite angles α and β the same?
Mathematics and the Aesthetic
experience similar to that of experiencing spherical geometry by means of handling a physical sphere. In other words, the experience of hyperbolic geometry available through the models did not directly include an experience of the intrinsic nature of hyperbolic geometry. Mathematicians looked for surfaces that would posess the complete hyperbolic geometry, in the same sense that a sphere has the complete spherical geometry. A little earlier, in 1868, Beltrami had described a surface (called the ‘pseudosphere’, see Figure 11), which has hyperbolic geometry locally. The pseudosphere also has a certain aesthetic appeal for us in the way (as with the Poincaré models) it points the imagination towards infinity. However, the pseudosphere allows only a very limited experience of hyper- bolic geometry, because any patch on the surface that wraps around the surface or extends to the circular boundary does not have the geometry of any piece of the hyperbolic plane.
Figure 10: M. C. Escher’s Circle Limit III (based on the Poincaré disc model)
Figure 11: The pseudosphere
Chapter 3 – Experiencing Meanings in Geometry
At the very beginning of the last century, David Hilbert (1901) proved that it is impossible to use real analytic equations to define a complete surface whose intrinsic geometry is the hyperbolic plane. In those days, ‘surface’ nor- mally meant something defined by real analytic equations and so the search for a complete hyperbolic surface was abandoned. And N. V. Efimov (1964) extended Hilbert’s result, by proving that there is no isometric embedding of the full hyperbolic plane into three-space, defined by functions whose first and second derivatives are continuous. Still, even today, many texts state incorrectly that a complete hyperbolic surface is impossible. However, Nicolas Kuiper (1955) proved the existence of complete hyper- bolic surfaces defined by continuously differentiable functions, although without giving an explicit construction. Then, in the 1970s, William Thurston described the construction of a surface (one that can be made out of identical paper annuli) that closely approximates a complete hyperbolic surface. (See Figure 12 and Thurston, 1997, pp. 49-50.) The actual hyperbolic plane is obtained by letting the width of the annular strips go to zero. In 1997, Daina worked out how to crochet the hyperbolic plane, following Thurston’s annular construction idea. (See Figure 13.) Directions for constructing
Thurston’s surface out of paper or by crocheting can be found in Henderson and Taimina (2005) or in Henderson and Taimina (2001). In these refer-
Figure 12: Construction of the annular hyperbolic plane
The geodesics (‘intrinsic straight lines’) on a hyperbolic surface can be found using the “symmetry” meaning of straightness discussed above: for example, the geodesics can be found by folding the surface (in the same way that folding a sheet of paper will produce a straight line on the paper). This folding also determines a reflection about that geodesic. Now, by interacting with these surfaces, we can have a more direct experience of meanings in hyperbolic geometry. And, very importantly, we can experience the connections between these meanings and the three nine- teenth-century models discussed above. These models can now be interpre- ted as projections (or maps) of the hyperbolic surface onto a region in the plane that distort the surface in a similar manner to the way projections (maps) of a sphere (such as the Earth) onto a region of the plane distort dis- tances, areas and/or angles. This is important, because these models are used to study hyperbolic geometry in detail, while the surface itself allows us direct experience with the intrinsic geometry. Before we had experience of these physical surfaces, our only experi- ences of hyperbolic geometry were through formal study with axiom systems and analytic study of the nineteenth-century models. The models provided aesthetic experiences that led our imagination to infinity, but this was not directly connected with geometric meanings. For example, the question that we (as well as most students) had was: why are geodesics in both Poincaré models represented by semi-circles or circular arcs? To us, the nineteenth-century models were more like artistic abstractions. But, after constructing the surfaces, we could see how and why the geodesics are represented in the way they are. (See Henderson and Taimina, 2005, or Henderson and Taimina, 2001, for more details of these connections, includ- ing proofs that the intrinsic geometry of each of the surfaces is the same geometry as that represented by all of the models.)
Radius and curvature of the hyperbolic plane
Since all hyperbolic planes are the same up to scale, most treatments of the hyperbolic plane consider the curvature to be –1. It is very difficult to give meaning to the effects of the change of curvature without looking at actual physical hyperbolic surfaces with different curvatures. Each sphere has a radius r (which is extrinsic to the sphere) and its (Gaussian) curvature (as defined in differential geometry) is 1/ r^2. In a similar way, each hyperbolic plane has a radius r, which turns out to be the ( extrinsic ) radius of the annuli that go into Thurston’s construction and the (Gaussian) curvature of the hyperbolic plane is –1/ r^2. We were not aware of any meaning for the radius of a hyperbolic plane before experiencing these surfaces. From a theoretical perspective, changing the radius or curvature is merely a change of scale and spheres, for example, of radii 4cm, 8cm and 16 cm look very much alike. However, we were shocked when we looked at the hyperbolic planes with these same radii (see Figures 15a, 15b and 15c, drawn with radii of 4 cm, 8 cm and 16 cm respectively).
Chapter 3 – Experiencing Meanings in Geometry
Mathematics and the Aesthetic
Figure 15a-c: Hyperbolic planes with different radii (crocheted by Daina Taimina, photographed by David W. Henderson)
In the next section, we turn to look at the design of machines in the nine- teenth century—at first sight, perhaps, a surprising leap. But in a curious way, these machines embody striking geometric principles and experiences in their design and the same questions we have been addressing (such as what is ‘straight’?) reappear in exciting ways and, perhaps unexpectedly, horocycles reoccur once more.
Experiencing Geometry in Machines
Recently, we have been working on an NSF-funded project to examine the mathematics inherent in a collection of nineteenth-century mechanisms, as well as to see to the inclusion of these mechanisms (along with commen- taries and learning modules) as part of the new National Science Digital Library (NSDL—see www.nsdl.org). Our experiences with these all of vari- ous mechanisms are offering us different perspectives on geometry, per- spectives that arise from motion. For example, this work has brought us back to the question: what is ‘straight’? When using a compass to draw a circle, we are not starting with a figure we accept as circular: instead, we are using a fundamental property of circles, namely that the points on a circle are at a fixed distance from the center, as the basis for the tool. In other words, we are drawing on a mathematical definition of a circle. Is there a comparable tool (serving the equivalent role to a compass) that will draw a straight line? If, in this case, we want to use Euclid’s definition (“a straight line is a line that lies evenly with the points on itself”), this will not be of much help. One could say: We use a straight-edge for constructing a straight line. To which a response might be:
Well, how do you know that your straight-edge is straight? How do you know that anything is straight? How can you check that some- thing is straight?
This question was important for James Watt. When he was thinking about improving steam engines, he needed a mechanism in order to convert cir- cular motion into straight-line motion and vice versa. In 1784, Watt found a practical solution (which he called “parallel motion”) that consisted of a link- age with six links. He described his parallel motion mechanism as being free of “untowardly frictions and other pieces of clumsiness”, claiming it to be “one of the most ingenious simple pieces of mechanisms that I have con- trived” (in Ferguson, 1962, p. 195). These expressions of smoothness and efficiency seem to be very close to what we are calling ‘aesthetic’. However, Watt’s mechanism produced only approximate straight-line motion: in fact, it actually produces a stretched-out figure of eight. Mathematicians were not satisfied with this approximate solution and worked for almost a hundred
Mathematics and the Aesthetic
Chapter 3 – Experiencing Meanings in Geometry
years to find exact solutions to the problem. A linkage that draws an exact straight line (see Figure 17a) was first reported by Peaucellier, in 1864. (See Henderson and Taimina, 2004 and 2005, for a discussion of relevant history.)
Why does the Peaucellier linkage draw a straight line? We suggest the reader connect to a web site where this linkage is depicted in motion (for example, see: KMODDL.library.cornell.edu). As an exercise in analytic geom- etry, one can verify that the point Q will always lie along a straight line—but this still does not answer the why- question. Especially difficult is being able to see any relationship with either the “shortest” or the “symmetric” meaning of straightness: is there perhaps a different meaning of straightness that is operative here? In the ‘inversor’ (that is, the links joining C, R, Q, S, and P in Figure 17b), the points P and Q are inverse pairs with respect to a circle with center C and radius r = √ s^2 – d^2. Analytically, this means that:
distance ( C to P ) × distance ( C to Q ) = r^2.
Here, the crucial property of circle inversion is that it takes circles to circles. (For details on circle inversion, see Chapter 16 of Henderson and Taimina, 2005.) After experiencing the motion of the linkage, we now see that P is constrained (by means of its link to the stationary point B ) to travel in a circle around B. Thus, Q must be traveling along the arc of a circle. The radius
Figure 17a: The Peaucellier linkage (photographed by Francis C. Moon)
