Int J Comput Math Learning (2008) 13:71–82 DOI 10.1007/s10758-008-9126-6 COMPUTER MATH SNAPSHOTS. EDITOR—URI WILENSKY*
Explorations with Sketchpad in Topogeometry Amanda Hawkins Æ Nathalie Sinclair
Published online: 5 March 2008 Ó Springer Science+Business Media B.V. 2008
In his report on the Fields medalist Grigory Perelman, the comedian Steven Colbert convincingly disproved mathematicians’ long-standing claim that the doughnut is not topologically equivalent to the sphere by squeezing a doughnut into a ball. While mathematicians may not accept this proof, Colbert certainly showed that analogies that may seem to provide non-mathematicians with more intuitive and perhaps amusing insights into mathematics can turn out to seem even more absurd than the arcane language that mathematicians speak amongst themselves. Nonetheless, the first author of this snapshot was quite compelled by the kinds of intuitive, visual, and interesting ideas that topology seemed to offer, like the fact that topologists call any two surfaces with just one hole, no matter how different they may look, the same. However, after an initial description of topology as ‘‘rubber band geometry,’’ she found that her university topology course went quickly the way of symbolism and abstraction. In this snapshot, we describe several microworlds of topological surfaces (the Mobius strip, the torus and the Klein bottle) that we created using The Geometer’s
*This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy. They could also include puzzles or brain-teasers involving the use of computers or computational theory. Snapshots are subject to peer review. This innovative snapshot uses Geometric Sketchpad to create a series of microworlds for representing and exploring standard topological surfaces such as the Torus, Klein bottle and Mobius strip. Using differing strategies of ‘‘wraparound’’, these surfaces can be explored in their 2D flat representations. Using a dot on the leading edge of a sketchpad ray as the driver for the exploration, the authors call this domain ‘‘topogeometry’’. Several interesting topogeometric questions are explored and visualized. From the Column Editor Uri Wilensky, Northwestern University. e-mail:
[email protected] A. Hawkins (&) Michigan State University, 8462 Pine Creek Drive, Shelby Township, MI 48316, USA e-mail:
[email protected] N. Sinclair Simon Fraser University, Burnaby, BC, Canada e-mail:
[email protected]
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Sketchpad. We originally designed the microworlds with the intent of providing students with opportunities to explore topological ideas using a representation that is sufficiently serious, mathematically, to move beyond standard analogies of doughnuts and coffee cups, without being so symbolic or abstract as to discourage elementary intuition and introductory reasoning. However, the constraints and affordances of Sketchpad provoked further exploration in a new domain that combines both geometric and topological ideas and techniques. We begin by describing how the microworlds’ representation of topological surfaces fundamentally differs from the conventional representations of textbooks and blackboards. We then discuss some of the interesting phenomena these microworlds allowed us to explore, some of which ended up being more geometric in nature.
1 Introduction: Representations of Topological Surfaces Introductory topology classes, like the one taken by the first author, often begin with representations such as those shown in Fig. 1—two-dimensional drawings of threedimensional solids, with the objects of focus being the surfaces of the solids. These images may help illustrate the objects of study, but their representational effectiveness becomes quickly challenged: try using the picture to find whether a path can be drawn from any one point to any other without crossing a boundary line! Of course, a more helpful representation might be a doughnut itself, on which one could imagine tracing a path with a pen. The actual doughnut has its own representational limitations: you have to keep turning it around in order to follow the fate of a line winding around its contours; you have to keep brushing sugar crumbs off your desk. Beyond these, handling the actual doughnut imposes an extrinsic experience of the topological surface, in which one leaves the doughnut and looks at it from above. Instead, topology favours a more intrinsic viewpoint in which one stays on the surface and makes do with the locally available information. Partially to address these limitations, topologists often prefer rectangular diagrams: flat (two-dimensional) rectangles that represent topological surfaces such as the as the torus and the Klein bottle. Such schematic diagrams provided a way of seeing the whole surface at once, and encode symbolically information about the ways in which edges of the rectangle are glued together to create the three-dimensional object. Figure 2a shows a torus
Fig. 1 Two-dimensional picture of a three dimensional torus
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Fig. 2 Rectangular representations of the (a) torus and (b) Klein bottle
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Fig. 3 ‘‘F’’ on the (a) torus and (b) Klein bottle
while Fig. 2b shows the rectangular diagram of a Klein bottle. The sides that are glued together are labeled with the same letter, and the direction of the arrows indicates whether there is a twist before the edges are glued. These representations have the benefit of picturing the whole surface at once. In Fig. 3a, the letter F has exited the rectangle on the right, which means it will come back in from the left, having preserved its orientation. Figure 3b shows the letter F in a similar situation, but this time on the Klein bottle, which contains a twist between the right and left edges. Thus the portion of the F extending to the right of our flat rectangular diagram re-appears on the left side upside-down. On a Mo¨bius strip (not pictured), the inverted letter F would look just like it does on the Klein bottle as long as extends off a left/right edge; however, extensions off the top/bottom edges would simply ‘‘fall off’’ the strip. The grammar of the rectangular diagram can itself be quite generative; we can simply permute edge letters and directions to produce new topological surfaces that may or may not have physical counterparts instantiable in three dimensions. Of course, we could also consider representations that are non-rectangular; the double-holed torus, for example, can be represented as an edge-connected octagon (Meyerhoff 1992). These diagrams are intriguing, in that they blend representational (literal) and symbolic (figurative) notational components. But they can be cumbersome to work with purely representationally: imagine having to trace the path of an ant beginning its walk at some starting position and traversing the Klein bottle, and having to figure out where it re-enters the fundamental region after each exit. Having done so, what if you wanted to have the ant walk off in a different direction or start at a different initial position? These questions are ones that require dynamic representations such as those familiar in other areas of mathematical visualization through interactive geometry software such as
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The Geometer’s Sketchpad (Jackiw 1991). In the next section, we describe briefly how we constructed dynamic flat rectangular diagrams using Sketchpad, and how they enabled us to interact with topological surfaces in new ways. We propose that this representation provides some of the intuitive understanding enabled by exploring the surface of realworld objects, while also benefiting from the generative potential and propositional nature of the ‘‘flat’’ rectangular diagrams.
2 Creating our Topological Microworlds We begin by describing the dynamic flat Torus microworld, and then we provide a brief explanation of how it was constructed in Sketchpad. Then we show examples of geometric objects constructed in Torus, Klein bottle and Mo¨bius strip microworlds. The dynamic flat Torus microworld includes a sketch that contains an empty rectangle representing the fundamental region of the torus, along with a set of custom tools to be used in that rectangle that allow users to construct geometric objects such as points, circles, and rays (see Fig. 4). (The geometric objects that are produced by these custom tools are quite different from the objects produced by ‘‘regular’’ Sketchpad tools—those objects naturally live in the Euclidean plane rather than on the Torus). The implementation paradigm of this set of microworlds is that of a Sketchpad ‘‘drawing world’’ (Jackiw 1997) where from the user’s perspective Sketchpad’s traditional tools are replaced by surrogate, authored tools. Behind the scenes (that is, from the author’s perspective), coordinate arithmetic and the generalization of a point construction through locus techniques to surrogate compass and straightedge constructions form the ingredients for each topological toolkit. The Torus microworld implementation requires that any point on the Euclidean plane be mapped to a corresponding point in the fundamental region—for instance, by hypothesizing a tessellation of fundamental region tiles covering the plane. The edge connections of the torus demand that such tiles be related through translational symmetry. Such a mapping can be accomplished in Sketchpad through geometric techniques, but for the purposes of authoring a surrogate toolkit, an analytic approach is preferable. For convenience, define the fundamental region to be the unit rectangle of the rectangular coordinate system. Any point P’ in the first quadrant can be thought of the image of some point P in the fundamental region whose coordinates are only the fractional components of the coordinates of P’. For example, consider the Euclidean point P’ at (5.1, 1.7). Such a point can be thought of as the image of some P in the fundamental region of
Fig. 4 The dynamic flat Torus microworld
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the torus, after it has moved through 5 full (and 0.1 partial) horizontal transits across the region, and after one full (and 0.7 partial) vertical transits. Since edges in the torus are untwisted, each full transit parallels the previous one, and so any point (5.1, 1.7) or (3.1, 6.7) or (10.1, 0.7) has the same fundamental pre-image (0.1, 0.7)—in other words, the value of the integer part of the coordinates does not matter. In the Klein bottle, however, horizontal transits have twisted edges. While an even number of twists cancel themselves out, the odd number of horizontal transits implied by the x = 5.1 requires us to ‘‘twist’’ the vertical remainder of y = 0.7 through 1-y to 0.3. Thus (5.1, 1.7) in the Euclidean plane images becomes (0.1, 0.3) in the Klein bottle’s fundamental region. Once a topologically-satisfactory point mapping has been established—the arithmetic is trickier outside the first quadrant—the construction of other geometric objects such as circles and rays can be achieved mechanically, by finding the locus of the pre-image points as the image point travels along the path of those other geometric objects. Thus a Torus ray is simply the locus of a ‘‘torus point’’ P’ defined as the corresponding Euclidean point P travels a Euclidean ray. To use the Torus Ray tool, click-and-drag that tool with the mouse to establish two points on the fundamental region. Since this fundamental region represents the whole torus, the ray can never escape it, but instead wraps around it like the striping on a candy cane. Figure 5 shows quite clearly that the flat representation of a ray looks nothing like the Euclidean ray, and nor does it have the curvature or continuity that the corresponding ray would have on a 3d torus. Now consider the Klein bottle microworld. The use of the Klein Ray is similar to that of the Torus Ray but the resulting image is quite different. Due to the Klein bottle’s twisted side, the candy cane stripes of the torus turn into a ‘‘criss-cross’’ on the Klein bottle, as seen in Fig. 6a—unless the ray is drawn parallel to the right or top edge of the fundamental region (Fig. 6b). In the Mo¨bius strip microworld, the Mobius Ray tool is somewhat less interesting as the ray eventually disappears whenever it touches the top or bottom edge of the fundamental region (falling off the strip).
3 Investigations in the Topogeometry Microworld While creating the tools was interesting, challenging and—on completion—satisfying, the real fun began as we explored the dynamics of these flat representations, and developed a Fig. 5 A ray on the flat torus
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Fig. 6 (a) A ray on the flat Klein Bottle. (b) A second Klein ray
sense for how different geometric objects behave as we changed their shapes and sizes. And at a reflective level, we were delighted by the degree to which being able to author these microworlds did not preclude our ability to be confounded, surprised, and illuminated by subsequent explorations within them. We present here a few of the ideas we explored that seem particularly influenced by the dynamics of this new, technological representation. 3.1 Infinitely big circles on the Torus and Klein bottle In the Torus microworld, one can construct a circle using the Torus Circle tool, which is defined by a center and radius point. Imagine placing your circle at the center of the fundamental region and steadily increasing the radius of the circle by dragging the radius point. What will happen once the circle gets as big as the fundamental region? Figure 7 shows a series of screenshots from this process of continuously increasing the radius of the circle. We see the reflectional symmetries that we might expect from the way the sides of the fundamental region are matched. The images produced using the Klein bottle (which we do not include here) look exactly the same due to the symmetry of the circle. However, by moving the location of the center, we were able to create quite different images. Figure 8a–d show congruent torus circles centered at different locations in the bottom right corner of the fundamental region, with each circle being the same size. Figure 9a–d show congruent Klein circles centered at the same respective locations as the torus circle. There are two different phenomena to notice in our sequence of diagrams. First, moving the centre along any horizontal line in the fundamental region will maintain the basic shape of the circle on both surfaces, as can be seen in the transition from 8b to 8c and 9b to 9c. In contrast, moving along any vertical line will change the basic shape of the circle on each surface, as can been seen in the transition
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Fig. 7 A series of screen shots showing a torus circle dilating on the fundamental region
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Fig. 8 (a–d) Varying the location of the centre of congruent Torus circles
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Fig. 9 (a–d) Varying the location of the centre of congruent Klein circles
from 8a to 8b and 9a to 9b. The second thing to notice is that any circle centered along the bottom, midpoint or top horizontal lines looks the same in both the torus and Klein (for example, in 8a and 9a as well as 8d and 9d), which makes sense since the horizontal sides are mapped to each other without a twist. In contrast, circles centered off those horizontal lines looked different, as can be seen in 8b and 9b as well as 8c and 9c. These different images are due to the twist introduced when mapping the vertical sides of the fundamental region. The twist breaks the fourfold symmetry of the torus, unless the circle is centered in the middle of the fundamental region. While the torus circle looks essentially the same as one drags its centre around, the Klein circle seems to fluctuate. This behaviour of the Klein circle is due to the fact that it gets represented on the fundamental region as reflections and translations of arcs whereas the torus circle consists only of regularly composed arc translations. So when the center of the Klein circle is in the middle of the fundamental region, the twist (reflection) that would occur to the top arc of the circle produces the same arc as the translated arc of the bottom of a torus circle. The particular images generated here would change, of course, if the sizes of the circles were increased or decreased, but their basic reflectional properties would remain the same. Creating and analyzing these designs involves using an extrinsic perspective, but we were also able to put a point in motion on each of the circle, and watch as it traversed the entire path in a constantly curving manner. Watching the point move is not the same as being actually on the path, but in watching, we found ourselves turning our torsos in unison with the point. 3.2 Introducing the New Torus Circle and the New Klein Circle After we had appreciated the designs produced in Figs. 7– 9, which use a Euclidean object (the circle) on the fundamental region, we began to wonder how those designs related to
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Fig. 10 (a–d) Enlarging the radius of a circle using the New Torus Circle
what we might actually see on a torus. This led us to consider more carefully the whole notion of a circle on the torus and, indeed, of using the circle in the context of a topological exploration—after all, a circle is defined in terms of distance (all points equidistant from a given point) and topology, with its tolerance for stretching and shrinking, doesn’t much care about measure. Thus, one way to interpret the diagrams in Fig. 7 is to imagine taking a (more or less circular) closed loop of string and applying it to the surface of a torus. One can make the loop as big as one wants, but eventually the loop will overlap previous pieces of string. It’s hard to call the resulting object a circle though. It certainly no longer looks like a circle, whether one sees it in the fundamental region or on the torus itself—though an ant walking along the piece of string might experience a sense of constant curvature. Another way to think about the circle in this microworld, however, is to return to the definition of constant distance to a fixed point. Since we were working with Sketchpad, we took our metric to be the usual Euclidean one (shortest distance on the plane). Figure 7a thus illustrates all the points equidistant from the center C. However, Fig. 7b shows points that are not all equidistant from C, since any point on each of the four arcs on each side of the fundamental region are closer to the centre than the points on the arcs that round each of the four corners. This observation prompted us to create a New Torus Circle tool that constructed only circles conforming to the usual Euclidean definition1. Figure 10a, b look just the same as they would with the regular Torus Circle, but as soon as the radius gets big enough (we keep C fixed and increase the radius) the New Torus Circle looks quite different, as in Fig. 10c, d. These new shapes are showing only the points equidistant from the given centre, taking into account that sometimes closer might involve ‘‘wrapping around’’ the torus to get from one point to another. As one keeps increasing the radius, the New Torus Circle disappears entirely. How big does that radius have to be for the circle to no longer exist? Well, as soon as ffiffiffiffiffiffiffi theffi radius is greater than the length of the half-diagonal (so, bigger than the ratio of p 2=2 to the side of the square), the circle disappears because every point on the fundamental region can be reached in less than that amount. The same process will also be true using the New Klein Circle, as shown in Fig. 11a–d. The pattern has a different symmetry than in the case of the Torus, but the circle also disappears as the radius increases. The actual patterns in both cases will change depending on the location of C in the fundamental region, but they will all have the same property that the length of the ‘‘circle’’ decreases after the radius reaches ½ the side of the square and will decompose into smaller arcs.
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This tool simply adds an additional constraint on the locus created for the Torus Circle, which is to say, the locus only exists when its’ closest path to the centre is equal to the given radius.
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Fig. 11 Enlarging the radius of a circle using the New Klein Circle
This investigation with the circle on the fundamental region, in which we preserve the Euclidean idea of a circle while drawing it on a topologically-inspired surface, suggests a new approach that we call topogeometry. This new approach arose in part because of the constraints of Sketchpad, which forces the use of geometric objects. However, the constraints gave rise to a unique set of questions that combines techniques and motivations from topology and geometry. 3.3 Path Connectedness The dynamic flat torus can be used to explore the notion of path-connectedness on the surface of the torus. Two points are path-connected on a topological surface if there exists a path joining them that does not cross a given boundary curve. On the sphere, if the boundary curve is a closed path, then a point inside the boundary and a point outside of it are not path-connected. Interestingly, any closed path on the sphere is non-homotopic because it can be continuously deformed, or shrunk, to a point. However, there exist closed paths on the torus that are not non-homotopic (cannot be shrunk to a point). It turns out that these non-homotopic closed paths, shown in Fig. 12, are of particular interest in the question of path connectedness. Surprisingly, these closed paths that are not non-homotopic do not create a meaningful inside and outside on the surface of the torus, as we shall see. First, though, what might these types of closed paths look like on the fundamental region of the flat torus? To create this special closed path, one could create a curve that starts and ends at the same point and travels around the whole circumference of the torus vertically or horizontally. Two such closed paths are shown in Fig. 13 (using the Torus Line tool). Looking at Fig. 13, one might well believe that these paths still seem to separate the torus into two disjoint regions (i.e. left and right or bottom and top). Additionally, at first glance, it seems that if A were placed in one of those seemingly disjoint halves and B in the
Fig. 12 Two closed paths that are not non-homotopic
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Fig. 13 Two closed paths that are not non-homotopic on the torus
Fig. 14 Points A and B appear to be in two disjoint regions of the torus
other half (see Fig. 14), then the path between A and B would certainly cross the boundary of the closed path. However, it is important to recall that this is not an ordinary rectangle but the fundamental region of the torus and so things behave differently. Multiple solutions can be explored with the dynamic option afforded to the user in Sketchpad. Here’s one (Fig. 15):
4 Concluding Remarks While these microworlds might well have been developed in other programming environments, the use of Sketchpad introduced interesting consequences. On the one hand, Sketchpad’s native language is geometry, and this made the creation and use of the geometrical tools for the microworlds both natural and meaningful. Of course, this also led to some departures from traditional topological explorations, since we were encouraged to use Euclidean objects on the fundamental region. The resulting topogeometry environment produced some fruitful investigations. In addition, the dynamic nature of Sketchpad allowed us to look for invariances as we dragged objects around on the screen, so that seeing, for example, the design created by a large Torus Circle as the center moved across the fundamental region, provided compelling insight and evidence. On the other, the constraint of working with geometric objects such as the circle and line prevented us sometimes from reasoning in a truly topological way. We were tempted at first by the
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Fig. 15 One path between A and B that will not cross the boundary of the closed path
designs produced by arcs and lines, slipping easily into the more extrinsic frame of mind. When we were able to set points in motion on given paths (using Sketchpad’s animation commands) the intrinsic viewpoint became much more palpable. Our explorations of the microworld were often inspired by possibly related topological problems and results, and we used the microworld to try to represent these. For example, we spent some time thinking about the closure of path. We were mesmerized by the candystripping path created by a line on the torus the criss-crossing paths of a line on the Klein bottle (see Figs. 5, 6a). We wondered why some of these paths seemed to close very quickly on themselves while others hinted to us that they may never close. Through this exploration we meandered through the land of winding numbers and found the answer to our question in the simple concept of rational versus irrational slopes. We also tried to implement the map colouring result on the torus which states that any map drawn on the torus requires at most seven colors. Although we were able to create and color these maps, the dynamics were harder to implement. We have described a new representation of several topological surfaces and have pointed to some of the topological ideas that these dynamic representations can help provide a more intrinsic view of the way in which geometric objects behave on these surfaces. Jeffrey Weeks has done much to make some very sophisticated ideas of topology accessible to a wide group of learners through his work on the Exploring the Shape of Space (2001) and the Jeffrey Weeks’ Geometry and Topology Website ( http://www.geometrygames.org/). This work aims to follow in the spirit of Weeks’ initiatives. In addition to constructing compelling microworlds for our own investigations with topological surfaces, we hope that our dynamic flat surfaces can both provide new approaches to topological ideas for those familiar with static representations, and provide accessible initial forays into topology for the newcomer2. Acknowledgments We would like to thank Ronald Fintushel and Robert R. Bruner for answering our many topological questions and Nicholas Jackiw for answering our many Sketchpad questions. We would also like to thank our anonymous reviewer for suggesting the term ‘topogeometry.’
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The Sketchpad file containing the three microworlds described in this paper can be found on-line at http://www.sfu.ca/*nathsinc/gsp/topology.zip
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References Jackiw , N. (1991, 2001). The Geometer’s SketchpadÒ. Emeryville, CA: Key Curriculum Press. Jackiw, N. (1997). Drawing worlds: Scripted exploration environments in the Geometer’s SketchpadÒ. In J. R. King & D. Schattschneider (Eds.), Geometry turned on!: Dynamic software in learning, teaching, and research (pp. 179–184). Washington, DC: The Mathematical Association of America. Meyerhoff, R. (1992). Geometric invariants for 3-manifolds. The Mathematical Intelligencer, 14(1), 37–53. Weeks, J. (2001). Exploring the shape of space. Emeryville, CA: Key Curriculum Press.
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