Navigating DaViD FraNcK DaViD

286 MatheMatics teacher | Vol. 105, No. 4 • November 2011

Copyright © 2011 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. between the Two classroom activities—the game and sliceforms—are useful vehicles for student exploration of the geometric interplay between the dimensions.

Julian F. Fleron and Volker ecke

Yet I exist in the hope that these memoirs, in some man- our three-dimensional world. Both activities are ner, I know not how, may find their way to the minds effective vehicles for students’ exploration of the of humanity in Some Dimensions, and may stir up a geometric interplay between the dimensions. race of rebels who shall refuse to be confined to limited As inhabitants of a three-dimensional world, Dimensionality.—A. Square, Flatland we authors feel that solid geometry could play a more prominent role in secondary school curricula enerations have been inspired by and that students’ experiences could move beyond Edwin A. Abbott’s profound tour of explorations of spheres, cones, cylinders, prisms, the dimensions in his novella Flat- pyramids, and the platonic solids. The activities land: A Romance of Many Dimensions introduced here allow students to explore our three- (1884). This well-known satire is the dimensional world in its full diversity, using any Gstory of a flat land inhabited by geometric shapes solid object as the focus of each activity. This libera- trying to navigate the subtleties of their geometric, tion, we believe, honors the spirit of A. Square of social, and political positions. (A new animated Flatland, who urged humans to work to have our version of this classic is now available as well; see minds “opened to higher views of things” (Abbott Johnson and Travis [2007].) In this article, we 1884, p. 4). In more concrete terms, the activities introduce the Flatland game and a related project, provide rich opportunities to support NCTM’s which inspire repeated trips between Flatland and Geometry Standards (NCTM 2000) (see fig. 1).

Vol. 105, No. 4 • November 2011 | MatheMatics teacher 287 C onnections to NCTM’s working; general three-dimensional modeling Geometry Standards such as Google SketchUp™ Among the many connections, our activities help • Building—Construction trades, landscapers, and students “use visualization, spatial reasoning, and regional planners; work from blueprints geometric modeling to solve problems,” address- • Three-dimensional scanning, three-dimensional ing the following grades 9–12 expectations: printing, and stereolithography • Holography, IMAX, and 3D movies • “Draw and construct representations of • Three-dimensional simulation and virtual reality two- and three-dimensional geometric objects CAVEs used in surgical training and flight training using a variety of tools” • Digital animation and computer gaming • “Visualize three-dimensional objects from different perspectives and analyze their cross These activities are related to a wealth of real-life sections” applications. We have used these activities in a vari- • “Use geometric models to gain insights into, ety of settings, including with high school and intro- and answer questions in, other areas of ductory college-level audiences as well as in teacher mathematics” professional development, with much success. Obvi- • “Use geometric ideas to solve problems in, and ously, there is a great deal here both to inspire and gain insights into, other disciplines and other to help prepare the next generation of students for areas of interest such as art and architecture” the mathematics they will need as architects, trades- people, engineers, artists, builders, graphic design- Fig. 1 The activities presented in this article have many ers, medical specialists, animators, and teachers. connections to NCTM’s Geometry Standards (NCTM 2000, p. 308). Flatland and Slicing The interplay between the dimensions can be Real-Life Applications investigated in many ways. Robbins (2006) argues Humans live in the three-dimensional world of strongly on behalf of projection, and Frantz and col- Spaceland, yet many of our means of investigation, leagues (2006) show how perspective drawing can visualization, and construction are two dimensional— be effectively explored in mathematics classrooms— images captured on paper and computer screens. both valuable approaches. Here we will focus on the Every day we spend a great deal of time trying to nav- slicing technique popularized in Flatland. igate between two and higher dimensions, playing the Teachers can introduce slicing and the mathe- roles of radiographers and builders described below. matics of cross sections in many ways, but the most To set the context for our activities and to help direct is simply to have students physically slice pique student interest, we briefly provide real-life real objects into cross sections. We suggest lining applications of the interplay between the dimen- tables with large sheets of paper or disposable table- sions. Some of these applications will be familiar to cloths for easier cleanup. Denser and drier objects you and perhaps to your students; others may not seem to work better; apples, zucchini, bagels, be. Space prohibits us from describing these appli- sponge cake, Play-Doh, cheese, and pineapple all cations in detail, but we have compiled informa- work well. Students can use either sharp knives or tion, short activities, and additional links related to thin wire to cut the cross sections (see fig. 2). each area online at http://www.westfield.ma.edu/ Once students have sliced the objects, their natu- ecke/flatland. We encourage you to use this mate- ral impulse is to arrange their cross sections linearly, rial in your lessons because it helps motivate stu- showing the solid object as a sequence of cross-sec- dents and enhance their understanding. tional slices. If the same object is sliced in a different Real-life applications that require the ability to direction, students will discover the dramatic impact navigate the dimensional cycle between two- and that this change in orientation has on the shape of the three-dimensional spaces successfully include these: cross sections. Encourage students to draw pictures of the sequences of cross sections in their notebooks. • Mapping—Topographic maps, topographical This activity provides a developmentally impor- profiles, and contour lines; a host of connections tant experience in understanding the interplay with modern geographic information including between the dimensions. However, as an analytical a GPS and Google Earth™ tool, physically slicing a few routine objects is only • Medical imaging—X-rays, CAT scans, and many a beginning. We need to expand this experience to other forms of medical applications encompass a more diverse collection of geometric • Design—Architecture and architectural design; objects and activities to help students to develop a blueprints; computer-aided design (CAD) in deeper, more sophisticated ability to navigate the product development, engineering, and wood- dimensional cycle shown in figure 3.

288 Mathematics Teacher | Vol. 105, No. 4 • November 2011 (a)

Fig. 3 the dimensional cycle relates two and three dimensions.

(a)

(b)

(b)

(c)

Fig. 4 Which solid objects produce these slices?

tions. Three examples are given in figure 4. Can you guess what three secret solids make these cross sections as they pass through Flatland? (Answers are provided at the end of the article.) (c) Determining the solid is the inverse problem

JULiaN F. FLerON JULiaN F. of determining the cross sections. If we combine Fig. 2 students can use knives to cut cross sections of a determining the solid with the direct problem of bagel (a), an apple (b), and a pear (c). slicing, we have a way to complete the dimensional cycle in figure 3. This is a developmentally impor- the Flatland gaMe tant activity that is easily made dynamic by turning Having toured the higher dimensions, our hero it into a game—the Flatland game. A. Square knows why Sphere appeared to him as In the Flatland game, a team of radiographers circles of continuously changing size: They were chooses a solid object and draws a sequence of the cross sections of Sphere. In the Flatland game, cross-sectional slices, one at a time. The radiog- we put ourselves in A. Square’s position. Our task raphers compete against teams of builders who is to determine the identity of a solid object passing attempt to determine the identity of the solid object through Flatland from a series of parallel cross sec- from the cross-sectional clues.

Vol. 105, No. 4 • November 2011 | MatheMatics teacher 289 Rules and Roles for the Flatland Game Suggestions for Playing the Flatland Game Instructions for the Flatland game follow: The following suggestions will help teachers imple- ment this game in the classroom: 1. Choose one or more radiographers. The radiog- rapher can be a single person or a small team • Teachers may introduce the game by providing the that has chosen an illustrator. cross sections, perhaps using the clues in figure 4, 2. Choose teams of builders. Teams can consist of and having the entire class work as builders. a single person, if necessary; it is best if a few • If students have drawn images of the real objects small teams compete. that they have physically sliced, they have 3. The game starts with the radiographers deter- already had practice in their roles as radiogra- mining among themselves a solid object from phers. Otherwise, have them practice before Spaceland; the identity of this solid will be the they play the game. focus of the game. • We are not really seeing the Flatland view of the 4. The illustrator for the radiographers begins play objects—this view is too hard to comprehend by drawing a single cross section, as viewed even for Flatlanders. The slices are drawn as a from above, of the secret solid. top view (as they would appear from above). 5. The builders attempt to guess the identity of the • The radiographers should be constrained some- secret solid. what in their choice of a solid; otherwise, their 6. The illustrator for the radiographers then draws objects may be too complicated. For example, a another cross section of the secret solid. This swarm of mosquitoes passing through Flatland is cross section must be parallel to earlier ones and not a good place to start. must be revealed consecutively as they would be • The importance of the illustrating component if the object actually passed through Flatland. should not be underestimated. Students’ ability 7. Steps 5 and 6 are repeated until— to render the slices often speaks directly to their (a) a team of builders correctly guesses the iden- abilities in geometric visualization. Encourage tity of the secret solid, in which case this students to make faithful slices but also recog- team is declared the winner, or nize their varying artistic abilities. (b) there are no more cross sections to draw and • As the radiographers work in a group, they the radiographers are declared the winner. should be encouraged to talk among themselves and, discreetly, to use actual three-dimensional This game encourages multiple circuits around objects to help them visualize the slices. the dimensional cycle. Each time, students draw, • Because the exact nature of these developmen- visualize, construct, slice, analyze, and recon- tal activities is different for the two different struct—in short, they develop critical tools for navi- roles, encourage students to exchange roles in gating between the dimensions. No longer are their subsequent games (the builders would become investigations of solid objects limited; chairs, coffee radiographers). cups, and people are the fodder for their geometric investigation. The interplay between cross sections Sliceforms: Physical three- and real-life solids provides a powerful opportunity Dimensional Reconstruction for reflection on the nature of the three-dimen- With success in the roles of radiographers and sional world we inhabit. builders in the Flatland game, students develop the ability to navigate the dimensional cycle shown in figure 3, an important developmental step. None- theless, students’ success is largely qualitative. Neither the drawings nor the reconstructed solid is exact. In the game, students seek only approxi- mately correct cross sections. Having done this, they are prepared to navigate the dimensional cycle more exactly with sliceforms—real, scaled, three- dimensional models of everyday objects. Several sliceform models are shown in figures 5 and 6 (the latter are original models designed and built by the authors’ students). Sliceforms are (a) (b) constructed from sequences of perpendicular cross sections that have been notched with slots and then Fig. 5 Sliceforms may include a paraboloid of revolution (a) and a Cassinian solid of joined together in a regular array. Operating much revolution (b). like the inserts that protect beer bottles in a case

290 Mathematics Teacher | Vol. 105, No. 4 • November 2011 from breaking, these card stock models are hinged In contrast with assembling a sliceform from a so that they can collapse to be completely flat. These template, designing and creating one’s own slice- magical collapsing objects were invented by the Dan- form of an everyday, solid object is a more signifi- ish mathematician Olaus Henrici in the late nine- cant undertaking. This activity will likely take the teenth century and have gained attention through equivalent of several periods of class time, although the popularizing work of John Sharp (1995, 2004). much can be done at home. Students are captivated Sliceforms enable students to take the Flatland by this activity, and their increase in understanding game one step further. Now they reconstruct three- of the dimensional cycle can be seen as they work. dimensional solids from their cross sections. Our goal is to design and then construct sliceform Creating Sliceforms models of real-life solids. Following are a number of guidelines for students As a first step, students should create a few as they create sliceforms: sliceforms from stock models to help them under- stand the design and construction concepts. We • Importance of design. Students should be encour- include templates for one stock model, a lightbulb aged to spend a significant amount of time at the (see figs. 7a and 7b). Many others can be found in outset on design. Poor design often cannot be Sliceforms: Mathematical Models from Paper Sections overcome in the construction phase. (Sharp 1995). • Experimental design. Have students experiment. Sliceforms are constructed from medium-weight They will need to take many laps around the card stock on which the templates have been copied dimensional cycle as they design. Encourage or printed. Have students cut around the outline of students to use Play-Doh to shape their solid and each slice. The slices labeled with an x are cross sec- then to slice mock sliceform cross sections. Bring tions of the solid along an x-axis. Analogously, the in profile or contour gauges for them to use. In slices labeled with a y are cross sections of the solid addition, have students use the free, three-dimen- along a perpendicular y-axis. After students cut out sional design tool Google SketchUp to build mock each cross section, the next step is to carefully cut versions. (The authors’ website, www.westfield slots along the indicated lines. These slots are not slits. .ma.edu/ecke/flatland, includes instructions and Students must remove a thin section of card stock to a .skp file that will help students convert Google make a slot; otherwise, the sliceform will not collapse SketchUp models into sliceform templates.) properly. Sharp scissors are essential for this step. • Slice regularity. Slices should be made at regular After students complete several slices, they can begin intervals and come together at right angles. Graph to assemble the model. Slices marked x and y meet paper of an appropriate size can be useful in help- at right angles with their respective slots to form a ing students visualize an aerial view of their slice- square lattice when viewed from above. The slices are form as well as the layout of their slices. numbered sequentially so that their order is clear. • Problem solving. The appropriate shapes and It should take students about twenty minutes to sizes of individual slices often lead to geometric construct the stock model we have provided. When problems that are perfect for consideration in students are finished, they will see what a mechanical mathematics classes. For example, the widths of marvel these models are. They should also begin to successive vertical cross sections of a cylinder have a sense of their many connections to geometry. can be computed using the Pythagorean theorem.

(a) (b) (c)

Fig. 6 Some of the best original student sliceforms included “The Face” (a), “Cactus” (b), and “Guitar” (c).

Vol. 105, No. 4 • November 2011 | Mathematics Teacher 291

(a) (b)

Fig. 7 These horizontal (a) and vertical (b) cross sections of a lightbulb can be used for the sliceforms activity.

• Guess and check. In the design of some objects—for References example, a coffee cup—the slices can be determined Abbott, Edwin A. Flatland. 1884. London: Seeley and fairly rigorously in both directions. In the design Co. www.ibiblio.org/eldritch/eaa/FL.HTM. of other objects—for example, a rainbow trout— Fleron, Julian F., and Volker Ecke. Companion site the slices in one direction can be made in a fairly to “Navigating between the Dimensions.” http:// obvious way, but it then becomes quite difficult to www.westfield.ma.edu/ecke/flatland. determine how to make the perpendicular slices Frantz, Marc, Annalisa Crannell, Dan Maki, and Ted correspond precisely. One strategy is to make the Hodgson. 2006. “Hands-On Perspective.” Math- perpendicular slices too large on purpose and then ematics Teacher 99 (8): 554–59. cut off the excess once they have been integrated. Johnson, Dano, and Jeffrey Travis. 2007. Flatland: • Depth of slots. Students often have trouble deter- The Movie. Austin, TX: Flat World Productions. mining the depth of their slots. Because the two National Council of Teachers of Mathematics integrated pieces have the same height along (NCTM). 2000. Principles and Standards for School their intersection, a good rule of thumb is to Mathematics. Reston, VA: NCTM. have each of their slots take up half this height Sharp, John. 1995. Sliceforms: Mathematical Models (see, e.g., figs. 7a and 7b). from Paper Sections. St. Albans, UK: Tarquin. ———. 2004. Surfaces: Explorations with Sliceforms. St. Cuoncl sion Albans, UK: Tarquin. The activities introduced here offer dynamic, develop- Robbins, Tony. 2006. Shadows of Reality: The Fourth mentally important ways to meet a number of crucial in Relativity, Cubism, and Modern goals set forth in NCTM’s Geometry Standards. In Thought. New Haven, CT: Yale University Press. addition, the depth of the connection to contemporary applications of obvious importance does a great deal to ANSWERS TO THE FLATLAND GAME motivate and inspire students. They see that special- Answers are from clues given in figure 4: (a) cube ists in areas such as medicine, engineering, art, digital or right rectangular prism; (b) lightbulb; (c) cone. technology, and architecture continue to make pro- found contributions based on the interplay between two-dimensional and three-dimensional worlds. By developing abilities to navigate the world of three-dimensional objects in their full diversity, stu- JULIAN F. FLERON, jfleron@westfield dents may be able to contribute to new advances. .ma.edu, and VOLKER ECKE, vecke@ Beyond this, perhaps the explorations here will westfield.ma.edu, are colleagues at inspire students to break free of the prejudices of Westfield State University in our human dimension and help us learn to visual- Massachusetts. This article is a ize exotic, higher-dimensional objects. sample of their work on discovering the art of mathematics (see http:// A cknowledgments artofmathematics.westfield.ma.edu/), Work on this article was supported by National Sci- a project focused on developing a large collection ence Foundation grant no. 0836943 and a generous of inquiry-based materials in mathematics. gift of Harry Lucas.

292 Mathematics Teacher | Vol. 105, No. 4 • November 2011