The Dry-erase Cube: Making Three-dimensional Visualization Easy

Transcription

1 The Dry-erase Cube: Making Three-dimensional Visualization Easy Yvette D. Kuiper Department of Geology and Geophysics, Boston College, Chestnut Hill, Massachusetts, 02467, ABSTRACT The 10 cm dry-erase cube is an effective tool for illustrating the three-dimensionality of geological features in maps, cross sections and block diagrams, and it aids in the understanding of stereographic projections and orientated samples. It is ideal for use in classes and labs of structural geology and geological field methods courses. The cube is useful for demonstrating the three-dimensional relationships among any planar or linear features in geoscience courses at any level. INTRODUCTION Many students are challenged by the three-dimensional problems they have to solve in courses such as structural geology and field mapping. Commonly, they have not had much training in three-dimensional thinking, and especially for students who do not have a natural aptitude for such thinking, structural geology, mapping and construction of cross sections and block diagrams can be a real struggle. There are many tools available to instructors from science education supply stores that are designed to assist them in teaching three-dimensional relationships, including paper models, wooden block diagrams, and plastic replicas. However, these tools are generally static and typically designed to illustrate a specific geological relationship. More dynamic computer modeling programs have been developed that are designed to aid in three-dimensional visualization of geological relationships in research and teaching (Jessell, 1981, 2001; Reynolds et al., 2001). Block diagrams can be rotated, and they can be made transparent or parts can be cut out of the blocks so that the internal structure of the blocks is visible. Three-dimensional visualization abilities and associated mental processes in humans have been studied by various authors (e.g. Shepard and Metzler, 1971; Shepard, 1978; Kali and Orion, 1996; Ishakawa and Kastens, 2005) and Piburn et al. (2005) demonstrated that students' three-dimensional visualization skills can be improved by using these computer modeling programs. Whereas the computer models have proven to be helpful, the diagrams are still two-dimensional projections of three-dimensional models and important concepts such as apparent dips of geological layers can still be difficult to understand. A simple cube constructed from white high-density plastic provides a solution to many of the limitations of the various teaching aids mentioned above. The dry-erase cube is a dynamic, three-dimensional, instructional tool that is capable of being adapted to demonstrations of multiple geological relationships. Dry-erase cubes can quickly be reconfigured to illustrate a limitless number of complex three-dimensional geological relationships. The cubes serve as a three-dimensional model for two-dimensional maps, cross sections and block diagrams and the relationships between them. They can be used in classroom and field settings. From my observations in class so far, students seem to solve problems involving block diagrams, maps and cross sections at greater ease when they are using the dry-erase cubes than they did without them. Perhaps of equal importance, they just love working with them! THE CONCEPT The dry-erase cube is simply a 10 cm cube made of a white opaque high-density plastic (Figure 1). Suitable materials include acrylic, acetal and PVC plastics that can be purchased at material supply stores. Plastic sheets can be cut, machined and glued or screwed together to form 10 cm cubes. The cubes shown in this paper were made of acetal plastic by the Boston College Scientific Instrumentation and Machining Services. We use wet- or dry-erase markers, which can easily be removed using paper towel and alcohol, windex, or other cleaning agents. Ideally the cubes should not have any seams or screws. However, material, facilities and price constraints may require them. For example, acetal plastic is durable but cannot be glued well. Acrylic plastic can be glued, but is more brittle and therefore easier to break than acetal plastic. As long as the cubes are carefully cleaned, screws and seams are not too problematic. Similar cubes could be fabricated from non-erasable materials, such as plastic or wood, or even cardboard or paper so that they can be made in class. Other cubes are commercially available such as plastic boxes, desk display cubes (with paper inserts) and solid clear acrylic cubes. Solid clear acrylic cubes would be suitable but are expensive. The cheaper cubes commonly have seams, holes, lids, or only five sides, for example if they were intended as boxes. Furthermore these cubes usually consist of plastics that are not as well erasable. Erasable material is most efficient, because it allows students to draw and erase their work repeatedly. This way, students can experiment with various solutions to a problem and various exercises can be given using the same cubes. The ideal cubes with six sides, made of a white opaque high-density plastic as explained above need to be custom-made. I am currently attempting to have dry-erase cubes manufactured and made available to the public. When they do become available I will notify the geoscience teaching community through appropriate list servers and websites. Whereas dry-erase cubes can be made in any size, 10 cm has proven to be most convenient for use with isometric block diagrams, maps and cross sections on paper. In an isometric block diagram, distances can be measured along, and parallel to, all edges of the block, the same way as they are measured on a map, correcting for scale. A 10 cm isometric block diagram is approximately the largest size diagram that can be printed on standard letter or A4 paper. Furthermore, cm maps and cross sections can also easily fit on a page. Comparisons between maps and isometric block diagrams, using a dry-erase cube, are easiest when they are the same size. Ideally, the block diagram has tick marks along all edges indicating a mm or cm scale, to facilitate the drawing of planar and linear features in the correct orientations. Kuiper - Making Three-dimensional Visualization Easy 261

2 Figure 1. Photograph of a 10 cm dry-erase cube. THE DRY-ERASE CUBE AS A MODEL FOR MAPS AND CROSS SECTIONS Students often encounter difficulties visualizing the representation of geological features on maps and those on cross sections, and the relationship between the two. One concept that is especially confusing to students is that of apparent dips. Apparent dips are essential in the construction of cross sections and block diagrams. An apparent dip (γ) can be calculated from the true dip (α) of a planar feature, and the angle between the dip direction of that planar feature and the trend of the cross section (β): tan(γ) = tan(α) cos(β) (derived below; cf. Marshak and Mitra, 1988; Rowland et al., 2007). An apparent dip can also be found by plotting the planar feature of interest and the cross section on a stereographic projection, and by taking the plunge of the line of intersection. Even students who have a good understanding of these methods may have some difficulty plotting the apparent dip on a cross section or block diagram. The dry-erase cube can make this much easier for them to visualize, because the map can be sketched on an actual horizontal plane and the cross section or vertical faces of the block diagram can be sketched on actual vertical planes. Therefore the entire plane can be made visible. The students' understanding can be further enhanced by the combination of several dry-erase cubes (Figure 2). This allows for e.g. (1) construction of several parallel cross sections, (2) construction of perpendicular cross sections, (3) extension of cross sections above the given map level by piling up blocks on top of the map, and (4) construction of several parallel maps at various elevations. The dry-erase cubes are easily incorporated into a wide variety of individual and group exercises. The advantage of having the students do the exercise in small groups is that it creates discussion amongst the students. Students can be provided with dry-erase cubes with 262 Figure 2. A) Map of folded surface drawn on top of six dry-erase cubes. B) Same as A with two additional cubes, and map drawn on top. East-west and north-south trending cross sections of the folded surface are also drawn on the cubes. C) Same as B, but with two cubes removed and folded surface indicated on newly exposed vertical sides. Journal of Geoscience Education, v. 56, n. 3, May, 2008, p

3 Figure 3. Projected planes in isometric block diagrams. Italicized numbers are actual dips and normal numbers are projected angles. A) Plane dipping 30 to the west, B) plane dipping 30 to the north. C) and D) are the same as A and B, respectively, drawn on the dry-erase cubes. maps already drawn on top (Figure 2), or they can be given blank cubes and asked to transfer data from an existing map onto the cubes. The map in Figure 2A shows an upright 40 east-plunging cylindrical fold. In this example, students could be asked to complete a number of tasks. First, they can be asked to construct cross sections on the vertical faces of the cubes. Several cross sections can be constructed parallel and/or Kuiper - Making Three-dimensional Visualization Easy perpendicular to each other by moving the cubes in Figure 2A apart. They need to determine the apparent dips using trigonometry or stereographic projection. They also need to take into account that the surface is curved around the hinge area, so they need to curve the surface on the cross sections according to the given map. The exercise can be made easier by giving a more angular folded surface such as a chevron fold, so that the limbs 263

4 Figure 4. A) Isometric block diagram showing projections of 45 and 10 intervals. Italicized numbers are actual angles and normal numbers are projected angles. The projected angles increase with decreasing difference in plunge of the two projected lines (relative to horizontal, or projection on paper). B) Photograph with dry-erase cube with the lines drawn as in A. are planar and the hinge area is very small. Subsequently, they can draw the folded surface on any faces of the cubes on which it would be visible (including the bottoms!), so that a three-dimensional visualization is created. Additional cubes can be added on top of the arrangement given in Figure 2A, so that the cross sections can be extended above the map level, and a new map can be drawn on top of the added cubes (Figures 2B and C). I give the students six dry-erase cubes on top of the ones shown in Figure 2A. The four cubes in the front (omitted in Figure 2B) do not show the folded surface on their top faces, but they do show it on some of their vertical faces. A great variety of maps of different levels can be given, including those showing inclined parallel planar geological contacts, angular unconformities, non-plunging or plunging folds with vertical, horizontal or inclined axial planes, or even refolded folds. When relatively simple maps are given, such as the example shown in Figure 2, the students can be asked to accurately construct the folded surface on various faces of the block diagrams by accurately determining apparent dips. If more complex geometries are given, such as refolded folds, it may be more suitable to ask them to sketch the surfaces of interest on the remaining faces of the cubes. THE DRY-ERASE CUBE AS A MODEL FOR ISOMETRIC BLOCK DIAGRAMS An isometric block diagram (all block diagrams in this paper) is a projection of a block, such that the angle between each edge of the block and the projected surface is equal. Students benefit from learning how to show geological features on (isometric) block diagrams in preparation for potential careers in the petroleum or mining industries, geotechnical consulting, or academia. However, perhaps more importantly, working with these diagrams greatly enhances the students' three-dimensional thinking skills. In an isometric block diagram, all angles and lengths of lines are distorted, except the lengths of lines measured parallel to the edges of the block (correcting for scale). On the dry-erase cube nothing is distorted, which makes it an excellent instructional aid for the three-dimensional visualization of two-dimensional projections. The students can draw planes on the surfaces of the cube, and then visualize how these planes will project through the cube, e.g. by holding a piece of paper in the right orientation close to the cube. To visualize what the drawings on the cube would look like on an isometric block diagram, the students can look at the cube in the direction of projection and see what it would look like on paper. The effectiveness of this exercise can be further reinforced by simply taking a photograph of the completed cube. This adds an additional challenge in that the students have to determine the vantage point from which the photograph must be taken to get the best correspondence between the photo and the isometric diagram. They will discover that the photograph is always different from the isometric block diagram, because the photograph shows a perspective projection, in which parallel lines converge away from the viewer, as illustrated by the wooden planks behind the cubes in Figures 3C and D. Some distortion in addition to the perspective projection is caused by camera lens effects, but these distortions are sufficiently small that the exercise can still be performed. The isometric block diagram is an orthogonal projection, in which parallel lines remain parallel (compare Figures 3A and B with Figures 3C and D). The best approximation to orthographic projection is achieved when the photograph is taken at a great distance from the 264 Journal of Geoscience Education, v. 56, n. 3, May, 2008, p

5 Figure 5. Example of a lab exercise. A) Geological map and legend. B) Isometric block diagram; on the students version, the vertical faces of the block diagram are blank. C) Illustrations to solutions to the problems. See text for discussion. D) Photograph of the dry-erase cube with the map on top and cross sections on the sides. cube, so that convergence of parallel lines away from the viewer is reduced to a minimum. Furthermore the photo should be taken so that the angles between the viewing direction and all edges of the cube are equal, as is true for an isometric block diagram. Kuiper - Making Three-dimensional Visualization Easy Example 1: Plotting true dips on an isometric block diagram - Whereas the plotting of true dips on cross sections is straightforward, because angles can be measured on the cross section, plotting true dips on the vertical faces of an isometric block diagram may be complicated, because of the distortion of angles. The

6 Figure 6. Example of lab exercise to show that the front face of this diagram is distorted, contrary to the students intuition. See text for discussion. angles between edges of a three-dimensional block appear as 60 or 120 on the two-dimensional isometric block diagram. A common mistake is that angles within the faces of the block diagram are plotted proportionally with the distortion of the 90 angles. For example, in Figure 3A, the angle of a plane dipping 30 to the west with the horizontal surface would be /90 = 20 on paper, where 60 is the angle between the two edges of the cube in the projection and 90 is the actual angle between the two edges of the cube. However, the correct projected angle is Similarly, in Figure 3B, the angle on paper of a plane dipping 30 to the north would be /90 = 40, but the correct projected angle is The correct angle can be found by trigonometry. Because distances can be measured along the edges of the block diagram, the length of Y in Figures 3A and B can be found by Y=X tan(30 ). The projected dips of the planes can then be measured off the paper. To convince the students that their projected dips represent a 30 dip, they can do the same trigonometry exercise on the dry-erase cubes and measure the dip off the cube (Figures 3C and D). It should be 30. Figure 4A can be used in combination with a dry-erase cube (Figure 4B) to make the projection problem more clear to the students. On the vertical faces of the diagram the angles between the lines are 10. By looking onto the corner of the dry-erase cube, the angles actually appear different in reality, the same way as they show on the isometric block diagram (compare Figures 4A and B). Example 2: Plotting apparent dips on an isometric block diagram - Plotting apparent dips on isometric block diagrams builds upon skills developed in the previous exercise. First, the students need to determine the apparent dip of a plane on a face of the block diagram (as explained above for cross sections) before they can calculate the projected angle on the isometric block diagram (as explained above). An example is given in Figure 5. A map with a legend (Figure 5A) and this same map on top of an isometric block diagram (Figure 5B) are given. Questions can be posed such as: (a) What is the nature of the surface between the Ordovician and the Permian rocks? (b) Determine the apparent dip of the geological contacts in faces (A) and (B), given that the Permian beds dip 28º to 210º and the Ordovician beds dip 72º to 315º. (c) Complete the block diagram and briefly describe the geological history of the area. The nature of the surface between the Ordovician and the Permian rocks is an angular unconformity. The apparent dips of the beds on the two vertical faces of the block diagram can be determined by using stereographic projection, or by using the formula tan(γ) = tan(α) cos(β) (symbols explained above). However, to avoid 'cook-book' style application of these methods without thoroughly understanding the methods, I prefer having the students develop their own method by using trigonometry. This approach refreshes their trigonometry skills, but most importantly forces them to think three-dimensionally. In Figure 5C, the triangle ABC is the same as in Figure 5A. BD can be measured off the map (Figure 5A) and BE can be calculated: BE = BD tan(28º) = 3.55 cm tan(28º) = 1.89 cm. The angular unconformity will therefore show on the front edge on the block diagram at 1.89 cm below the front corner of the diagram. The plane of the unconformity can then be drawn without calculating the apparent dip. The other contacts between the Permian rocks can be drawn parallel to the unconformity. The apparent dips of the Ordovician beds can be determined the same way. A more general way to calculate apparent dips is essentially a derivation of the formula tan(γ) = tan(α) cos(β) (e.g. Marshak and Mitra, 1988; Rowland et al., 2007), which is a useful exercise to do in a lab. Nothing is measured off the map. Combination of BE = BD tan(28º) and BE = BF = AB tan(γ) yields tan(γ)/tan(28º) = BD/AB. Also, cos(60º) = BD/AB and therefore tan(γ)/tan(28º) = cos(60º) or tan(γ) = tan(28º) cos(60º) = 0.266, yielding γ = 14.9 º, which is the apparent dip on face B of the isometric block diagram. The other apparent dips can be found the same way. The apparent dip cannot be measured on the block diagram. Therefore distances along the vertical edges of the block diagram still need to be calculated, e.g., BF = AB tan(14.9º) = 7.1 cm tan(14.9º) = 1.89 cm. The dry-erase cube (Figure 5D) is illustrative in this case, because the apparent dips can be measured and drawn directly on the cube. The map can be cut out and be placed on top of the cube (Figure 5D). Using the cube makes it easier to visualize what the contacts look like in three dimensions. On the cube it is also possible to draw the contacts on the back and the bottom of the cube, so the contacts can be followed all around it, further enhancing the three-dimensional visualization. Example 3: Distorted angles and distances on a 'front' section of the isometric block diagram - As another exercise, I use the isometric block diagram in Figure 6, given with a geological map on top and blank vertical faces. I ask the students to finish the block diagram. By this time they know how to construct faces Journal of Geoscience Education, v. 56, n. 3, May, 2008, p

7 Figure 7. Dry-erase cubes used as an analogue for orientated rock samples. A) An S-C fabric that is especially clearly shown on the right side indicates that the lineation on top is an intersection lineation. Foliation dips 85 degrees towards 165. The shear sense is sinistral. B) Dry-erase cube placed in its correct orientation according to the measurement on the foliated surface (dipping 70 degrees towards 350 ). Marker points north. The lineation is a stretching or mineral lineation and the shear sense is dextral, north-side-up. and 3. However, completion of face 2 is a surprise. It looks like the face is parallel to the plane of its projection, but in reality, the front face is dipping away from the viewer in the same way as the vertical edges of the block diagram are plunging away from the viewer. Also, the horizontal and vertical scales on this face are different due to the projection. The vertical edges of the front face are parallel to the edges of the block diagram and therefore distances can be measured parallel to these edges. The horizontal edges actually lie within the plane of projection, but because the scale is adjusted such that distances can be measured along the edges of the block diagram, distances measured parallel to these horizontal edges are longer on paper than they are in reality (compare the real and projected diagonals of the map and the isometric block). The width of the front face is 8.7 cm on paper. However, in reality it is 7.1 cm. The students can convince themselves of this by measuring the width on paper and on the dry-erase cube. The horizontal 'stretch' of this face on paper also explains why the apparent dip of 35.3º actually measures 30.0º on paper. This exercise emphasizes once more that on an isometric block diagram, distances can only be measured parallel to the edges of the diagram and all angles are distorted (the exception is when the corner of the cube is cut such that the new face is parallel to the plane of projection; in this case angles are not distorted, but distances are). THE DRY-ERASE CUBE AS AN AID IN UNDERSTANDING STEREOGRAPHIC PROJECTIONS A common goal in structural geology labs is to train students to relate planar and linear structures in various formats, including: (1) symbol on a map, (2) dip direction and dip (3) strike and dip, (4) planes or lines on an isometric block diagram and (5) planes and lines in reality (three dimensions) and (6) planes and lines in stereographic projection. To aid the students in their attempts to understand stereographic projections, I have experimented with numerous three-dimensional models for stereographic projections varying from sophisticated plastic models with rotatable planes and lines to simple soup bowls with home-made card-board planes and lines drawn on them. Still, the students have difficulty relating the planar and linear features drawn on stereographic projections, to those in reality. The dry-erase cube with the plane or line of interest drawn on it can be viewed side by side with the stereographic projection and any of their three-dimensional models. Seeing the plane in three dimensions helps the students relate stereographic projections to the measurements, symbols and diagrams they have on paper. THE DRY-ERASE CUBE AS AN ANALOGUE FOR ORIENTATED SAMPLES The dry-erase cubes can also be used as an analogue for orientated samples, and to illustrate the relationship between lineations on the foliation plane and shear sense indicators on sections perpendicular to the lineation (Figure 7). The students can be asked to place the 'sample' in a sandbox in the correct position, using an orientation already drawn on the cube (Figure 7B). Alternatively, a dry-erase cube can be placed in a certain position, so that a student can determine its orientation. For the latter example, a second student can try to place that same 'sample' in the correct position and test if the first student's orientation was correct. Students can also be asked to indicate on the cubes where they would cut a thin section if they were interested in shear sense indicators. In Figure 7A, the lineation is an intersection between an S-foliation (oblique on the right vertical face) and a C-foliation (top face), and therefore perpendicular to the shear direction. A thin section would need to be perpendicular to both Kuiper - Making Three-dimensional Visualization Easy 267

8 Figure 8. Configuration of dry-erase cubes with extra cut, allowing for diagonal cross sections. Grey faces indicate vertical faces. foliation and lineation. In Figure 7B, the lineation is a stretching or mineral lineation, which is parallel to the shear direction. The best orientation for a thin section would thus be perpendicular to the foliation and parallel to the lineation. With the help of a piece of paper, which would represent a rock chip, students could practice what to write on the rock chip or thin section to be able to orient it relative to the rock sample. Once the students are able to identify the shear sense indicators, the type of lineation and the orientation of the sample, they should be able to find the shear sense. In Figure 7A the lineation is down dip and the shear sense is sinistral. In Figure 7B, the lineation is oblique and the shear sense is dextral, north-side-up. CONCLUDING REMARKS AND ALTERNATIVE IDEAS The dry-erase cube is an excellent tool for students in the three-dimensional visualization of geological features on maps, cross sections, block diagrams and stereographic projections. The cube is simple and small, so it can easily be employed in the classroom and in the field. Modifications of the isometric block diagram can also be made. For example, an extra cut can be made so that the cube consists of two pieces: a model as shown in Figure 6 and the missing triangular piece. This helps in the type of exercise explained in example 3, but additionally several two-piece cubes can be arranged so that a diagonal cross section can be drawn (Figure 8). This extra cross section orientation can also be achieved by cutting the cubes along their diagonals, so that one cube consists of two triangular pieces. More importantly, the concept of true and apparent dips can be shown more clearly on a triangular piece. For example, if BD in Figure 5C would be at 45 with AB, then the true dip can be shown on the diagonal cut of the block diagram, and the apparent dip on the side (i.e. A, B, D, E and F could all be indicated on one triangular piece). Triangular pieces can be assembled into cubes by embedding magnets within the triangular pieces, or simply by using tape. Many more variations can be designed. The possibilities are endless. For example, it is possible to construct a transparent plastic cube and project planes through it with a planar light beam, such as a laser level, so that the plane is visible within the cube as well as on its faces. Alternatively, the cube may be made of magnetic material with separate faces of dry-erase material that can be attached to the cube. This way the faces can be taken of the cube and placed along a cross section or block diagram on paper. These ideas may be the subject of future experiments and publications. ACKNOWLEDGEMENTS Constructive comments by Kurt Burmeister (University of the Pacific), Paul Williams (University of New Brunswick), journal reviewers Stephen Harlan and Sven Morgan, and associate editor Kristen St. John, helped me improve the manuscript significantly. Figures 5 and 6 are based on labs taught by Paul Williams, while I was his teaching assistant. Alan Vachon of Boston College Scientific Instrumentation and Machining Services produced the dry-erase cubes shown in this paper. Students in several of my classes at Boston College are thanked for enthusiastically using my dry-erase cubes in class and for demonstrating that these cubes help them in developing their three-dimensional thinking skills. REFERENCES Ishikawa, T., and Kastens, K.A., 2005, Why Some Students Have Trouble with Maps and Other Spatial Representations, Journal of Geoscience Education, v. 53, p Jessell, M.W., 1981, Noddy - an interactive map creation package. MSc Thesis, University of London, England, program available at encom.com.au/template2.asp?pageid=20 (4 April, 2008) Jessell, M.W., 2001, Three-dimensional geological modelling of potential-field data, Computers & Geosciences, v. 27, p Kali, Y., and Orion, N., 1996, Spatial abilities of high-school students in the perception of geologic structures, Journal of Research in Science Teaching, v. 33, p Marshak, S., and Mitra, G., 1988, Basic methods of structural geology, Prentice Hall, 446p. Piburn, M., Reynolds, S., McAuliffe, C., Leedy, D., Birk, J., and Johnson, J., 2005, The role of visualization in learning from computer-based images, International Journal of Science Education, v. 27, p Reynolds, S.J., Leedy, D.E., and Johnson, J.K., 2001, GeoBlocks 3D - Interactive 3D Geologic Blocks, (4 April, 2008) Rowland, S.M., Duebendorfer, E.M., Schiefelbein, I., 2007, Structural Analysis and Synthesis: A Laboratory Course in Structural Geology, Wiley-Blackwell, 320p. Shepard, R.N., 1978, The mental image, American Psychologist, v. 33, p Shepard, R.N., and Metzler, J., 1971, Mental rotation of three-dimensional objects, Science, v. 171, p Journal of Geoscience Education, v. 56, n. 3, May, 2008, p