Folding, Cutting and Joining DERIVE 5 Style

Peter Schofield Trinity and All Saints College (University of Leeds), UK

email: p_schofield@tasc.ac.uk Introduction

A basic requirement of specialist mathematics modules for College primary education stu-dents (following a BA honours degree) is the relevance of the degree course topics to Key Stage 1 and Key Stage 2 (approx. 5-11 yrs) of the UK National Curriculum for Mathematics. This was stated in Annex A (2f) of the DfEE circular 4/98 as follows:

for any specialist subject(s), have a secure knowledge of the subject to at least a stan-dard approximating to GCE Advanced level in those aspects of the subject taught at KS1 and KS2;

This presentation describes a number of Key Stage 1 and Key Stage 2 activities relating to concepts of transformation geometry. It then describes how, with the help of DERIVE, these can be translated into challenging 2D- and 3D-coordinate and transformation geometry topics for mathematics primary education degree students. The links between classroom and degree course activities are obvious. The main resource used is a DERIVE Users File 2D-&3D-Transformations.dfw. This file is included in the users folder of DERIVE 5 which can be downloaded from the DERIVE Web Site. 1. Paper Folding and Cutting Activities

2D-reflections and rotations can be investigated at Key Stages 1 and 2 by paper folding and cutting to make attractive patterns. There follows a number of examples of such activities to-gether with their corresponding DERIVE simulations. Example 1.1 A Pattern with Dihedral Group D4 Symmetry

Take a square piece of paper and hold it so that its diagonals are vertical and horizontal. Fold the paper in half from left-to-right along the vertical diagonal, then in half again (from bottom to top) along the horizontal diagonal to form a smaller isosceles right-angled triangle. The third fold is made along the median of this triangle (diagonally from top to bottom). You should finish with a small triangle of folded paper. See figure 1. Now (for example) cut out four triangular shapes (shaded areas) from the folded piece of paper as in the figure.

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When you unfold the paper you should have a cutout pattern similar to figure 2. This pattern has rotational symmetry of order four and four axes of reflection, spaced at angle intervals of π/4 about its centre. It is an example of a pattern with Dihedral Group D4 symmetry. Simulating Example 1.1 using 2D-&3D-Transformations.dfw

Open 2D-&3D-Transformations.dfw, select the 2D-plot Window and Tile Vertically. Select the 2D-plot settings: Options> Approximate before Plotting ON; Op-tions>Display>Points>Connect (YES), Small.

The outline and cuts in figure 3 can be set up using instructions:

The outline is simply a connected network of points forming an isosceles triangle. To work out the Boolean expression for each triangular cut, students first have to calculate the equa-tions of the boundary lines and then express the criteria for an interior point. For example, the boundary lines for cut1 have equations: . It is clear that an interior point lies to the right of the first line, below the second line and above the third line. Whence the Boolean expression for cut1. The outline and cuts can be been collected together in a list and stored under a single variable name as follows:

2; 4 ; 2 / 2x y x y x= = − = −

patt := [outline, cut1, cut2, cut3, cut4 ]

The first reflection (unfolding) takes place about the line y = x. This can be performed by en-tering and 2D-plotting patt1 := AREF(patt, 45deg) . See figure 4. (The 2D-&3D-Transformations.dfw instruction AREF(v, α) appends v and its image formed by a reflection in a line making an angle α with the positive direction of the x-axis.) The next unfolding (fig-ure 5) is a reflection in the x-axis, which can be performed by entering and 2D-plotting patt2 := AREF(patt1, 0) . The final unfolding (figure 6) is a reflection in the y-axis performed by

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entering and 2D-plotting patt3 := AREF(patt2, π/2). Note that, these three operations can also be combined into the single composition instruction

AREF(AREF(AREF(patt, π/4), 0), π/2).

(The axes can be turned off using Options>Display>Axes, etc.) Example 1.2 A Pattern with Dihedral Group D6 Symmetry

Take a regular hexagonal piece of paper and fold, from vertex to vertex, along one of its di-agonals. Make two folds in the half-hexagon, each fold along a line from the mid-point of its long side to an opposite vertex. You should have piece of paper, shaped like an equilateral triangle, and folded over into six layers. Finally, fold this triangle in half along its median from the vertex corresponding to the centre of the original hexagon to the opposite side. You should finish with a small triangle of folded paper. See figure 7. Now (for example) cut out four parts of circles (shaded areas) from the folded piece of paper as in the figure.

When you unfold the paper you should have a cutout pattern similar to figure 8. This is an example of a pattern with Dihedral Group D6 symmetry. Rotation and reflection symmetry can be investigated. Simulating Example 1.2 using 2D-&3D-Transformations.dfw

Open 2D-&3D-Transformations.dfw, select the 2D-plot Window and Tile Vertically. Check the 2D-plot settings: Options> Approximate before Plotting ON; Op-tions>Display>Points>Connect (YES), Small.

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The edges and circular cuts in figure 9 can be set up using instructions:

Note that parts of the circular Boolean areas fall outside the edges of the outline. This can be rectified (see figure 10) when all of these are collected together to form the basic pattern unit as follows:

The first unfolding (figure 11) can be performed by entering and 2D-plotting cpatt1:=AREF(cpatt, 0), the next double unfolding (figure 12) can be performed by entering and 2D-plotting cpatt2:=AREF(cpatt1, π/6, −π/6) and the final unfolding (figure13) is per-formed by entering and 2D-plotting cpatt3:=AREF(cpatt2, π/2). I leave it as an exercise to express these three operations as a single composition instruction.

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Example 1.3 Part of a Strip-Repeating Symmetry Group Pattern

Take a long strip of paper and fold it in half lengthwise three times to form a small rectangular shape (folded over into eight layers). See fure 14. Make (for example) two cuts on either side of the shape (shaded areas) as in the figure.

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When you unfold the paper you should have a cutout pattern similar to:

Figure 15 is an example of part of a strip-repeating symmetry group pattern. The true strip-repeating symmetry group pattern would repeat itself an infinite number of times in the direc-tion of the strip. Rotation, reflection and displacement symmetry can be investigated. Simulating Example 1.3 using 2D-&3D-Transformations.dfw

Open 2D-&3D-Transformations.dfw, select the 2D-plot Window and Tile Horizontally. Check 2D-plot settings: Options> Approximate before Plotting ON; Op-tions>Display>Points>Connect (YES), Small.

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In figure 16, note the change in the aspect ratio caused by tiling horizontally. The outer edges and cuts can be set up as

This example uses an instruction PARA which restricts the Boolean area plot to the finite rec-tangular region Both the outer edges and cuts can be stored using a single list instruction panel := [outer, zpat]. The first unfolding (figure 17) is a reflection in the y-axis. That is, panel1:=AREF(panel, π/2). The next unfolding (figure 18) is a composition of a displacement followed by a reflection in the y-axis as follows: panel2:=AREF(DIS(panel1, [1,0]), π/2) . The final unfolding (figure 19) is also a composition of a displacement and re-flection in the y-axis:

0 1; 2x y≤ ≤ − ≤ ≤ 2.

panel3:=AREF(DIS(panel2, [2,0]), π/2) .

Note that, the three unfoldings can be performed by the single composition instruction:

AREF(DIS(AREF(DIS(AREF(panel, π/2), [1,0]), π/2), [2,0]), π/2) .

Example 1.4 Part of a Plane-Filling Symmetry Group Pattern

Take a piece of A3 paper and fold it into quarters along its horizontal and vertical axes to form an A5 piece of paper folded into four layers. Repeat with the A5 piece to form a small rectangle (A7 size) of paper folded into 16 layers. See figure 20. Make (for example) two cuts on one corner of the shape (corresponding to the centre of the A5 piece) as in the figure.

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When you unfold the paper you should have a cutout pattern similar to:

Figure 21 is an example of part of a plane-filling symmetry group pattern. The true plane-filling symmetry group pattern would repeat itself an infinite number of times in both hori-zontal and vertical directions. Rotation, reflection and displacement symmetry of the pattern can be investigated. Simulating Example 1.4 using 2D-&3D-Transformations.dfw

Open 2D-&3D-Transformations.dfw, select the 2D-plot Window and Tile Vertically. Check the 2D-plot settings: Options> Approximate before Plotting ON; Op-tions>Display>Points>Connect (YES), Small. (It is simpler to build up the pattern in the (square) tiled vertically 2D-plot Window and maximize the finished result.)

In figure 22, the outer frame and cuts can be set up using:

Both the frame and cuts can be stored using a single list instruction pane := [frame, cuts]. The first double unfolding (figure 23) is with respect to reflections in both axes. That is, pane1:=AREF(AREF(pane,0),π/2). The next unfolding (figure 24) is with respect to a dis-placement and reflections in both axes as follows: pane2:=AREF(AREF(DIS(pane1,[2,2]),0),π/2) .

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I leave it as an exercise to express these operations in terms of a single composition instruc-tion.

Final Remarks on Section 1

By computer standards, DERIVE�s 2D-Boolean area plotter is fairly slow. Providing the basic patterns are kept simple, this is not a disadvantage. The observer has time see the patterns un-folding in the 2D-plot Window. There are many other ways of folding and cutting sheets pa-per, each with their own symmetrical properties. There are also many other concrete methods at Key Stage 1and Key Stage 2 for making patterns based on reflections, rotations and dis-placements. For example, mirrors, kaleidoscopes, ink blots, etc. With the powerful general-purpose tools of 2D-&3D-Transformations.dfw many of these can be simulated in DERIVE. Consider the following example on rotational symmetry: Example 1.5 Constructing a simple windmill

The classic method of constructing a simple windmill is to take a square piece of paper and cut along the diagonal from each corner almost to the centre. Then each half-cut corner is folded to the centre, to make a sail, and secured with a wire and bead axis. (A stick is usually added to hold the windmill in the wind to make it turn.) Simulating Example 1.5 using 2D-&3D-Transformations.dfw

Open 2D-&3D-Transformations.dfw, select the 3D-plot Window and Tile Vertically. Select the 3D-plot setting: Options> Approximate before Plotting ON and, using the arrow keys, manipulate the 3D-plot window to view along the z-axis from above. Enter and 3D-plot PARA([s, t(5 - s), 0], [0, 5], [0, 1]) (to construct a triangular sheet for mak-ing a sail) using Insert>Plot>Number of Panels 16 and 16>Apply parameters to rest of plot list ON. See figure 25. To turn this into a sail, the sheet (instead of lying entirely in the Oxy plane) can be wrapped around a semi-ellipse (parallel to the Oxz plane) by entering and 3D-plotting sail := PARA([2.5SIN(s), t(5 - 5s/π), - COS(s)], [0, π], [0, 1]). See figure 26. This can be clearly illustrated by tipping the 3D-plot window to view along the y-axis but, when

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you have finished, return to the z-axis viewpoint. The final stage, is to rotate and copy sail a number of steps around a complete turn to form a windmill as follows: windmill(n) := VECTOR(ROT(sail, [0, 0, 1], 2πi/n), i, 1, n) Figure 27 shows a 3D-plot of windmill(6). Note that, with DERIVE, you get more than the initial four-sail windmill model. Blowing on the VDU and using the 3D-plot rotation button to start and stop the windmill can provide an amusing twist to this example!

As well as basic work on transformations, in the degree course module, these activities are used as a prelude to a systematic study of symmetry groups, including: cyclic and dihedral groups (together with their Cayley Tables); the 7 strip-repeating symmetry groups and the 17 plane-filling symmetry groups. A good introduction to this topic can be found in Bell and Fletcher. Also, you can download and consult the users file Patts2D.dfw from the users folder on the DERIVE Web Site.

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2. Constructing Polyhedrons from Nets of Faces Another interesting activity is constructing polyhedrons from nets of their faces. This can lead to pupils to investigating properties of 3D-objects and shapes. For example, axes and planes symmetry, faces, edges and surfaces, area and volume. Example 2.1 Making a Cube

Folding the faces, and joining the edges to-gether can construct a cube from the network osquare faces in figure 28.

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This can be simulate in DERIVE, by first con-structing an open box from the four faces around the central square and then adding a base and lid.

Simulating Example 2.1 using 2D-&3D-Transformations.dfw

Open 2D-&3D-Transformations.dfw, select the 3D-plot Window and Tile Vertically. Select: Options> Approximate Before Plotting ON; Change Plot Colors OFF. Enter and 3D-plot face := SQUARE(3,[−1.5,−1.5,−1.5]) using Insert>Plot>Number of Pan-els 1 and 1> Apply parameters to rest of plot list ON. A network of faces for the open box can be generated by entering and 3D-plotting AROT(DIS(face,[3,0,0]), [0,0,1], π/2, π, 3π/2) .

However, face has been carefully positioned at z = −1.5. Hence, rotating face a quarter-turn about the x-axis and then rotating and copying the result in quarter turn steps about the z-axis can form the open box. Therefore, enter and 3D-plot

box := AROT(ROT(face,[1,0,0],π/2), [0,0,1], π/2, π, 3π/2) .

Figure 29 shows a composite of the box and its network of faces.

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The final stage is to add the base and lid to form the cube. This can be completed by entering and 3D-plotting cube := [box, AROT(face,[1,0,0],π)] . See figure 30. Things to try

(i) If you 3D-plot cube using: Insert>Plot>Number of Panels 1 and 1> Scheme - Auto Plot Color, then the cube will be plotted in a single outer colour. Delete one of the faces of the cube so that you can see its inside, which should be a different colour. Delete the whole cube and enter and 3D-plot INV(cube). The colours of outside and inside of the centrally inverted cube should now be reversed! Can you explain this?

(ii) Instead of assigning a square lamina to face, enter and 3D-plot face:= [[-1.5, -1.5, -1.5; -1.5, 1.5, -1.5; 1.5, 1.5, -1.5; 1.5, -1.5, -1.5; -1.5, -1.5, -1.5]].

A skeletal network for the cube (figure 31) can now be formed by 3D-plotting box. (iii) A stellated cube can be formed by assigning face:= PARA([(1.2 - r)COSθ, (1.2 - r)SINθ, 3·r + 0.6·√2], [0, 1.2], [−π/4, 7π/4]), and 3D-plotting box, followed by cube using Inset>Plot>Set Number of Panels to 1

and 4. See figure 32.

Example 2.2 Making a Dodecahedron

A network of regular pentagon faces for a do-decahedron can be formed from two copies of the network of six pentagon faces in figure 33.

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Simulating Example 2.2 using 2D-&3D-Transformations.dfw

Open 2D-&3D-Transformations.dfw, select the 3D-plot Window and Tile Vertically. Select: Options> Approximate Before Plotting ON; Change Plot Colors ON. To construct a pentagon face, enter and 3D-plot pent := CIRCLE(2,[0,0,−2.618]) using In-sert>Plot>Number of Panels 1 and 5, etc. Although the z-coordinate of the center of pent looks a little odd, it has been calculated to be the value of the radius (to three decimal places) of the inner sphere of a dodecahedron with pentagon faces inscribed on a circle of radius 2.

The exact value can be expressed as 1 12cos( / 5)cot / 2 sin2sin( / 5)

π ππ

− −

.

In the 3D-plot Window one of the sides of the pentagon is parallel to the y-axis. We can make use of this to form a network of six pentagons. First reflect pent in the Oyz plane and displace to match up edges by entering pent1 := DIS(REF(pent, [1; 0; 0]), [− 4COS(π/5); 0; 0]) . Now, rotate and copy pent1 about the z-axis in steps of 2π/5 by entering and 3D-plotting

AROT(pent1, [0; 0; 1], 2π/5, 4π/5, 6π/5, 8π/5) . See figure 34.

To make the bottom half of the dodecahedron, first rotate pent through a half-turn about an axis passing through the origin and the mid-point of the side parallel to the x-axis. Because the z-position of pent has been carefully chosen, this will match up the edges and also provide the correct dihedral angle for the faces of the dodecahedron. Therefore, enter

pent2 := ROT(pent, [2cos(π/5); 0; 2.618]) , and form a half dodecahedron by entering and 3D-plotting halfd := [pent, AROT(pent2,[0; 0; 1], 2π/5, 4π/5, 6π/5, 8π/5)]. For a composite of halfd and its net see figure 34. Finally, the full dodecahedron of figure 35 can be generated by appending and rotating halfd a half turn about an axis through the origin in the Oxy plane. For figure 35, enter and 3D-plot dodecahedron := [halfd, ROT(halfd, [0, π/5, 0], π)]. (Note: the image of halfd is twisted through π/5 to mesh the top and bottom halves together.)

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Things to try

(i) The dodecahedron has been constructed using rotations about the origin so that, if you select Options> Change Plot Colors OFF, and 3D-plot dodecahedron using In-sert>Plot>Number of Panels 1 and 5> Scheme - Auto Plot Color, then it will be plot-ted in a single outer colour. Again, you can delete faces to see the alternative inside colour and experiment with INV as well as other transformers.

(ii) Instead of assigning a pentagon lamina to pent, enter and 3D-plot pent := [VECTOR([2COS(2πi/5), 2SIN(2πi/5), −2.618], i, 0, 5)]

A skeletal network of the dodecahedron (figure 36) can now be formed by 3D- plot-ting, in turn, pent2, halfd and dodec.

(iii) The stellated dodecahedron of figure 37 can be generated by first assigning pent := PARA([(1 - r)COS(θ), (1 - r)SIN(θ), 3r + 1.309], [0, 1], [−π, π])

then 3D-plotting, in turn, pent2, halfd and dodec.

Final Remarks on Section 2

Students can base their own examples on networks of faces for pyramids, prisms, cuboids and other polyhedrons. I can foresee these two examples leading to DERIVE constructions of more and more sophisticated polyhedrons and 3D-shapes. The users folder of DERIVE 5 now contains the file RegularPolyhedrons.dfw. This file illustrates how to construct and 3D-plot facial, skeletal and stellated forms of the five regular polyhedrons. That is: tetrahedron; cube; octahedron; dodecahedron; icosahedron. Conclusion

The activities described in sections 1 and 2 make up only a small part of the content of the College degree course module on linear algebra and transformation geometry. Their main aim is to establishment explicit links between classroom activities at Key Stages 1 and 2 and de-gree course work.

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Using 2D-&3D-Transformations.dfw students have in their possession an extremely versatile resource for investigating linear transformations from both an intuitive and mathematical viewpoint. You may agree that some of the DERIVE applications outlined in this presentation are spectacular. However, the applications have been made possible by the general-purpose tools of the DERIVE Users File 2D-&3D-Transformations.dfw. For more information about this file, consult P. Schofield (2002) in the workshop notes of these proceedings. References DfEE 4/98 Summary Web Site: http://www.dfes.gov.uk/publications/guidanceonthelaw/4_98/summary.htm DERIVE Web Site: http://www.derive.com. A Bell & T Fletcher, Symmetry Groups, Association of Teachers of Mathematics. P. Schofield (2002), Some General-Purpose Tools for Carrying out 2D- and 3D-Linear Trans-formation Geometry Plots with DERIVE 5, Proceedings of the 5th Int. DERIVE & TI-89/92 Conference.

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