Origami, geometry and art

Origami, geometry and art

The Common Core State Standards put a tremendous amount of emphasis on conceptual understanding [1]. Origami provides an engaging context for conceptual understanding of mathematical ideas. Among other things, origami gives our students handy manipulatives that can be used to visualize abstract mathematical ideas in a concrete way [2,3]. For instance, when one creates a box from a flat sheet of paper, the box becomes the object that can be manipulated and analyzed, and abstract concepts like length, width, height, volume and surface area become something that one can 'touch' . When students have objects that they have created, students communicate better with one another and with their teacher. Moreover, paper folding in general is mathematics in action. When one is folding paper she or he is playing with mathematical concepts like perpendicular bisection, angle bisection, properties of right isosceles triangles, properties of squares, just to name a few of the mathematical concepts that are inextricably connected to paper folding. In fact, it is fairly difficult to separate paper folding from mathematics. Due to the strong link between origami and art, origami can additionally be used to inspire artistic-minded students to think mathematically. Lastly, origami creates a powerful context for the application of Howard Gardner's theory of multiple intelligences [4,5]. Gardner's theory of multiple intelligences incorporates several other dimensions of intelligences besides linguistic and logical-mathematical intelligence. Gardner identified the following nine intelligences: linguistic, logical-mathematical, bodily-kinesthetic, spatial, musical, interpersonal, intrapersonal, naturalist and existential intelligence [4].

In this paper we learn to fold an origami box and discuss the mathematics embedded in the box. No experience in origami is needed to construct this box. However, it is important to make the creases sharp and accurate. Figure 1 illustrates the two types of creases that are formed when a piece of paper is folded. The constructed box will be a prism with a rectangular base. Figure 2 shows a photograph of the box that we will be making.

Figure 1

Figure 2

We will use two sheets of paper of the same size to construct the box. We will choose 11 inches by 8.5 inches sheets. One sheet of paper will be used to construct the box; we will call that sheet the origami sheet. The other sheet of paper will be used as a measuring tool;

CONTACT Arsalan Wares awares@valdosta.edu http://dx.doi.org/./X.. . This is a picture of the box that we will be making.

we will call that sheet the measuring sheet. Ideally the origami sheet should have at least one fancy side. However, feel free to use an 11 inches by 8.5 inches sheet straight out of the recycling bin to fold the box. The measuring sheet will be discarded during the process so it does not have to be fancy. Let us follow the 18 steps to construct the box. Pictures for steps 1-6 are shown in Figure 3. Pictures for steps 7-13 are shown in Figure 4. Pictures for steps 14-18 are shown in Figure 5. A video showing how to fold this box is also available at the following link: https://youtu.be/1i8zlVkly30

Figure 3

Figure 4

Figure 5

Step 1: Start with the measuring sheet. By folding and unfolding the two shorter edges of the measuring sheet onto each other create a valley crease right through the middle of the measuring sheet. This crease must be parallel to the shorter edges of the rectangular measuring sheet.

Step 2: Use one of the shorter edges of the measuring sheet and the crease created in step 1 to make another crease that splits the measuring sheet into ¼ and ¾ parts. This crease must also be parallel to the shorter edges of the rectangular measuring sheet.

Step 3: Fold ¼ of the measuring sheet over the crease created in step 2.

Step 4: Take the origami sheet with the plain side up and slide it all the way between the two layers of the measuring sheet. In other words, one of the shorter edges of the origami sheet is pushed against the crease created in step 2. Fold and unfold the uncovered shorter edge of the origami sheet against the shorter edge of the measuring sheet that is on top of the origami sheet to make another valley crease on the origami sheet.

Step 5: Take the other shorter edge of the origami sheet with the plain side up and push it against the crease created in step 2. Fold and unfold the uncovered shorter edge of the origami sheet against the shorter edge of the measuring sheet that is on top of the origami sheet to make another valley crease on the origami sheet. Discard the measuring sheet. The measuring sheet will not be needed any more.

Step 6: Start with the plain side of the origami sheet up. Fold and unfold the left shorter edge onto the crease on the left-hand side of the origami sheet to make one more valley crease. Fold and unfold the right shorter edge onto the crease on the right-hand side of the origami sheet to make one more valley crease.

Step 7: With the plain side up, fold the left-hand side of the top edge of the origami sheet over the crease on the right-hand side to make a short slant valley crease as shown in the picture for step 8. This slant crease should be between the two creases on the right-hand side as shown in the picture for step 8.

Step 8: Unfold the origami sheet so that the plain side is up.

Step 9: With the plain side up, fold the right-hand side of the top edge of the origami sheet over the crease on the left-hand side to make a short slant valley crease as shown in the picture for step 10. This slant crease should be between the two creases on the right-hand side as shown in the picture for step 10.

Step 10: Unfold the origami sheet so that the plain side is up. Step 11: Rotate the shape around its centre by 180°, and repeat steps 7 through 10 to make two more short, slant valley creases.

Step 12: Make a horizontal crease by folding and unfolding the top strip of the origami sheet that contains the two slant creases. This horizontal crease must pass through the terminal points of the two top slant creases. Make another horizontal crease by folding and unfolding the bottom strip of the origami sheet that contains the other two slant creases. This horizontal crease must also pass through the terminal points of the two bottom slant creases.

Step 13: Flip the origami sheet over so that the fancy side is up. Make valley creases along the dotted lines. Mountain creases already exist along these dotted lines, just change the orientation of these creases from mountain to valley.

Step 14: Start with the plain side of the origami sheet up. Lift the top and bottom parts of the sheet along the creases that are parallel to the longer edges of the origami sheet. These vertical strips will form two of the walls of the box that is being constructed.

Step 15: Fold along two of the short slant creases as shown and lift up the part of the sheet that contains one of the shorter edges of the origami sheet.

Step 16: Fold the parts of the paper that contain the two short slant creases inward as shown.

Step 17: Tuck the excess paper in carefully so that the creases line up neatly inside the partially constructed box. This should complete the third wall of the box.

Step 18: Use steps very similar to steps 15-18 with the other lose end of the partially folded origami sheet to complete the box. Figure 6). One of the questions that comes to mind is how can we determine the dimensions of the constructed box, if we know the dimensions of the rectangular sheet.

Figure 6

Let us first show if the width of the folded portion of the measuring sheet is x inches (see the picture on the left in Figure 7), then the distance between the first two creases of the origami sheet is also going to be x inches (see the picture on the right in Figure 7). Suppose the width of the folded portion of the measuring sheet is x inches (see the picture on the left in Figure 7), then due to the nature of the fold we can establish the following (see Figure 7): m = n = 11−x 2 , and p = 11 − 2 11−x 2 = x Due to the nature of the fold, since we started with an 11 inches by 8.5 inches measuring sheet, x = 11/4 = 2.75 inches. Take another 11 inches by 8.5 inches measuring sheet, and fold it so that x is slightly less than 2.75 inches. Use this measuring sheet to fold a second origami box as above. The second box will smoothly slide into the first box. These two boxes together can be used to cover a small gift in an artistic and personal way.

Figure 7

We know that the dimensions of the rectangular origami sheet are 11 inches by 8.5 inches. Let the dimensions of the rectangular base of the constructed box be a inches by b inches (a < b), and let the height of the constructed box be c inches (see Figure 8). Let us also observe that the distance between the first two creases made on the origami sheet is 2.75 inches (see Figure 8). Let us now determine the exact values of a, b and c.

Figure 8

We will use Figure 6 to determine the exact values of a, b and c. Due to the nature of the fold, in Figure 6, the following can be established: Note UT turns out to be the shorter dimension of the rectangular base of the constructed box, ST turns out to be the longer dimension of the rectangular base of the constructed box and TY turns out to be the height of the constructed box. Therefore, in Figure 8, a = HG = 2.75 inches, b = ST = 4.375 inches, and c = TY = 2.0625 inches. Now we can easily determine the volume of the constructed box. Which turns out to be 2.75 × 4.375 ×2.0625 = 24.81445313 cubic inches. By changing the value of x in the original measuring sheet (see Figure 7), we can change the dimensions of the constructed box. For an 11 inches by 8.5 inches measuring sheet x has to be less than 11 inches. Students can use different values of x and construct different boxes in the class. They can also use algebra as discussed above to find the volume of the constructed box. Let V(x) represent the volume of the box, where x is the length of one of the shorter sides of the rectangular base of the constructed box. We can establish V (x) = x( One of the strengths of paper folding activities is that they can be easy to understand and model, yet at the same time the solutions are not always obvious. Since students have a concrete object (the folded paper) in front of them, the level of intellectual engagement with the task is heightened and the quality of communication in the classroom becomes richer during the lesson. By its nature, origami creates a context for rich discourse.

Since mathematics is a cultural endeavour, it is ideal to teach mathematics as it manifests in the context of various cultures [6]. Appreciation of cultural diversity is not only important for the minority groups, but it is also important for the dominant ethnic group in any society because even the members of the dominant ethnic group will be working in an environment that is increasingly getting more and more diverse [7]. The use of origami in mathematics classroom can provide a powerful context for the appreciation of cultural diversity of our world.