The Origami of a Tiny Cube in a Big Cube. Emily Gi. Mr. Acre & Mrs. Gravel GAT/IDS 9C

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1 The Origami of a Tiny Cube in a Big Cube Emily Gi Mr. Acre & Mrs. Gravel GAT/IDS 9C 12 January 2016

Gi 1 The Origami of a Tiny Cube in a Big Cube It is exhilarating to finish a seemingly impossible project. In this case, that impossible project is creating a cube inside another cube.

2 Gi 1 The Origami of a Tiny Cube in a Big Cube It is exhilarating to finish a seemingly impossible project. In this case, that impossible project is creating a cube inside another cube. Being made out of mere paper, the project must be constructed carefully. And in the end, the work of mathematics and calculations are utilized to find out just how and why this cube had been able to fit inside of the other. The method in which the answer is discovered has been used by people in many specific jobs for a long time, especially ones that require perfection, much like an architect or even a mathematician. With that concept in mind, not only learning how to construct a cube, but also finding the surface area and the volume of a cube inside a cube can be deemed quite important. Figure 1. Step 1 To start constructing the outer cube, take a piece of paper with equal sides. In this case, a 6 in by 6 in piece of paper is used.

If there is one color all throughout both sides, the need to pay attention to these details should not be bothered with.

3 Gi 2 Figure 2. Step 2 Fold the piece of paper in half, hamburger style; with the color inside. This only applies if the piece of paper used has one colored side and one white side, however. If there is one color all throughout both sides, the need to pay attention to these details should not be bothered with. Once the paper is folded in half, its width will end up being 3 in. Figure 3. Step 3 Unfold the piece of paper and using that crease as a guideline, fold two flaps to stop at the center, or where the first crease had ended up being. Those two flaps widths became 1.5 in.

4 Gi 3 Figure 4. Step 4 Now, take the corner of each flap and fold them up until the side of the width is lined up against the top of the paper. Flip the paper 180 degrees and do the same to the other flap. The width of 1.5 in is now folded up, but the measure does not change. Even so, the triangle created by the fold can now be considered a , which is a special right triangle. The mentioned triangles are highlighted in Figure 4. This causes for the side opposite of the 90 degree angle to become in. <NOTE: This student decided to show a step by step, showing how she folded the paper to come up with each exact side length. Her remaining steps have been deleted so that you can figure these out for yourself>

In the very end, one face will consist of four pieces and one edge will consist of one piece.

5 Gi 4 <From here she decided to show how to put the cube together. This is OPTIONAL> Figure 10. Step 9 Gather together the twelve pieces needed to assemble the outer cube. In the very end, one face will consist of four pieces and one edge will consist of one piece. Now take two pieces and grab the outermost triangle tab (the triangle that is pointing towards the right when the piece is held pointing up) of piece one and slide it into piece two s innermost pocket, which is the pocket nearest to what might be considered the center of the piece. Figure 11. Step 10 As an end result, it should look like Figure 11 above.

When the cube is completed, each pocket should have one tab inside it at the very end, no less, no

6 Gi 5 Do the same to all the other pieces to make one face of the outer cube. Consistently, the outermost tab must go inside an innermost pocket. When the cube is completed, each pocket should have one tab inside it at the very end, no less, no more. Likewise, every tab should be inside a pocket. Figure 12. Step 11 Once one face is finished, the face should resemble Figure 12. Figure 13. Step 12

Step 13 Don t forget to connect the second tab of the piece that was inserted in Step 12 to the other side if there is one.

7 Gi 6 To further extend from a corner, take another piece that has not been used yet and put it into an innermost pocket that is free and empty. Once in the pocket, turn it so that the piece that will become perpendicular to the part that it is next to. Figure 14. Step 13 Don t forget to connect the second tab of the piece that was inserted in Step 12 to the other side if there is one. If not, just repeat the steps over and over again until another face is formed and so on. If it is needed, relay back to the previous steps for assistance. Figure 15. Step 14

Once the tab or pocket is located, take that tab that could be hiding underneath the connected pieces and bring it to the top like in Figure 15. Figure 16.

8 Gi 7 Though, once everything has been connected, the structure may still feel not sturdy. Look around to see if there are any tabs sticking out which do not have a pocket or if there is a pocket that does not have a tab. Once the tab or pocket is located, take that tab that could be hiding underneath the connected pieces and bring it to the top like in Figure 15. Figure 16. Step 15 Continue to take that flap and slide it into the corresponding pocket. The ending result will look like Figure 16. If there is still uncertainty that all tabs have been found out, just make sure every corner of the cube resembles the picture above. There should be three tabs. Figure 17. Completed Outer Cube

Though, instead of folding it with the color inside, fold it with the color outside. Figure 19.

9 Gi 8 In the very end, the outer cube would look like Figure 17. Figure 18. Step 16 Do the same steps as in the first few steps for the outer cube but this time, for the inner cube. Though, instead of folding it with the color inside, fold it with the color outside. Figure 19. Step 16 Continued All of the steps are the same up until the step after Step 16. The same measurements will apply all throughout. <She does the same thing again for the inner cube. Again the remaining steps have been deleted.>

(different colored) piece s pockets with both of them going into opposite pockets. Figure 24.

10 Gi 9 Figure 23. Step 19 Take two pieces of the same color (if different colors were used) and insert them into another (different colored) piece s pockets with both of them going into opposite pockets. Figure 24. Completed Inner Cube Do the same throughout the entire cube, making sure that every tab and every pocket is full or used alike to the outer cube. <Again she decides to show how to put the inner cube together this is optional>

11 Gi 10 Figure 25. Completed Cube in a Cube completed. Finally, just insert the inner cube into the outer cube s hole and the cube in a cube will be Area (A) = s 2 Formula for the Area of a Square Multiplication Property Formula for the Area of a Square Multiplication Property Figure 26. Area of One of the Outer Cube s Sides The chart above shows how to find the area of one of the outer cube s faces, not yet excluding the space that lies in the middle of the square. It shows to use the area formula, in which one edge is??? in, so square it and the answer will become??? in 2. The second problem displays how to find the space where the small cube would be able to fit through, or the empty square that cannot be touched. Following the same formula as the first problem, but with the side being??? in 2, the answer becomes??? in 2.

12 Gi 11 Surface Area = (A of One Side) (6) S 6 S Subtraction Property Formula for the Surface Area of a Cube Multiplication Property Figure 27. Surface Area of the Outer Cube The first part of Figure 27 demonstrates the steps to find the area of one of the outer cube s sides, which is to subtract the area of the entire face, in this case??? in 2, with the part of the square that is empty, as it is??? in 2, gaining the area of the face that can actually be touched on the origami cube. The answer ends up becoming??? in 2. To get the surface area of the entire outer cube, just take the area of one side without the small square in the middle,??? in 2, and multiply it by six. The surface area is??? in 3. A = s 2 SA = (A of One Side) (6) S 6 S Formula for the Area of a Square Multiplication Property Formula for the Surface Area of a Cube Multiplication Property Figure 28. Area of Inner Cube s Side and Surface Area of the Inner Cube Finding the area and surface area of the inner cube is now simpler as it follows the same steps as to find the outer cube s area and surface area; the only difference is that it has one less step. Instead of subtracting the empty space in the middle of the square, just skip that step entirely. The figure above shows how to do that.

13 Gi 12 Find the area of square, which has already been found as the side of the square is the same length as the side of the empty square in Figure 26, and multiply that??? in 2 with six and now the surface area of the smaller cube is known as??? in 2. SA = (Outer Cube SA) + (Inner Cube SA) S +??? S Formula of Surface Area of Entire Cube in a Cube Addition Property Figure 29. Total Surface Area of the Cube in a Cube Now that the surface area of the outer cube and the inner cube are both solved, all that needs to be done is adding those two numbers together and the total surface area of the cube in a cube will be found out. Figure 29 says that all that needs to happen is to add??? in 2, the outer cube s surface area, and??? in 2, the inner cube s surface area, to obtain the total surface area, that becomes??? in 2. Volume (V) = s s s V =??? V =??? V =??? V =??? Formula for the Volume of a Cube Multiplication Property Multiplication Property Figure 30. Volume of the Outer Cube and the Inner Cube In Figure 30, the volumes of both the outer and inner cubes are found. Taking the one side of the outer cube (??? in) and multiplying with itself three times gains the volume of the outer cube. The volume of the outer cube is??? in 3. This is because multiplying one side with itself will gain the area of one face, or the base as any of the faces of a square can be its base. After that, multiply the answer with the side once

14 Gi 13 again and the volume will be the answer because all of the square s sides are equal and all of its angles are perpendicular, so henceforth, the side can be used as the height Continue to use the same method to find the inner cube s volume, using its side of??? in 2 to end up with??? in 3. V =??? -??? V =??? Subtraction Property Figure 31. Volume of the Cube in a Cube How to get the total volume of the outer cube with the smaller cube inside it is shown above. Gathering the volume of the outer cube and the inner cube, subtract them both to obtain the final volume of the cube in a cube. Subtracting??? in 3 and??? in 2 gives the final volume of??? in 3. Because the outer cube s total volume is??? in 3 and the inner cube sits inside it, taking up space that adds up to be??? in 2, subtracting the outer cube s volume with the inner cube s volume will finish up with the cube in a cube s volume. In conclusion, throughout the cube in a cube project, its process can help people understand 3-dimensional shapes and its measurements better. With the knowledge of how the measurements of a cube can be found, it will be easier to explain the solutions to many other problems as long as the correct information is known. And with that, the folding can make this information pass through easier as it assists those visual learners in seeing the steps. This knowledge can be applied to many instances and even to moments where one would need to find out a certain distance or length for a job if one had wanted to be an architect or an engineer of sorts. Even scientists can use methods like so. But of course, there were problems that occurred with the origami folding. Firstly, the directions had been read incorrectly, and that caused the folding and the fitting of the pieces to be completely messed up, which jeopardized the accuracy of the measurements. Another problem was that the creases were not neat and none of the edges

15 Gi 14 were really lined up like shown. That causes the measuring to be wrong, but if one were to use mathematics and logic, those same measurements will end up being precise and accurate as if the structure was built perfectly. In example, the number that started as 6 in ended up becoming a complete??? in 3 in volume once figured out. Along with that, the surface area was found to be??? in 2 even though the only measurement given was, in fact, 6 in. This statement shows just how powerful applying geometry skills to real life can be with almost no information. Math actually is everywhere.

Cubigami. Knicko Mojica. Mr. Acre & Mrs. Gravel GAT/IDS 9C

Cubigami. Knicko Mojica. Mr. Acre & Mrs. Gravel GAT/IDS 9C Cubigami Knicko Mojica Mr. Acre & Mrs. Gravel GAT/IDS 9C 12 January 2016 Mojica 1 Cube in a Cube When creating a three-dimensional figure out of paper, it may simply be seen as an inanimate object. However,