The Origami of a Tiny Cube in a Big Cube

Emily Gi

Mr. Acre & Mrs. Gravel

GAT/IDS 9C

12 January 2016

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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.

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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.

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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 45 – 45 – 90, 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 1.5√2 in.

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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.

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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

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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

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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

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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.

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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.

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Figure 25. Completed Cube in a Cube

Finally, just insert the inner cube into the outer cube’s hole and the cube in a cube will be

completed.

Area (A) = s2 Formula for the Area of a Square

A = ??? Substitution Property

A = ??? Multiplication Property

A = ??? Formula for the Area of a Square

A = ??? Substitution Property

A = ??? 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 ??? in2.

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 ??? in2, the answer becomes ??? in2.

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A = ??? Substitution Property

A = ??? Subtraction Property

Surface Area = (A of One Side) · (6) Formula for the Surface Area of a Cube

SA = ??? · 6 Substitution Property

SA = ??? 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 ??? in2, with the part of

the square that is empty, as it is ??? in2, gaining the area of the face that can actually be touched

on the origami cube. The answer ends up becoming ??? in2.

To get the surface area of the entire outer cube, just take the area of one side without the

small square in the middle, ??? in2, and multiply it by six. The surface area is ??? in3.

A = s2 Formula for the Area of a Square

A = ??? Substitution Property

A = ??? Multiplication Property

SA = (A of One Side) · (6) Formula for the Surface Area of a Cube

SA = ??? · 6 Substitution Property

SA = ??? 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.

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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 ??? in2 with six and

now the surface area of the smaller cube is known as ??? in2.

SA = (Outer Cube SA) + (Inner Cube SA) Formula of Surface Area of Entire Cube in a Cube

SA = ??? + ??? Substitution Property

SA = ??? 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 ??? in2, the outer

cube’s surface area, and ??? in2, the inner cube’s surface area, to obtain the total surface area,

that becomes ??? in2.

Volume (V) = s · s · s Formula for the Volume of a Cube

V = ??? Substitution Property

V = ??? Multiplication Property

V = ??? Substitution Property

V = ??? 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 ??? in3.

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

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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 ??? in2

to end up with ??? in3.

V = ??? - ??? Substitution Property

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 ??? in3 and ??? in2 gives the final volume

of ??? in3. Because the outer cube’s total volume is ??? in3 and the inner cube sits inside it,

taking up space that adds up to be ??? in2, 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

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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 ??? in3 in volume once figured out. Along with that, the surface area was found to be ???

in2 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.