Tumbling Cubes:Modular Origami that Relates to Geometry

Zachary Mahalak

Geometry, Algebra, and Transformations

Mr. Acre

January 12, 2013

9A

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Although many people do not realize it, volume and surface area are very influential in

people’s lives. For example, when someone buys a car they look for volume. Some people say

that they want a bigger trunk or more leg room. That is another way of saying they want a car

with more volume. Also, if a car has less surface area on the leading edge and top of the car, the

car will be more aerodynamic. If a car is aerodynamic, it will increase fuel economy. Lastly,

when you mail a package, you buy the smallest box that will fit the item. Nobody will buy a box

seven times the size they need. Buying the smallest box that will work saves money. This is

how volume and surface area influence everyone’s lives. The project assigned was to build a

tumbling cube and find the surface area and volume.

In order to make a tumbling cube, you need to make nine units. The first step is to lay the

paper down with the colored side facing up (Figure 1). Then, fold the diagonals (Figure 2).

Your paper should look like figure 2. The next step is to rotate the paper 90° and fold two

corners to the center. There should be two colored diamonds and two white triangles (Figure 3).

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Figure 1 Figure 2

Figure 3

Next, fold the two triangles into the center (Figure 4).

After that, the next step is to reverse the edge of the white trapezoids so that the color is facing

up (Figure 5).

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Figure 4

Figure 5

Now fold the two triangles on the edges in half (Figure 6).

Take the bottom right corner and fold it to the top right of the trapezoid. After, take the top left

corner and fold it to the bottom left of the trapezoid (Figure 7). To get from figure 6 to figure 7

fold the red dots on to the yellow dots. The red dots are the corners that are folded; the yellow

dots are where the red dots are folded to.

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Figure 6

Figure 7

Next, take the white triangle and tuck it underneath the trapezoid. Do this for both sides (Figure

8).

Flip the shape over and fold one corner to the angle straight across from it. The red dot is the

corner that you fold to the yellow dot (Figure 9).

Lastly, flip the unit back over and crease the diagonal. Repeat figures 1 – 10 eight more times,

but only crease the diagonal on three units (Figure 10).

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Figure 8

Figure 9

Figure 10

To attach these units, there are two rectangles in the center of the square. Take one of the

triangles and tuck it under the rectangle (Figure 11).

Next, make two corners each containing three units. There should be three extra units (Figure

12).

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Figure 11

Figure 12

Now take one corner unit and add on the three extra units. Looking at it from the front, the

opening should form a triangle (Figure 13).

Lastly, join the 3 units with the triangles by inserting them into the 6 other pieces. Once one unit

is fit in, the rest naturally fit into place (Figure 14).

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Figure 13

Figure 14