by Arsalan Wares
Edited by Jane Rosemarin

In this article we will learn to fold an origami box with a square base. Origami is an old artform in which square sheets of paper are folded into three-dimensional objects. Among many things, origami teaches us how to focus, how to appreciate beauty in simple things, how to accept imperfections, how to embrace impermanence, how to play with creative ideas and how to think mathematically. Above all, origami teaches us how to embrace solitude and enjoy it. In this article we will learn to fold a simple origami box made from four identical rectangular sheets. We will also discuss some mathematics that emanates from the construction process.

## The Construction Process

The box we will be making is shown above. We will need five rectangular sheets of the same dimensions. I used five 8½-by-11-inch sheets to make the box. One of the five sheets will be used as a measuring tool. This measurement sheet will be discarded and will not be a part of the constructed box. Four of the five sheets must be origami sheets, ideally with at least one fancy side. The four origami sheets will constitute the entire box. Two of the four sheets will be used to make the top of the box and two will be used to make the bottom. Even though the top and the bottom parts of the box are identical, the top part expands as the bottom part is slid inside. Consequently, the top part holds the bottom part snugly, and the halves can also be separated easily. You will find a video that shows how the box is made at the end of this article, or use this link. In the video, we make only the top portion of the box with two origami sheets and the measurement sheet. The bottom portion is made in a similar manner with two additional identical origami sheets and the same measurement sheet.

The video is about 12 minutes long, but it will probably take about 30 minutes to make a decent box. No prior experience in origami is needed to make the box, but a little bit of mindfulness and patience can only help make the box more crisp, beautiful, and presentable. Mindfulness is about being aware of the present moment. In this context, it is about immersing oneself sincerely with the box-making process in a relatively distraction-free environment. The box shown above was made with 20-pound regular printing paper. However, we can also use 65-pound cardstock sheets to make a sturdy version of this box that can be reused many times. However, I would encourage readers to make their first box with regular printing paper, which is easier to fold than cardstock, especially for beginners.

## Some Mathematics

Suppose each rectangular sheet that we started with was $$a$$ by $$b$$, with $$b \geq a$$. Before we delve into the mathematics, we need to understand the role of the measurement sheet in the process. Suppose the distance between the third crease on the measurement sheet and the nearest parallel edge of the measurement sheet is $$x$$. Then we can show the distance between the first two creases on each origami sheet is also $$x$$ (see figure 2). This follows from the way the first two creases were made on the origami sheets using the measurement sheet. Since the longer edge of the measurement sheet is $$b$$, $$x = \frac{3b}{8}$$. This follows from the way the three creases were made on the measurement sheet sequentially.

Figure 3 shows the crease marks on each of the four origami sheets. The pink square forms the square base of the box and the grey rectangle forms one of the walls of the box. If $$h$$ is the height of the constructed box, we can show $$h = \frac{1}{2}\left(a-\frac{3b}{8}\right)=\frac{a}{2}-\frac{3b}{16}\;.$$ Since the constructed box is a prism with a square base with length $$x = \frac{3b}{8}$$, the volume of the box is: $$x^2h = \frac{9b^2}{64}\left(\frac{a}{2} - \frac{3b}{16}\right) in^3 .$$ As mentioned earlier, $$a$$ and $$b$$ are the lengths of the edges of each sheet, with $$b$$ > $$a$$. We can use this formula to determine the exact volume of the box we constructed. Recall that we used 8½-by-11-inch sheets to construct the box. Therefore, $$a$$ = 8.5 inches and $$b$$ = 11 inches. With these values of $$a$$ and $$b$$, the volume of the box turns out to be $$\frac{9(11)^2}{64}\left(\frac{8.5}{2}-\frac{3(11)}{16}\right)\approx 37.22\; in^3.$$ In summary, the height of the constructed box is: $$\frac{a}{2}-\frac{3b}{16} = \frac{35}{16}\approx2.1875\;in,$$ and the length of each side of the square base of the constructed box is: $$\frac{3b}{8} = \frac{33}{8} = 4.125\ in$$ (see figure 3). This gives us an idea what can be put inside the box. We can easily fit a couple of bars of soap or several bars of candy inside.

The use of the measurement sheet in the construction process adds a new dimension to this box. By changing the value of $$x$$, the distance between the third crease on the measurement sheet and the nearest parallel crease, infinitely many boxes can be constructed in a very similar manner. All these boxes will have different dimensions.

## The Video

November 23, 2021 - 7:14am clandesman

This is a lovely box. Is the measuring sheet necessary? I folded the box by dividing the rectangular sheets into thirds. It worked beautifully. Also why are the pdf instructions not a uniform set of drawings? Odd to have instructions start off as drawings and end up as photographs. Note that the word "emerging" is used when the word "immersing" is intended. "It is about immersing oneself..." I really enjoyed folding this box!

November 26, 2021 - 9:42pm Awares

The measurement sheet adds some flexibility. The box you made has different dimensions than the one shown here. It's probably just as good. The measurement sheet makes the math more interesting as well.

November 26, 2021 - 9:56pm Awares

Thank you for catching the typo. Hopefully it will be fixed.

November 26, 2021 - 11:08pm jane.rosemarin

Thanks for pointing out the typo. It's fixed. The PDF diagrams were meant to be a reminder of how to fold the box rather than the primary instructions. Arsalan originally submitted the text and the video. The diagrams were compiled by me from the material on hand: Arsalan's drawings, and, where there were none, from screen shots of the video. - Jane Rosemarin, managing editor.

November 23, 2021 - 8:43am jmetzger

Great box! Just wanted to bring to your attention a small error in the diagrams text. In step 1 (3), it says fold right edge to center, when it should say fold right edge to fold made in previous step 1 (2).

November 23, 2021 - 9:00am jmetzger

Excellent box! Thanks so much for sharing!

November 23, 2021 - 10:04am jmetzger

Error Fixed! Thank you!