by Arsalan Wares
Edited by Jane Rosemarin

In this article, we will learn to fold an origami box with a hexagonal base. The box is made from four identical rectangular sheets. We will also discuss some mathematics related to the box.

Origami is an old art form that teaches us how to appreciate beauty in simple things, how to accept imperfections, how to embrace impermanence, how to enjoy solitude, how to play with creative ideas and how to think mathematically. Above all, origami teaches us how to transfer complex ideas from one mind to another.

## 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. 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 bottom portion of the box with two origami sheets and the measurement sheet. The top portion is made in a similar manner, with two additional identical origami sheets and the same measurement sheet.

The video is about 13 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. After all, it is a gift box! The box shown above was made with 20-pound regular printing paper. However, we can also use 65-pound cardstock to make a sturdy version that can be reused many times. Cardstock sheets of two contrasting colors can be used to make stunningly beautiful boxes. 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

This box is very malleable. It can be made with 8½-by-11-inch sheets (U.S. Letter), 8½-by-14-inch (U.S. Legal) sheets, and 210-by-297-mm (A4) sheets. If we use five a by b ( a < b) rectangular sheets, then the height of the constructed box will be a/4, and the length of each side of the hexagonal base of the box will be b/4. The box shown above was made from 8½-by-11-inch sheets. Therefore, it has a height of 2⅛ inches, and the length of each side of the hexagonal base is 2¾ in. Alternately, if we use 8½-by-14-inch sheets, the height of the constructed box will be 2⅛ inches and the length of each side of the hexagonal base will be 3½ inches. We can also show that the base of the box is a regular hexagon. That is, each interior angle of the hexagonal base is 120°, and all the six sides are congruent to one another.

Readers may wonder why I chose to use a measurement sheet. The figure above shows (see the video at time mark 4:54) the three slant creases that were created with the help of the measurement sheet (shown in the figure above by the three green line segments). One may wonder why I didn’t create a crease along the dotted line and use it instead of the edge of the measurement sheet. There are two reasons. Firstly, a crease along the dotted line couldn’t be used to make the third slant crease (represented by the green line segment closest to the bottom). Secondly, a crease along the dotted line would leave “frivolous” crease marks on the hexagonal base of the box. I personally don’t find this aesthetically pleasing.

One final note, no actual dollar bills were harmed during the construction of the box shown in the picture or the video!