Diagrams and Video: A Triangular Gift Box

by Arsalan Wares
Edited by Jane Rosemarin
Here is the box we will be making. See PDF diagrams. Also see the instructional video at the end of this article.

In this article we will learn to make a triangular origami box.

Origami may be an old art form, but it is thriving today. Ninety-eight percent of the innovations in origami came about in the last two percent of the art’s existence.1 Origami has the potential to play a crucial role in creative teaching and learning.2 More specifically, origami can provide an engaging context for conceptual understanding of significant mathematical ideas.3,4,5 Moreover, paper folding in general is mathematics in action.

The Construction Process

The box we will be making is shown above. We will need six rectangular sheets of the same dimensions. I used six 8½-by-11-inch sheets to make the box. This is a very malleable box; it can be made with A4 (210-by-297-mm) sheets, A5 (148-by-210-mm) sheets, U.S. legal (8½-by-14-inch) sheets and square sheets.

Three of the sheets will be used to make the top of the box and three will be used to make the bottom. Even though the top and the bottom portions of the box are identical, the top portion expands as the bottom portion is slid inside. Consequently, the top portion holds the bottom portion snugly, and the halves can also be separated easily. The video showing how the box is made is embedded at the end of this article. In the video, we make the top portion using three sheets. The bottom portion of the box is made in the same manner with three additional identical sheets.

Two printable PDFs created by the author. Left: California in pink and white. See PDF. Right: Blue Concentric Circles. See PDF.

The video is about 16 minutes long, but it will probably take about 60 minutes to make a decent box (Editor’s note: much less for a longtime folder). 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 emerging oneself sincerely in the box-making process in a relatively distraction-free environment. The box shown above was made with 65-pound cardstock. This is a sturdy box that can be reused many times. But I would encourage readers to make their first box with printer paper. Printer paper is easier to fold than 65-pound cardstock, especially for beginners.

geometry
The relationship between the dimensions of the box and the dimensions of the rectangular sheets.

Some Mathematics

Suppose each rectangular sheet that we started with was \( a \) by \( b \), with \(b \ge a \). Let \( h \) and \( l \) be the height and the length of the triangular base of the box. With the help of high-school geometry, we can establish the following: \( h = \frac{a}{4} \) and \( l = \frac{2b}{3} \) (see the figure above).

Since the constructed box is a prism with a triangular base and the triangular base is an equilateral triangle, the volume of the constructed box is: \[ \frac{\sqrt{3}ab^2}{36}. \] This is a useful formula because it allows us to calculate the volume of the box if we know the dimensions of each of the six rectangular sheets used. For instance, if \( a = 8½ \textrm{ inches} \) and \( b = 11 \textrm{ inches} \), the volume of the constructed box is: \[ \frac{\sqrt{3}(8.5)(11)^2}{36} \approx 49.48 \textrm{in}^3 . \]

The Video

Endnotes

1. Robert Lang, Origami Design Secrets, (Boca Raton, Fla. CRC Press, 2012). [back]

2. Kazuo Haga, (2008). Origamics: Mathematical Explorations Through Paper Folding, (Hackensack, N.J.: World Scientific, 2008) [back]

3. Arsalan Wares and Iwan Elstak, (2017) “Origami, geometry and art,” International Journal of Mathematics Education in Science and Technology 48, no. 2 (2017): 317–324. [back]

4. Arsalan Wares, “Mathematical Art or Artistic Mathematics?” Math Horizons 27, no. 3 (2020): 13–15, https://doi.org/10.1080/10724117.2019.1676557. [back]

5. Arsalan Wares, (2020b). “Mathematical Art and Artistic Mathematics,” International Journal of Mathematical Education in Science and Technology 51, no. 1 (2020): 151–156, https://doi.org/10.1080/0020739X.2019.1577996. [back]

Comments

As an exercise for the reader, show that L=2B/3.
Hint: In the diagram, L is the distance from the bottom of the paper to the horizontal crease.

Good exercise!