Star Origami

Year: 2017 Authors: Joy Hsiao

Core claim

The decagonal star’s modular pieces contribute about 35.54° each, so ten pieces nearly complete a circle and require accurate folding.

Topics

modular origami, angle calculation, GeoGebra, symmetry

Domains

geometry, trigonometry, algebra, origami, paper folding, visual arts, craft

Methods

hand calculation, GeoGebra construction, Pythagorean Theorem, Law of Sines

Media

square paper, GeoGebra, crease patterns, diagrams

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2017 Conference Proceedings

Star Origami

Joy Hsiao Dept. of Mathematics, Stuyvesant High School 345 Chambers Street, New York, NY 10282, USA jhsiao@schools.nyc.gov

Abstract

A modular pentagonal star, whose creator is unknown, and a modular decagonal star designed by Tomoko Fuse will be folded using five and ten pieces of square paper, respectively. Angle measures of the decagonal star will be calculated to show how well the modular pieces fit together. Does each piece contribute exactly $3 6^{\circ}$ to form a $36 0^{\circ}$ circle? Participants will calculate angle measures in two different ways: by hand using high school mathematics, and on GeoGebra, a web-based graphing tool.

Introduction

Figure 1: Pentagonal and decagonal stars

Star origami models are popular, especially around some holidays. They are often admired for their symmetries and repetitive patterns. Some star origami models are folded with one square piece of paper to form a four-point or eight-point star. Pentagonal paper may be folded to form a five-point star, and hexagonal paper may be folded to form a six-point star. Modular origami stars are made by folding multiple sheets of square paper and then assembled to form a multi-pointed star. It is therefore less obvious as for why square pieces of paper can be folded to form angle measures that meet perfectly to form a five-point or a ten-point star.

In this workshop, we will begin by folding both models of star origami. I learned how to fold the five-point modular star origami from a former student who learned it from her grandmother. At first, I was annoyed by how the pieces did not fit perfectly and questioned my folding skills. I then set out to prove that the design does not produce a perfect $7 2^{\circ}$ angle as required in a five-point star. As a result of this study, I noticed how the star pops slightly into 3D when assembled. The five-point star model was analyzed in my 2015 article in Mathematics Teacher [1]. I will focus on the ten-point star in this workshop.

The decagonal star designed by Tomoko Fuse is made from folding and assembling ten pieces of square paper. Unlike the five-point star, I was intrigued by how these modular pieces appear to fit together perfectly, and again, I set out to find out if the measure of the angle from each modular piece is exactly $3 6^{\circ}$ to better understand this model. During the workshop, we will calculate the angle measure by hand and with GeoGebra, an online graphing tool. Participants can follow along on their own computer or sit back and enjoy a demonstration.

Hsiao

As an educator, I highly recommend origami explorations where there are numerous opportunities for students to discover patterns, pose original questions, and apply their mathematical knowledge to find original solutions. In the process of finding angle or line segment measurements, all three branches of high school mathematics, algebra, geometry, and trigonometry, are almost always applied. In addition, in order to construct the crease pattern using a graphing tool, knowledge of the use of compass and straight-edge constructions and geometric properties are required. Once constructed, measurements can also be calculated on the graphing tool to confirm calculations done by hand. These exercises create real-life applications for geometric constructions and the use of mathematics in general. They provide many opportunities for students to be creative in finding their own solutions through various possible methods – pure mathematical calculations on paper, use of a graphing tool, or the actual folding of paper that often makes symmetries and congruence obvious. It is a fun way to put one’s mathematical knowledge to practice.

Diagrams

The diagrams below are shown from a mathematical perspective where bisected line segments and angles are indicated and each bisector is a crease line. The folding instructions only lead up to the establishment of the angle used to assemble the decagonal star – the angle that helps us determine the “fit” of this model. Complete instructions on folding the decagonal star can be found in Fuse’s book [2] or at this Bridges workshop.

In the diagrams below, solid line segments represent the edges of the square paper and dotted line segments represent a crease, a hidden crease, or an auxiliary line.

Star Origami

Figure 9: Crease $\overline{J K}$ intersects $\overline{I F}$ at $L$ . Bisect $∠ L GE$ (not drawn) by folding $\overline{GE}$ to overlap $\overline{G L}$ as shown in Figure 10

Figure 10: $\overline{GM}$ is the bisector of $∠ L GE$ , and it creates $∠ A GM$ which is half of one modular angle used to make the decagonal star

When ten modular pieces, as shown in Figure 10, are assembled, they make the following diagram:

Figure 11: Assembled decagonal star

Calculations

Let’s calculate the measure of $∠ A GM$ shown in figure 12 below. This is half of the angle from each of the modular pieces used to form the ten-point star. Without loss of generality, let the side length of the original square paper be 1. We can calculate the following:

Diagonal $A C = 2$ (see Figure 3)
Segments $C^{'} G = GF = 2 - 1$ (see Figure 6)
Segment $H G = \frac{2 - 1}{2}$ (see Figure 7)
Using the Pythagorean Theorem in right triangle, $△ H GF$ from Figure 12 on the right, we can find $H F = \frac{15 - 10 2}{2}$

Figure 12

Hsiao

In Figure 13, $\overline{L K}$ is the perpendicular bisector of $\overline{H F}$ . In $Δ H GF$ and $Δ K NF$ , by using right triangle trigonometry, we can find the measures of $∠ H FG$ and $∠ N K F$ as follows:

m ∠ H FG = tan^{- 1} (\frac{H G}{GF}) = tan^{- 1} (\frac{2 - 1}{2}) = 26.5 7^{\circ}

m ∠ N K F = 9 0^{\circ} - m ∠ H FG = 63.4 3^{\circ}

Figure 13

Since $\overline{L K}$ is the perpendicular bisector of $\overline{H F}$ , we can find the measures of $∠ N KH$ and $∠ HK G$ :

m ∠ N KH = m ∠ N K F = 63.4 3^{\circ} m ∠ HK G = 18 0^{\circ} - m ∠ HK F = 53.1 3^{\circ}

Figure 14

In right $Δ H G K$ , we can calculate $G K$ and $HK$ as follows:

G K = \frac{2 - 1}{2 t a n 53.1 3 ^{\circ}} = 0.1553 HK = \frac{2 - 1}{2 s i n 53.1 3 ^{\circ}} = 0.2589

Figure 15

Since $G K = 0.1553$ , $K F = GF - G K = 0.2589$

Figure 16

Star Origami

In $Δ K L F$ ,

m ∠ K L F = 18 0^{\circ} - 4 5^{\circ} - 63.4 3^{\circ} = 71.5 7^{\circ} .

Using the Law of Sines, we can find

K L = \frac{0.2589 \cdot sin 4 5 ^{\circ}}{sin 71.5 7 ^{\circ}} = 0.1930

L F = \frac{0.2589 \cdot sin 63.4 3 ^{\circ}}{sin 71.5 7 ^{\circ}} = 0.2441

Figure 17

In $Δ G L F$ , we can calculate side $G L$ by using the Law of Cosines:

G L = 0.244 1^{2} + 0.414 2^{2} - 2 \cdot 0.2441 \cdot 0.4142 \cdot cos 4 5^{\circ} = 0.2969

Figure 18

In $Δ G L K$ , we can use the Law of Sines to find

m ∠ L G K = sin^{- 1} (\frac{0.1930 \cdot s i n 116.5 6 ^{\circ}}{0.2969}) = 35.5 4^{\circ} .

Figure 19

Since $\overline{MG}$ bisects $∠ L GE$ ,

m ∠ L GM = \frac{18 0 ^{\circ} - 35.5 4 ^{\circ}}{2} = 72.2 3^{\circ} m ∠ L G H = 9 0^{\circ} - 35.5 4^{\circ} = 54.4 6^{\circ}

Figure 20

Hsiao

Finally,

m ∠ H GM (o r ∠ o) = m ∠ L GM (o r ∠ v) - m ∠ L G H (o r ∠ ξ) = 72.2 3^{\circ} - 54.4 6^{\circ} = 17.7 7^{\circ}

Figure 21

When ten modular pieces are assembled, we now have a better understanding of how they fit:

Figure 22: Assembled decagonal star with measurements

Conclusion

Each modular piece contributes an angle of $17.7 7^{\circ} \times 2$ , or $35.5 4^{\circ}$ , for a total of $355. 4^{\circ}$ when ten modular pieces are assembled together. This is quite close to the $36 0^{\circ}$ needed to make a complete circle. It allows little room to accommodate the thickness of the paper and inaccuracies in folding. These calculations show that this model has a tight fit and therefore accuracy in folding is important to properly assemble the pieces.

With a thoughtful selection of origami models, various origami activities can be readily accessible to students of all ages. Students can observe, hypothesize, and verify mathematical properties through folding. For students with more content knowledge, they can perform calculations by hand or constructions on a graphing tool to reinforce their understanding of mathematics. Through these activities, both mathematics and art are created and exercised on multiple levels and in multiple forms. From the beginning to the end, we practice the art of posing good questions, the art of problem solving, the art in the physical folding, and the art in the final product, the origami stars.

References

[1] Joy Hsiao, “Finding Fifths in Origami,” Mathematics Teacher, 109(1): 71-75, 2015. [2] Tomoko Fuse, Home Decorating with Origami, Japan Publications Trading Company, 2000.

Jusur / Bridges Research Atlas

Explorer

Star Origami

Star Origami

Core claim

Topics

Domains

Methods

Media

Paper text

Star Origami

Abstract

Introduction

Diagrams

Calculations

Conclusion

References