5-Pointed Star: Exact Method 1
Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay
Exact methods for creating 5-pointed stars depend on handy identities like cos 36 = (1+√ 5)/4 and cos 72 = (√ 5 -1)/4. Since the diagonal of a 1 x 2 rectangle is √ 5, it's easy to find cos 36 and cos 72 by repeatedly folding a square sheet of paper.
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| Start with a square piece of paper folded in half as shown. The crease is at bottom. |
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| Fold it in half again |
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| Unfold it to reveal the center crease |
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| Fold one half inward so the end of the paper meets the center crease |
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| Unfold it to reveal the new crease |
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| The key to the construction is that if the edge of the original paper is 1, the diagonal of one of the quarter strips is (√ 5)/4. If we can swing an arc from the lower end of the diagonal down to the lower edge of the paper, the distance from the lower right corner to the arc is (1+√ 5)/4, which equals cos(36) |
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| Fold the lower left corner up as shown and mark where the upturned bottom edge crosses the top edge. Or use a scrap of paper to measure and mark the distance. |
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| The paper unfolded. |
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| Fold the mark inward to the center line and crease the paper. |
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| Unfold the paper. The distance from the left-most crease to the center is [(1+√ 5)/4 -1/2)]/2 = (√ 5 -1)/8 = cos(72)/2 |
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| Fold the paper to bring the lower left corner over to the leftmost crease. Since the distance from the left-most crease to the center is cos(72)/2, and the upturned lower edge is 1/2, the angle of the upturned edge is 72 degrees and the diagonal crease is at a 36 degree angle. The obtuse angle formed by the diagonal crease and the right bottom edge is 180-36 = 144 degrees. |
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| Bisect the obtuse angle to create a 72 degree wedge. |
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| Bisect the wedge one last time to get a 36 degree angle. |
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| Cut the wedge as desired and unfold as shown below. |
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Created 22 March 2006, Last Update20 January 2020