Folding a Strip to Converge to an Equilateral Triangle

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Begin with a long strip of gummed tape. Fold it up in any way you like, then unfold it, fold it down, and repeat as in the photo. The number of steps can be selected with the slider.

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This gives a sequence of isosceles triangles with summits at and . The sequence of triangles converges to an equilateral triangle.

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Contributed by: Izidor Hafner (February 2016)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Folding down means bisecting the angle . If the initial angle at is , then the angle at is , which rapidly approaches as . Since that is the angle at the top of an isosceles triangle, the other two angles approach as well. The same argument is valid for the triangles [1, pp. 11–12, 57].

Reference

[1] P. Hilton and J. Pedersen, Build Your Own Polyhedra, Menlo Park, CA: Addison-Wesley, 1994.



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