By Robert J. Lang (ed.)
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Additional resources for Origami 4: Fourth International Meeting of Origami Science, Mathematics, and Education
2 Hausdorff Dimension of the Pyramid Curve The calculation of the Hausdorﬀ dimension dimH is generally diﬃcult. But in this case, it is relatively easy and the Pyramid curve turns out to be a fractal set. Let C be a Pyramid curve of base length and height 1. C is self-similar, because there exist similarity transformations f (x, y) = 1 1 x, y , g(x, y) = 2 2 1 1 1 − x, y , h(x, y) = 2 2 1 1 x, 1 − y 2 2 such that C = f (C) ∪ g(C) ∪ h(C). Take an open set A as shown in Figure 11. ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 36 I.
5(lv)(rv], vvvv m : outer Beam,front : m 30 modules 12 modules Figure 10. Variations of modular. ”. , strap, diagonal, and beam. A shape of a module that belongs to the Celes family is uniquely described by this three-line formula, which we call the crease pattern formula for the module. 9 Variations of Modules and Assembly One of the merits of using this formula is that it leads to automatic design of a module directly from its symbol. Figure 10 shows some models from the Celes family, along with the names that I have given them.
Beware the diﬃculty of the twist. Both top and bottom of this fraction tend toward 0 as α tends toward π2 , so we may use L’Hˆ opital’s method from calculus to determine this limit. Thus, lim h = lim n→∞ α→π/2 1 −2 sin(α) + (π − 2α) cos(α) = . −2 cos(α) − (3π − 2α) sin(α) π ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ Constructing Regular n-gonal Twist Boxes 5 49 Conclusion In this paper we have shown that for any n, we can construct the crease pattern in Figure 1 and use it to build an n-gonal twist box. Many origami enthusiasts have doubtless wished for easier ways to construct twist boxes with odd numbers of sides.