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Rigid Foldability of the Augmented Square Twist

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Rigid Foldability of the Augmented Square Twist Rigid foldability of the augmented square twist Thomas C. Hull∗ Michael T. Urbanski∗ Abstract Define the augmented square twist to be the square twist crease pattern with one crease added along a diagonal of the twisted square. In this paper we fully describe the rigid foldability of this new crease pattern. Specifically, the extra crease allows the square twist to rigidly fold in ways the original cannot. We prove that there are exactly four non-degenerate rigid foldings of this crease pattern from the unfolded state. 1 Introduction The classic square twist, shown in Figure1(a) where bold creases are mountains and non-bolds are valleys, is well-known in origami art from its use in the Kawasaki rose [Kasahara and Takahama 87] and in origami tessellations [Gjerde 08]. However, this is also an example of an origami crease pattern that is not rigidly-foldable, meaning that it cannot be folded along the creases into the flat, square-twist state without bending the faces of the crease pattern along the way. (See [Hull 13] for one proof of this.) In fact, this lack of rigid foldability has been used by researchers to study the mechanics of bistability in origami [Silverberg et al. 15]. Interestingly, there are ways one can change the mountain-valley assignment of the square twist to make it rigidly foldable; they require the inner diamond of Figure1(a) to be either MMVV, MVVV, or VMMM [Evans et al. 15]. 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sha1_base64="K2m1izUlPrk39prXdOjLEllAYzk=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBi8eKxhbaUDbbTbt0swm7k0IJ/QlePKh49R9589+4bXPQ1gcDj/dmmJkXplIYdN1vZ219Y3Nru7RT3t3bPzisHB0/mSTTjPsskYluh9RwKRT3UaDk7VRzGoeSt8LR7cxvjbk2IlGPOEl5ENOBEpFgFK30MO55vUrVrblzkFXiFaQKBZq9yle3n7As5gqZpMZ0PDfFIKcaBZN8Wu5mhqeUjeiAdyxVNOYmyOenTsm5VfokSrQthWSu/p7IaWzMJA5tZ0xxaJa9mfif18kwug5yodIMuWKLRVEmCSZk9jfpC80ZyokllGlhbyVsSDVlaNMp2xC85ZdXiV+v3dTc+8tqo16kUYJTOIML8OAKGnAHTfCBwQCe4RXeHOm8OO/Ox6J1zSlmTuAPnM8fcUmNYQ==</latexit> 1 arXiv:1809.04899v1 [math.MG] 13 Sep 2018 Figure 1: (a) The classic square twist with mountain-valley assignment. (b) The augmented square twist, with vertices and folding angles labeled. A logical question to ask is: Could we add creases to the classic square twist to make it rigidly foldable? The answer is, \Yes!" and the simplest way is to add a single crease along a diagonal of the inner square diamond, as shown in Figure1(b). In this paper we will prove the rigid-foldability ∗Western New England University, fthull, [email protected] 1 e2 ρ1 e 3 α π β ρ2 mode 1: − e1 β ρ3 π α − e4 ρ4 e2 ρ3 e 3 α π β ρ1 mode 2: − e1 β ρ2 π α − e4 ρ4 Figure 2: The two modes of a rigid folding of a degree-4 flat-foldable vertex. of this augmented square twist crease pattern and characterize its configuration space (i.e., its kinematics). 2 Preliminaries There are a number of tools that we will be using in this paper. One is the set of folding angle equations for a degree-4, flat-foldable vertex to rigidly fold and unfold [Evans et al. 15, Tachi and Hull 16]. If we let α and β be consecutive plane angles between the creases with α < β and α + β < π, and we let ρ1; : : : ; ρ4 be the folding angles as illustrated in Figure2, then there are equations for the mode 1 and mode 2 folding angles: ρ = ρ ; ρ = ρ in mode 1, ρ = ρ ; ρ = ρ in mode 2, 1 − 3 2 4 1 3 2 − 4 α+β α β ρ cos ρ ρ sin − ρ tan 1 = 2 tan 2 , and tan 2 = 2 tan 1 : 2 α β 2 2 α+β 2 cos −2 sin 2 For the degree-4 vertices in the augmented square twist, we have α = 45◦ and β = 90◦, which gives us 135◦ 45◦ cos sin − 45 2 2 ◦ p 45◦ = 135◦ = tan = 2 1: cos − 2 − sin 2 2 − Thus we get, for mode 1 and mode 2 respectively, ρ ρ ρ = 2 arctan((p2 1) tan 2 ) and ρ = 2 arctan((p2 1) tan 1 ): 1 − 2 2 − − 2 The other major tool we will use is Balkcom's procedure for deriving the kinematic equations for rigidly folding a vertex of any degree [Balkcom 04]. We'll apply this to the degree-5 vertices in the augmented square twist as follows: Impose a coordinate system where the unfolded paper is in 2 ( 1, 1, 0) <latexit sha1_base64="DYp3tUyARvWIOgz8LTdHmeevWQk=">AAAB7nicbVBNSwMxEJ31s9avqkcvwSJUqGVTBPVW8OKxgmsL7VKyabYNzWbXJCuUpX/CiwcVr/4eb/4b03YP2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHDzpOFWUejUWs2gHRTHDJPMONYO1EMRIFgrWC0c3Ubz0xpXks7804YX5EBpKHnBJjpXblHFdx1T3rlcpuzZ0BLROckzLkaPZKX91+TNOISUMF0bqD3cT4GVGGU8EmxW6qWULoiAxYx1JJIqb9bHbvBJ1apY/CWNmSBs3U3xMZibQeR4HtjIgZ6kVvKv7ndVITXvkZl0lqmKTzRWEqkInR9HnU54pRI8aWEKq4vRXRIVGEGhtR0YaAF19eJl69dl3DdxflRj1PowDHcAIVwHAJDbiFJnhAQcAzvMKb8+i8OO/Ox7x1xclnjuAPnM8fwgKN9g==</latexit>
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