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Branches of Triangulated Origami Near the Unfolded State

Chen, Bryan Gin-ge, Santangelo, Christian D.·2018·Physical Review X
DOI 10.1103/physrevx.8.011034
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Abstract

Origami structures can potentially lead to the development of self-assembling devices, and understanding the many ways that such a structure can end up in an undesired configuration is essential to that goal. New theoretical work analyzes an origami structure composed of triangles and characterizes the various ways it can unfold.

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Figures

FIG. 1 A neighborhood of the unfolded state in the configuration space of a two-vertex origami structure (inset) projected onto the fold angles of 3 folds (thick lines). This was computed by solving numerically the length constraint equations (Eq. 1). Locally, there are 4 branches, labeled I-IV, each a one-dimensional configuration space, all intersecting at a single point: the flat, unfolded configuration. Supplementary Movies 1-4 show animations of these branches. We did not attempt to compute the global structure of the configuration space, e.g. how the branches join.
FIG. 1 A neighborhood of the unfolded state in the configuration space of a two-vertex origami structure (inset) projected onto the fold angles of 3 folds (thick lines). This was computed by solving numerically the length constraint equations (Eq. 1). Locally, there are 4 branches, labeled I-IV, each a one-dimensional configuration space, all intersecting at a single point: the flat, unfolded configuration. Supplementary Movies 1-4 show animations of these branches. We did not attempt to compute the global structure of the configuration space, e.g. how the branches join.CC-BY · From the PDF
FIG. 2 (a) An origami in the unfolded state; the vector u n represents the in-plane displacement of vertex n . (b) A cross-sectional side view near vertex n ; the scalar h n gives the vertical displacement of vertex n . (c) A small deformation of this origami away from the unfolded state.
FIG. 2 (a) An origami in the unfolded state; the vector u n represents the in-plane displacement of vertex n . (b) A cross-sectional side view near vertex n ; the scalar h n gives the vertical displacement of vertex n . (c) A small deformation of this origami away from the unfolded state.CC-BY · From the PDF
FIG. 3 (a) Extracting a single internal vertex from a larger origami structure. (b) The 'wheel stress' for each single vertex, represented as either extensional or compressional arrows, is also present as a self stress in the larger origami structure of (a). Indeed, such wheel stresses form a basis for the space of self stresses of a generic unfolded triangulated origami, thus the dimensionality of that space is equal to the number of internal vertices. The labels in (b) are associated with Eqs. (7).
FIG. 3 (a) Extracting a single internal vertex from a larger origami structure. (b) The 'wheel stress' for each single vertex, represented as either extensional or compressional arrows, is also present as a self stress in the larger origami structure of (a). Indeed, such wheel stresses form a basis for the space of self stresses of a generic unfolded triangulated origami, thus the dimensionality of that space is equal to the number of internal vertices. The labels in (b) are associated with Eqs. (7).CC-BY · From the PDF
FIG. 4 (a) The configuration space of a five-fold origami vertex structure, for the crease pattern in (c), where the three unlabeled vertices are pinned to the xy -plane. The axes correspond to the vertical displacements of the three vertices labeled 4-6. (b) The eigendeformation of the negative eigenvalue would, on its own, give the vertex positive Gaussian curvature. (c) The planar crease pattern with the three labeled vertices, 4-6.
FIG. 4 (a) The configuration space of a five-fold origami vertex structure, for the crease pattern in (c), where the three unlabeled vertices are pinned to the xy -plane. The axes correspond to the vertical displacements of the three vertices labeled 4-6. (b) The eigendeformation of the negative eigenvalue would, on its own, give the vertex positive Gaussian curvature. (c) The planar crease pattern with the three labeled vertices, 4-6.CC-BY · From the PDF
FIG. 5 (a) A 5-fold origami vertex in a popped up configuration; the vertical line depicts the 'upwards' normal of the origami surface. (b) The same vertex in a popped down configuration.
FIG. 5 (a) A 5-fold origami vertex in a popped up configuration; the vertical line depicts the 'upwards' normal of the origami surface. (b) The same vertex in a popped down configuration.CC-BY · From the PDF
FIG. 6 An example of a crease pattern containing a fold joining two external vertices. Such folds do not couple to any others and we will not consider any crease patterns containing these (except for the individual butterflies in Section IV). This crease pattern also contains a 'triangulated triangle.' All of the red folds above the upper left diagonal of the square lie within a triangle, hence the three vertices interior to the triangle are rigid even though they each have four folds.
FIG. 6 An example of a crease pattern containing a fold joining two external vertices. Such folds do not couple to any others and we will not consider any crease patterns containing these (except for the individual butterflies in Section IV). This crease pattern also contains a 'triangulated triangle.' All of the red folds above the upper left diagonal of the square lie within a triangle, hence the three vertices interior to the triangle are rigid even though they each have four folds.CC-BY · From the PDF
FIG. 7 A view of the configuration space shown in Fig. 1 embedded in the space of height coordinates, rather than fold-angle coordinates, with the four folded configurations from branches I-IV superimposed. Only the second-order approximations to the branches are plotted here; since at this order they consist of intersecting lines, the height units are arbitrary. The upper / lower sides of the representative configurations have been colored blue / red, respectively, to make the popping at each vertex more evident. The space is divided into four wedge-shaped 3D chambers (unshaded, red, purple, blue) corresponding to the four possible pairs ((+ , +) , (+ , -) , ( -, -) , ( -, +)) of signs of the two quantities s 5 = e 5 · h and s 6 = e 6 · h , where e 5 , e 6 (not drawn) are the negative-eigenvalue eigenvectors corresponding to the internal vertices 5,6, respectively. Thus for example, the unshaded chamber (+ , +) consists of all height deformations h where vertices 5 and 6 are both popped up. The viewpoint has been chosen carefully to be 'edge-on' to the planes s 5 = 0 and s 6 = 0. From this angle, the two planes are projected onto the lines separating the colored regions in the figure, and the line s 5 = s 6 = 0 along which these planes intersect becomes the line passing through the origin that is normal to the plane of the figure.
FIG. 7 A view of the configuration space shown in Fig. 1 embedded in the space of height coordinates, rather than fold-angle coordinates, with the four folded configurations from branches I-IV superimposed. Only the second-order approximations to the branches are plotted here; since at this order they consist of intersecting lines, the height units are arbitrary. The upper / lower sides of the representative configurations have been colored blue / red, respectively, to make the popping at each vertex more evident. The space is divided into four wedge-shaped 3D chambers (unshaded, red, purple, blue) corresponding to the four possible pairs ((+ , +) , (+ , -) , ( -, -) , ( -, +)) of signs of the two quantities s 5 = e 5 · h and s 6 = e 6 · h , where e 5 , e 6 (not drawn) are the negative-eigenvalue eigenvectors corresponding to the internal vertices 5,6, respectively. Thus for example, the unshaded chamber (+ , +) consists of all height deformations h where vertices 5 and 6 are both popped up. The viewpoint has been chosen carefully to be 'edge-on' to the planes s 5 = 0 and s 6 = 0. From this angle, the two planes are projected onto the lines separating the colored regions in the figure, and the line s 5 = s 6 = 0 along which these planes intersect becomes the line passing through the origin that is normal to the plane of the figure.CC-BY · From the PDF
FIG. 8 (a) Crease pattern of a triangulated square with V i = 5 with a pair of branches where the MV assignments to the edges are identical. Edges colored red and blue represent mountain and valley folds, respectively. Vertices with red and blue outlines represent vertices that are popped up and popped down in both branches, respectively. The state of the green vertex (labeled 7) distinguishes the two branches; i.e. it is popped up in one branch and popped down in the other. (b), (c) give images of folded configurations on the two branches. Supplementary Movies 5-6 show animations of the folding motions along these branches from the unfolded state.
FIG. 8 (a) Crease pattern of a triangulated square with V i = 5 with a pair of branches where the MV assignments to the edges are identical. Edges colored red and blue represent mountain and valley folds, respectively. Vertices with red and blue outlines represent vertices that are popped up and popped down in both branches, respectively. The state of the green vertex (labeled 7) distinguishes the two branches; i.e. it is popped up in one branch and popped down in the other. (b), (c) give images of folded configurations on the two branches. Supplementary Movies 5-6 show animations of the folding motions along these branches from the unfolded state.CC-BY · From the PDF
FIG. 9 (a) Crease pattern of a triangulated square with V i = 5 and a pair of branches with coinciding MV assignments and vertex sign patterns, with coloring as in Fig. 8. There is a short fold between vertices 5 and 7 which is a valley fold. In this pair of examples, all vertices are popped the same way in both branches. (b), (c) give images of folded configurations on the two branches. Supplementary Movies 78 show animations of the folding motions along these branches from the unfolded state.
FIG. 9 (a) Crease pattern of a triangulated square with V i = 5 and a pair of branches with coinciding MV assignments and vertex sign patterns, with coloring as in Fig. 8. There is a short fold between vertices 5 and 7 which is a valley fold. In this pair of examples, all vertices are popped the same way in both branches. (b), (c) give images of folded configurations on the two branches. Supplementary Movies 78 show animations of the folding motions along these branches from the unfolded state.CC-BY · From the PDF
FIG. 10 (color) (a) An illustration of the distancemaximization property of the unfolded state for a 'butterfly'. (b) An H1 triangulated origami with V i = 4. One possible reduction sequence is depicted, with butterflies and their associated boundary degree-3 vertices colored and labeled in order: (1) red, (2) orange, (3) yellow, (4) green, and ending on (5) the blue seed.
FIG. 10 (color) (a) An illustration of the distancemaximization property of the unfolded state for a 'butterfly'. (b) An H1 triangulated origami with V i = 4. One possible reduction sequence is depicted, with butterflies and their associated boundary degree-3 vertices colored and labeled in order: (1) red, (2) orange, (3) yellow, (4) green, and ending on (5) the blue seed.CC-BY · From the PDF
FIG. 11 The spherical polygon spanned by a single vertex. The angle between a fold and the ˆ z axis is ψ i , the planar angle between adjacent folds in the plane is α i,i +1 while the angle between adjacent folds at the north pole is β i,i +1 . Note that ∑ i β i,i +1 = 2 π .
FIG. 11 The spherical polygon spanned by a single vertex. The angle between a fold and the ˆ z axis is ψ i , the planar angle between adjacent folds in the plane is α i,i +1 while the angle between adjacent folds at the north pole is β i,i +1 . Note that ∑ i β i,i +1 = 2 π .CC-BY · From the PDF
FIG. 12 Trigonometric diagram for verification of force balance at spoke vertex i along the direction parallel to the spoke. The 'crossed ladder' identity relates the lengths of the three parallel dashed lines, 1 /A +1 /B = 1 /h .
FIG. 12 Trigonometric diagram for verification of force balance at spoke vertex i along the direction parallel to the spoke. The 'crossed ladder' identity relates the lengths of the three parallel dashed lines, 1 /A +1 /B = 1 /h .CC-BY · From the PDF
FIG. 13 Portion of the force polygon for the hub vertex considered during the induction. Force balance at the hub vertex is equivalent to the closure of the force polygon. Each edge in the figure is a vector parallel to the spoke edges whose length is equal to the magnitude of the stress component in that edge. Here we only show the forces along spokes i -1 , i, i +1 and how they change if spoke i were to be removed.
FIG. 13 Portion of the force polygon for the hub vertex considered during the induction. Force balance at the hub vertex is equivalent to the closure of the force polygon. Each edge in the figure is a vector parallel to the spoke edges whose length is equal to the magnitude of the stress component in that edge. Here we only show the forces along spokes i -1 , i, i +1 and how they change if spoke i were to be removed.CC-BY · From the PDF
FIG. 14 The probability distribution for the angles between branch vectors in random origami with V i = 6 (black) compared to randomly distributed points on a 6-dimensional sphere (gray). The bin sizes are π/ 32. The solid gray line is an analytical prediction using Eq. 5 of Ref. [38].
FIG. 14 The probability distribution for the angles between branch vectors in random origami with V i = 6 (black) compared to randomly distributed points on a 6-dimensional sphere (gray). The bin sizes are π/ 32. The solid gray line is an analytical prediction using Eq. 5 of Ref. [38].CC-BY · From the PDF
FIG. 15 (color) The probability distribution that any two pairs of branches will be a given angle apart for random origami with V = 2 - 8 vertices. The bin sizes are π/ 32.
FIG. 15 (color) The probability distribution that any two pairs of branches will be a given angle apart for random origami with V = 2 - 8 vertices. The bin sizes are π/ 32.CC-BY · From the PDF

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Cites

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Research works printed but not linked: 8

  1. T. Tachi, 2010, Proceedings of the Tenth International Association for Shell and Spatial Structures (IASS) Symposium
  2. K. Miura, 1980, Proceedings of the 31st Congress of the International Astronautical. Federation
  3. E. D. Demaine, 2016, Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium
  4. I. Streinu, 2005, Proceedings of the Japanese Conference on Discrete and Computational Geometry (JCDCG), Tokyo 2004, Revised Selected Papers
  5. H. Gluck, 1975, Proceedings of the Conference on Geometric Topology, Park City, UT, 1974
  6. T.-S. Tay, 1985, Struct. Top., 11, 21
  7. A. Nixon, 2014, Rigidity and Symmetry
  8. T. Cai, 2013, J. Mach. Learn. Res., 14, 1837

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Metadata

Authors
Chen, B. G.-G., & Santangelo, C. D.
Venue
Physical Review X
Publisher
American Physical Society (APS)
Published
2018-02-27
Volume
8
Issue
1
ISSN
2160-3308
DOI
10.1103/physrevx.8.011034
Cited by
21
References
44
Language
en
Licence
creativecommons.org/licenses/by/4.0/ (vor)
Funders
  • National Science Foundation (NSF PHY-1125915)
  • Division of Emerging Frontiers in Research and Innovation (EFRI ODISSEI-1240441)

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Chen, B. G.-G., & Santangelo, C. D. (2018). Branches of Triangulated Origami Near the Unfolded State. Physical Review X. https://doi.org/10.1103/physrevx.8.011034

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