Tomohiro TACHI

We give a necessary and sufficient condition that characterizes rigidly foldable single vertex crease patterns. We believe that this serves as the basic theorem for rigid-foldable origami, as an equivalent of Kawasaki's and Maekawa's theorems for flat-foldable origami. In addition, we introduce the concept of a forcing set for rigidly-foldable vertices. A forcing set is a subset of the creases whose folded state uniquely determines the folded state of the remaining creases.

We investigate the continuous rigid folding motion of an origami tessellation structure when their pattern is innitely tessellating a plane with periodic symmetry. We analytically and numerically describe the kinematics of triangular origami tessellation by assuming that the fold configuration, as well as the pattern, is periodic. Owing to the geometric constraints, the global forms of these tessellations are always restricted to cylindrical configurations unlike the folding of a finite open origami tessellation, which allows a double curved surface. In such triangulated patterns, the number of fold angles representing the conguration of the fundamentalfigure is equal to the number of equations from rigid origami constraints. However, this forms a two degrees of freedom mechanism instead of being statically determinate. One of the degrees of freedom represents the folding-unfolding motion, and the other represents the change in the orientation of the rolling axis.

- Tomohiro Tachi "Rigid Folding of Periodic Origami Tessellations", to appear in Origami
^{6}, 2015.

We describe a general family of curved-crease folding tessellations consisting of a repeating "lens" motif formed by two convex curved arcs. The third author invented the first such design in 1992, when he made both a sketch of the crease pattern and a vinyl model (pictured below). Curve fitting suggests that this initial design used circular arcs. We show that in fact the curve can be chosen to be any smooth convex curve without inflection point. We identify the ruling configuration through qualitative properties that a curved folding satisfies, and prove that the folded form exists with no additional creases, through the use of differential geometry.

- Erik D. Demaine, Martin L. Demaine, David A. Huffman, Duks Koschitz, Tomohiro Tachi "Characterization of Curved Creases and Rulings: Design and Analysis of Lens Tessellations" arXiv
- Erik D. Demaine, Martin L. Demaine, David A. Huffman, Duks Koschitz, Tomohiro Tachi "Characterization of Curved Creases and Rulings: Design and Analysis of Lens Tessellations" to appear in Origami
^{6}, 2015.

A novel origami cellular material based on a deployable cellular origami structure is described. The structure is bi-directionally flat-foldable in two orthogonal (x and y) directions and is relatively stiff in the third orthogonal (z) direction. While such mechanical orthotropicity is well known in cellular materials with extruded two dimensional geometry, the interleaved tube geometry presented here consists of two orthogonal axes of interleaved tubes with high interfacial surface area and relative volume that changes with fold-state. In addition, the foldability still allows for fabrication by a flat lamination process, similar to methods used for conventional expanded two dimensional cellular materials. This article presents the geometric characteristics of the structure together with corresponding kinematic and mechanical modeling, explaining the orthotropic elastic behavior of the structure with classical dimensional scaling analysis.

- Kenneth C Cheung, Tomohiro Tachi, Sam Calisch and Koryo Miura, "Origami interleaved tube cellular materials", Smart Materials and Structures, 23(9), 2014 doi:10.1088/0964-1726/23/9/094012

In this study, we show a family of multilayered rigid-foldable and flat-foldable vault structures, which can be designed by constructing rigid-foldable curved folded tubular arches and assembling the arches to construct cellular structures. The resulting vault structure form an overconstrained mechanism.

- Tomohiro Tachi, "Composite Rigid-Foldable Curved Origami Structure", in Proceedings of Transformables 2013, Seville, Spain, September 18-20, 2013. [pdf]

In this research, we propose a novel computational method to simulate and design origami whose form is governed by the equilibrium of forces from the elastic bending of each panel. In special, we explore statically indeterminate origami structure that can be manipulated by pin supporting finite number of vertices. The computational method is proposed so that we can interactively explore the design space of such origami form. The concept of kinematic origami tessellation based on bending of panels is introduced.

- Tomohiro Tachi, "Interactive Form-Finding of Elastic Origami", in Proceedings of IASS 2013, Wroclaw, Poland, September 23-27, 2013. [pdf]

In this research, we study a method to produce families of origami tessellations from given polyhedral surfaces. The resulting tessellated surfaces generalize the patterns proposed by Ron Resch and allow the construction of an origami tessellation that approximates a given surface. We will achieve these patterns by first constructing an initial configuration of the tessellated surfaces by separating each facets and inserting folded parts between them based on the local configuration. The initial configuration is then modified by solving the vertex coordinates to satisfy geometric constraints of developability, folding angle limitation, and local nonintersection. We propose a novel robust method for avoiding intersections between facets sharing vertices. Such generated polyhedral surfaces are not only applied to folding paper but also sheets of metal that does not allow 180◦ folding.

- Tomohiro Tachi, "Designing Freeform Origami Tessellations by Generalizing Resch's Patterns", Journal of Mechanical Design, Special Issue on Origami and Tessellation in Design, to appear.
- Tomohiro Tachi, "Freeform Origami Tessellations by Generalizing Resch's Patterns", Proceedings of ASME 2013 IDETC/CIE (Symposium on Origami-Based Engineering Design), DETC2013-12326, Portland, USA, August 4-7, 2013 [pdf]

collaboration with Koryo Miura

In this research, we present newly explored families of rigid-foldable cylinders and the cellular structures constructed from these cylinders; the families include zonogon extrusion cells, bi-directionally flat-foldable cells, and a novel type of cells, i.e., woven cylinder cells. We show the geometry of such structures to demonstrate their validity, their parametric design method, and their kinetic behaviors. These types of structures exhibit continuous rigid-foldability as well as flat-foldability in one or two directions; further, they have different kinetic properties that are potentially applicable for different purposes. The newly proposed woven cylinder cellular structure is a bi-directionally flat-foldable one-DOF rigid-foldable structure and has a distinctive geometric property: structural stiffness against compression in one of three directions.

- Tomohiro Tachi and Koryo Miura, "Rigid-Foldable Cylinders and Cells", Journal of the International Association for Shell and Spatial Structures (IASS), 53(4), pp. 217--226, 2012. [pdf](draft)

We investigate novel computational design method for infinitesimally and finitely foldable rigid origami based on solving a first-order folding mode, which can be represented by a reciprocal figure. We derive these graphical conditions from a matrix representation of rigid origami, and extend the conditions to cases when the surface includes holes. We propose an algorithm to obtain forms that satisfy the conditions and an interactive system to freely design infinitesimally foldable forms. We show design examples of shaky polyhedron and origami, and finitely foldable quadrivalent mesh origami. The graphical equivalence between Tensegrity and Origami is also mentioned.

We propose a novel interactive method for flexibly designing tensegrity structures under valid force equilibriums. Unlike previous form-finding techniques that aim to obtain a unique solution by fixing some parameters such as the lengths of elements and force densities, our method provides a design system that allows a user to continuously interact with the form within a multidimensional solution space. First, a valid initial form is generated by converting a given polygon mesh surface into a strut-and-cable network that approximates the mesh, and the form is then perturbed to attain an equilibrium state through a two-step optimization of both node coordinates and force densities. Then, the form can be freely transformed by using a standard 2D input device while the system updates the form on the fly based on the orthogonal projection of a deformation mode into the solution space. The system provides a flexible platform for designing a tensegrity form for use as a static architectural structure or a kinetic deployable system.

collaboration with Motoi Masubuchi, and Masaaki Iwamoto

We have been studying rigid origami structures. The following paper is a brief general introduction to rigid origami studies.

- Tomohiro Tachi, "Geoemtric Considerations for the Design of Rigid Origami Structures", in Proceedings of IASS Symposium 2010, Shanghai, China, pp. 771-782, 2010.[pdf]

- Rigid Origami Structure with Vacuumatics
- Designing One-DOF Mechanisms by Rationalizing Curved Folding
- Rigid-Foldable Cylindrical Polyhedra
- Freeform Rigid-Foldable Structure using Bidirectionally Flat-Foldable Planar Quadrilateral Mesh
- Thick Rigid Origami
- General Rigid-Foldable Quadrilateral Mesh Origami
- One-DOF Rigid-Foldable Tubes
- Rigid Origami Simulation

We show a novel design method of one-DOF deployable mechanism based on a space curve, through creating a curved folding and discretizing the folding into rigid origami. By interpreting constant angle curved folding as a flat-foldable quadrilateral mesh origami, we design novel irregular tessellated, cylindrical, and cellular flatly collapsible structures, whose behavior is easily controlled by space curves.

- Tomohiro Tachi, "One-DOF Rigid Foldable Structures from Space Curves", in Proceedings of the IABSE-IASS Symposium 2011, London, UK, September 20-23, 2011. [pdf]

collaboration with Gregory Epps

We propose a modeling method based on rationalizing curved folding in order to find the form variations of 1DOF origami mechanism.
We interact with a physical paper model of curved folding and then discretize a curved folding by identifying and fixing the rulings.
The discretized form is a rigid origami structure with at most one degree of freedom.
The form adjustment follows the discretization so that it is sure to realize a mechanism.
The workshop performed by the authors based on the proposed design method is reported.
The objective of the workshop was to utilize the 1DOF characteristic of discretized curved folds as a constraint in the design of dynamic architectural components.
The results showed the feasibility of the method and suggested a novel methodology for designing.

- Gregory Epps and Tomohiro Tachi, "One DOF linkage manipulation using rigid origami", workshop held at Advances in Architectural Geometry 2011, Wien, Austria, September 18-19, 2010.[url]
- Tomohiro Tachi and Gregory Epps, "Designing One-DOF Mechanisms by Rationalizing Curved Folding", in Proceedings of ALGODE 2011, Tokyo, Japan, March 14-16, 2011.[pdf]

collaboration with Koryo Miura

We present a family of rigid-foldable collapsible cylindrical polyhedra which is of great interest of structural engineering field. The symmetry operations in order to synthesize the cylindrical structures and their space filling tessellation are shown.

**presented at ISIS-Symmetry 2010, Gmuend, Austria**

- Koryo Miura and Tomohiro Tachi, "Synthesis of Rigid-Foldable Cylindrical Poyhedra", Journal of the International Society for the Interdisciplinary Study of Symmetry (ISIS-Symmetry), Special Issues for the Festival-Congress Gmuend, Austria, August 23-28, pp. 204-213, 2010. [pdf]

We present a computational design method to obtain collapsible variations of rigid-foldable surfaces, i.e., continuously and finitely transformable polyhedral surfaces, homeomorphic to disks and cylinders. Two novel techniques are proposed to design such surfaces: a technique for obtaining a freeform variation of a rigid-foldable and bidirectionally flat-foldable disk surface, which is a hybrid of generalized Miura-ori and eggbox patterns, and a technique to generalize the geometry of cylindrical surface using bidirectionally flat-foldable planar quadrilateral mesh by introducing additional constraints to keep the topology maintained throughout the continuous transformation.

- Tomohiro Tachi, "Freeform Rigid-Foldable Structure using Bidirectionally Flat-Foldable Planar Quadrilateral Mesh", in Advances in Architectural Geometry 2010, Vienna, Austria, September 18-21, pp. 87-102, 2010. [pdf]

We present a novel method to obtain a 3D freeform surface that can be constructed by folding a sheet of paper. Specifically, we provide a design system within which the user can intuitively vary a known origami pattern in 3D while preserving the developability and other optional conditions inherent in the original pattern. The system successfully provides designs of 3D origami that have not been realized thus far.

- Tomohiro Tachi, "Freeform Variations of Origami", Journal for Geometry and Graphics, 14(2), pp. 203-215, 2010.
- Software is available now.

collaboration with Erik D. Demaine

We consider the construction of points within a square of paper by drawing a line (crease) through an existing point with angle equal to an integer multiple of 22.5 degrees, which is a very restricted form of the Huzita-Justin origami construction axioms. We show that a point can be constructed by a sequence of such operations if and only if its coordinates are both of the form (m + n*sqrt(2))/(2^l) for integers m, n, and l, and that all such points can be constructed efficiently. This theorem explains how the restriction of angles to integer multiples of 22.5 degrees forces point coordinates to degenerate into a reasonably controlled grid, i.e., Maekawa-gami.

Abstract [url]

We propose a novel geometric method to implement a general rigid-foldable origami as a structure composed of tapered or non-tapered (constant-thickness) thick plates and hinges without changing the mechanical behavior from that of the ideal rigid origami.

Presented at 5OSME, Singapore, July 2010.

In general, a quadrilateral mesh surface does not enable a continuous rigid motion because an overconstrained system is constructed. We generalize the geometric condition for enabling one-DOF rigid motion in general quadrilateral mesh origami without the trivial repeating symmetry. This yields a variety of unexplored generalized shapes of quadrilateral mesh origami that preserve finite rigid-foldability in addition to developability and flat-foldability.

Presented at IASS Symposium 2009, Valencia 28 September - 2 October 2009, Universidad Politecnica de Valencia, Spain

for details refer

- Tomohiro Tachi, "Generalization of Rigid Foldable Quadrilateral Mesh Origami," Journal of the International Association for Shell and Spatial Structures, 50(3), pp. 173-179, 2009.
- Tomohiro Tachi, "Generalization of Rigid Foldable Quadrilateral Mesh Origami," Proceedings of the IASS Symposium 2009, pp. 2287-2294, 2009. pdf (1.4MB)

We present a novel cylindrical deployable structure and variations of its design with the following characteristics:

- Flat-foldable: The shape flattens into a compact 2D configuration.
- Rigid-foldable: Each element does not deform throughout the transformation.
- One-DOF: The mechanism has exactly one degree of freedom.
- Thick: Facets can be substituted with thick or multilayered panels without introducing the distortion of elements.

Presented at IASS Symposium 2009, Valencia 28 September - 2 October 2009, Universidad Politecnica de Valencia, Spain

for details refer

- Tomohiro Tachi, "One-DOF Cylindrical Deployable Structures with Rigid Quadrilateral Panels," in Proceedings of the IASS Symposium 2009, 2009, pp. 2295-2305. pdf (3.2MB)

This is a novel interactive system that enables a user to design crumpled papers.

Software will be available soon.

The first practical method for "origamizing" or obtaining the folding pattern that folds a single sheet of material into a given polyhedral surface without any cut is shown.

For the details

- Tomohiro Tachi, "Origamizing Polyhedral Surfaces," IEEE Transactions on Visualization and Computer Graphics, 16(2), pp. 298-311, 2010. pdf (1.2MB)
- Tomohiro Tachi, "3D Origami Design based on Tucking Molecule" in Origami^4:Proceedings of 4OSME, pp. 259-272, 2009.
pdf(Draft) (1.3MB)
folded models and screenshot

Software is available here

The method for calculating the kinematics of rigid origami from general crease pattern is presented.

- "Simulation of Rigid Origami" in Origami^4:Proceedings of 4OSME, pp. 175-187, 2009. pdf(Draft) (0.4MB)
- Power Point Presentation (Part) (6MB)
- Software is available here

I propose a method for making a smooth and comprehensible origami animation from crease pattern to folded base, by adding and adjusting crease lines on an origami model.

Poster(PDF)I propose a method for restoring Spectral Power Distribution (SPD) data from RGB image of skylight and calculating reflected color of synthetic objects lit by skylight. The algorithm is based on basis functions of skylight spectra given by light scattering model in atmosphere so that measurement of the SPD is not necessary. The method can be used to implement real-time environment mapping. Precise simulation of lighting with skylight enables designer to interactively design the color of an outdoor visual environment.

Abstract(PDF)The reflected color by this method is more bluish than the color calculated by just multiplying RGBs, because skylight's spectrum width is small in visible range.