Folding Polyominoes into (Poly)Cubes

We study the problem of folding a polyomino $P$ into a polycube $Q$, allowing faces of $Q$ to be covered multiple times. First, we define a variety of folding models according to whether the folds (a) must be along grid lines of $P$ or can divide squares in half (diagonally and/or orthogonally), (b) must be mountain or can be both mountain and valley, (c) can remain flat (forming an angle of $180^\circ$), and (d) must lie on just the polycube surface or can have interior faces as well. Second, we give all the inclusion relations among all models that fold on the grid lines of $P$. Third, we characterize all polyominoes that can fold into a unit cube, in some models. Fourth, we give a linear-time dynamic programming algorithm to fold a tree-shaped polyomino into a constant-size polycube, in some models. Finally, we consider the triangular version of the problem, characterizing which polyiamonds fold into a regular tetrahedron.Comment: 30 pages, 19 figures, full version of extended abstract that appeared in CCCG 2015. (Change over previous version: Fixed a missing reference.

Industrial product design by using two-dimensional material in the context of origamic structure and integrity

Thesis (Master)--Izmir Institute of Technology, Izmir, 2004Includes bibliographical references (leaves: 115)Text in English; Abstract: Turkish and English.xiii, 118 leavesThroughout the history of industrial product design, there have always been attempts to shape everyday objects from a single piece of semi-finished industrial materials such as plywood, sheet metal, plastic sheet and paper-based sheet. One of the ways to form these two-dimensional materials into three-dimensional products is bending following cutting. Similar concepts of this spatial transformation are encountered in the origami form, which has a planar surface in unfolded state, then transforms to a three-dimensional state by folding or by folding following cutting. If so, conceptually it may be useful to think of one-axis bending, which is a manufacturing technique, is somewhat similar to folding paper.In this regard, the studies in the scope of computational origami, which light the way for real-world problems such as how sheets of material will behave under stress, have applications especially in .manufacturing phase. of industrial product design.Besides manufacturing phase, origami design is also used as a product design tool either in .concept creating phase. (in the context of its concepts) or in 'form creating phase' (in the context of its design principles).In this thesis, the designing of industrial products, which are made from sheet material, is presented in a framework that considers the origami design. In the theoretical framework, evolutionary progression of origami design is discussed briefly in order to comprehend the situation of origami design in distinct application fields.Moreover, the elements, principles, basics of origami design and origamic structures are generally introduced. The theoretical framework is completed with the descriptions of the concepts on origami design and origamic structures. In the practical framework, typical applications that have origamic structures in distinct industrial product fields are exemplified. Furthermore, sheet materials and their bending process are taken up separately. By means of its excessive advantages, sheet metal bending is particularly emphasized. The practical framework is completed with several case studies base on sheet metal bending. Finally, the study is concluded with the evaluation of the origamic-structured product in respect of good design principles. Furthermore, designing by considering origami design is recommended to designer to design a good industrial product

Modeling and Programming Shape-Morphing Structured Media

Shape-morphing and self-propelled locomotion are examples of mechanical behaviors that can be "programmed" in structured media by designing geometric features at micro- and mesostructural length scales. This programmability is possible because the small-scale geometry often imposes local kinematic modes that are strongly favored over other deformations. In turn, global behaviors are influenced by local kinematic preferences over the extent of the structured medium and by the kinematic compatibility (or incompatibility) between neighboring regions of the domain. This considerably expands the design space for effective mechanical properties, since objects made of the same bulk material but with different internal geometry will generally display very different behaviors. This motivates pursuing a mechanistic understanding of the connection between small-scale geometry and global kinematic behaviors. This thesis addresses challenges pertaining to the modeling and design of structured media that undergo large deformations. The first part of the thesis focuses on the relation between micro- or mesoscale patterning and energetically favored modes of deformation. This is first discussed within the context of twisted bulk metallic glass ribbons whose edges display periodic undulations. The undulations cause twist concentrations in the narrower regions of the structural element, delaying the onset of material failure and permitting the design of structures whose deployment and compaction emerge from the ribbons' chirality. Following this discussion of a periodic system, we study sheets with non-uniform cut patterns that buckle out-of-plane. Motivated by computational challenges associated with the presence of geometric features at disparate length scales, we construct an effective continuum model for these non-periodic systems, allowing us to simulate their post-buckling behavior efficiently and with good accuracy. The second part of the thesis discusses ways to leverage the connection between micro/mesoscale geometry and energetically favorable local kinematics to create "programmable matter" that undergo prescribed shape changes or self-propelled locomotion when exposed to an environmental stimulus. We first demonstrate the capabilities of an inverse design method that automates the design of structured plates that morph into target 3D geometries over time-dependent actuation paths. Finally, we present devices made of 3D-printed liquid crystal elastomer (LCE) hinges that change shape and self-propel when heated.</p

Developing Design and Analysis Framework for Hybrid Mechanical-Digital Control of Soft Robots: from Mechanics-Based Motion Sequencing to Physical Reservoir Computing

The recent advances in the field of soft robotics have made autonomous soft robots working in unstructured dynamic environments a close reality. These soft robots can potentially collaborate with humans without causing any harm, they can handle fragile objects safely, perform delicate surgeries inside body, etc. In our research we focus on origami based compliant mechanisms, that can be used as soft robotic skeleton. Origami mechanisms are inherently compliant, lightweight, compact, and possess unique mechanical properties such as– multi-stability, nonlinear dynamics, etc. Researchers have shown that multi-stable mechanisms have applications in motion-sequencing applications. Additionally, the nonlinear dynamic properties of origami and other soft, compliant mechanisms are shown to be useful for ‘morphological computation’ in which the body of the robot itself takes part in performing complex computations required for its control. In our research we demonstrate the motion-sequencing capability of multi-stable mechanisms through the example of bistable Kresling origami robot that is capable of peristaltic locomotion. Through careful theoretical analysis and thorough experiments, we show that we can harness multistability embedded in the origami robotic skeleton for generating actuation cycle of a peristaltic-like locomotion gait. The salient feature of this compliant robot is that we need only a single linear actuator to control the total length of the robot, and the snap-through actions generated during this motion autonomously change the individual segment lengths that lead to earthworm-like peristaltic locomotion gait. In effect, the motion-sequencing is hard-coded or embedded in the origami robot skeleton. This approach is expected to reduce the control requirement drastically as the robotic skeleton itself takes part in performing low-level control tasks. The soft robots that work in dynamic environments should be able to sense their surrounding and adapt their behavior autonomously to perform given tasks successfully. Thus, hard-coding a certain behavior as in motion-sequencing is not a viable option anymore. This led us to explore Physical Reservoir Computing (PRC), a computational framework that uses a physical body with nonlinear properties as a ‘dynamic reservoir’ for performing complex computations. The compliant robot ‘trained’ using this framework should be able to sense its surroundings and respond to them autonomously via an extensive network of sensor-actuator network embedded in robotic skeleton. We show for the first time through extensive numerical analysis that origami mechanisms can work as physical reservoirs. We also successfully demonstrate the emulation task using a Miura-ori based reservoir. The results of this work will pave the way for intelligently designed origami-based robots with embodied intelligence. These next generation of soft robots will be able to coordinate and modulate their activities autonomously such as switching locomotion gait and resisting external disturbances while navigating through unstructured environments

Methods for Building Self-Folding Machines

Origami can turn a sheet of paper into complex three-dimensional shapes, and similar folding techniques can be used to build structures and mechanisms. However, folding by hand can be difficult and time consuming. This has led to the development of self-folding materials that transform themselves from flat sheets into 3D shapes by bending themselves along hinges. A variety of self-folding methods have been demonstrated at a variety of length scales, but they have not yet been used to build complex machines. This dissertation demonstrates that self-folding can produce functional machines with a new laminate we refer to as a shape memory composite. We characterize the behavior of this composite with models and experimental data, and use this information to develop design rules for self-folding. We apply these rules to create devices at multiple length scales, including a model crane, a crawling robot, a lamp, and a model ship.Engineering and Applied Sciences - Engineering Science

Geometry Synthesis and Multi-Configuration Rigidity of Reconfigurable Structures

Reconfigurable structures are structures that can change their shapes to change their functionalities. Origami-inspired folding offers a path to achieving shape changes that enables multi-functional structures in electronics, robotics, architecture and beyond. Folding structures with many kinematic degrees of freedom are appealing because they are capable of achieving drastic shape changes, but are consequently highly flexible and therefore challenging to implement as load-bearing engineering structures. This thesis presents two contributions with the aim of enabling folding structures with many degrees of freedom to be load-bearing engineering structures. The first contribution is the synthesis of kirigami patterns capable of achieving multiple target surfaces. The inverse design problem of generating origami or kirigami patterns to achieve a single target shape has been extensively studied. However, the problem of designing a single fold pattern capable of achieving multiple target surfaces has received little attention. In this work, a constrained optimization framework is presented to generate kirigami fold patterns that can transform between several target surfaces with varying Gaussian curvature. The resulting fold patterns have many kinematic degrees of freedom to achieve these drastic geometric changes, complicating their use in the design of practical load-bearing structures. To address this challenge, the second part of this thesis introduces the concept of multi-configuration rigidity as a means of achieving load-bearing capabilities in structures with multiple degrees of freedom. By embedding springs and unilateral constraints, multiple configurations are rigidly held due to the prestress between the springs and unilateral constraints. This results in a structure capable of rigidly supporting finite loads in multiple configurations so long as the loads do not exceed some threshold magnitude. A theoretical framework for rigidity due to embedded springs and unilateral constraints is developed, followed by a systematic method for designing springs to maximize the load-bearing capacity in a set of target configurations. An experimental study then validates theoretical predictions for a linkage structure. Together, the application of geometry synthesis and multi-configuration rigidity constitute a path towards engineering reconfigurable load-bearing structures.</p

Origami as a Tool for Mathematical Investigation and Error Modelling in Origami Construction

Origami is the ancient Japanese art of paper folding. It has inspired applications in industries ranging from Bio-Medical Engineering to Architecture. This thesis reviews ways in which Origami is used in a number of fields and investigates unexplored areas providing insight and new results which may lead to better understanding and new uses. The OSME conference series arguably covers most of the research activities in the field of Origami and its links to Science and Mathematics. The thesis provides a comprehensive review of the work that has been presented at these conferences and published in their proceedings. The mathematics of Origami has been explored before and much of the fundamental work in this field is presented in chapter 3. Here an attempt is made to push the bounds of this field by suggesting ways in which Origami can be used as a mathematical tool for in-depth exploration of non trivial problems. A particular problem we consider is the 4-colour theorem and its proof. Looking at some well known methods for producing angles and lengths mathematically the thesis also explores how accurate these might be. This leads to the surprisingly unstudied field of error modelling in Origami. Errors in folding processes have not previously been looked at from a mathematical point of view. The thesis develops a model for error estimation in crease patterns and a framework for error modelling in Origami applications. By introducing a standardised error into alignments, uniform error bounds for each of the one-fold constructions are generated. This defines a region in which a crease could lie in order to satisfy the alignments of a given fold within a specified tolerance. Analysis of this method on some examples provides insight into how this might be used in multi-fold constructions. An algorithm to that effect is introduced

Paper in Architecture:

Paper is a fascinating material that we encounter every day in different variants: tissues, paper towels, packaging material, wall paper or even fillers of doors. Despite radical changes in production technology, the material, which has been known to mankind for almost two thousand years, still has a natural composition, being made up of fibres of plant origin (particularly wood fibres). Thanks to its unique properties, relatively high compression strength and bending stiffness, low production costs and ease of recycling, paper is becoming more and more popular in many types of industry. Mass-produced paper products such as special paper, paperboard, corrugated cardboard, honeycomb panels, tubes and L- and U-shapes are suitable for use as a building material in the broad sense of these words – i.e., in design and architecture. Objects for everyday use, furniture, interior design elements and partitions are just a few examples of things in which paper can be employed. Temporary events such as festivals, exhibitions or sporting events like the Olympics require structures that only need to last for a limited period of time. When they are demolished after a few days or months, their leftovers can have a significant impact on the local environment. In the context of growing awareness of environmental threats and the efforts undertaken by local and international organisations and governments to counter these threats, the use of natural materials that can be recycled after their lifespan is becoming increasingly widespread. Paper and its derivatives fascinate designers and architects, who are always looking for new challenges and trying to meet the market’s demands for innovative and proecological solutions. Being a low-cost and readily available material, paper is suited to the production of emergency shelters for victims of natural and man-made disasters, as well as homeless persons. In order to gain a better understanding of paper’s potential in terms of architecture, its material properties were researched on a micro, meso and macro level. This research of the possible applications of paper in architecture was informed by two main research questions: What is paper and to what extent can it be used in architecture? What is the most suitable way to use paper in emergency architecture? To answer the first research question, fundamental and material research on paper and paper products had to be conducted. The composition of the material, production methods and properties of paper were researched. Then paper products with the potential to be used in architecture were examined. The history of the development of paper and its influence on civilisation helped the author gain a better understanding of the nature of this material, which we encounter in our lives every day. Research on objects for everyday use, furniture, pavilions and architecture realised in the last 150 years allowed the author to distinguish various types of paper design and paper architecture. Analysis of realised buildings in which paper products were used as structural elements and parts of the building envelope resulted in a wide array of possible solutions. Structural systems, types of connections between the various elements, impregnation methods and the functionalities and lifespan of different types of buildings were systematised. The knowledge thus collected allowed the author to conduct a further exploration of paper architecture in the form of designs and prototypes. To answer the second research question, the analysed case studies were translated into designs and prototypes of emergency shelters. During the research-by-design, engineering and prototyping phases, more than a dozen prototypes were built. The prototypes differed in terms of structural systems, used materials, connections between structural elements, impregnation methods, functionality and types of building. The three versions of the Transportable Emergency Cardboard House project presented in the final chapter form the author’s final answer to the second research question. Paper will never replace traditional building materials such as timber, concrete, steel, glass or plastic. It can, however, complement them to a significant degree

Paper in architecture: Research by design, engineering and prototyping

Paper is a fascinating material that we encounter every day in different variants: tissues, paper towels, packaging material, wall paper or even fillers of doors. Despite radical changes in production technology, the material, which has been known to mankind for almost two thousand years, still has a natural composition, being made up of fibres of plant origin (particularly wood fibres). Thanks to its unique properties, relatively high compression strength and bending stiffness, low production costs and ease of recycling, paper is becoming more and more popular in many types of industry. Mass-produced paper products such as special paper, paperboard, corrugated cardboard, honeycomb panels, tubes and L- and U-shapes are suitable for use as a building material in the broad sense of these words – i.e., in design and architecture. Objects for everyday use, furniture, interior design elements and partitions are just a few examples of things in which paper can be employed. Temporary events such as festivals, exhibitions or sporting events like the Olympics require structures that only need to last for a limited period of time. When they are demolished after a few days or months, their leftovers can have a significant impact on the local environment. In the context of growing awareness of environmental threats and the efforts undertaken by local and international organisations and governments to counter these threats, the use of natural materials that can be recycled after their lifespan is becoming increasingly widespread. Paper and its derivatives fascinate designers and architects, who are always looking for new challenges and trying to meet the market’s demands for innovative and proecological solutions. Being a low-cost and readily available material, paper is suited to the production of emergency shelters for victims of natural and man-made disasters, as well as homeless persons. In order to gain a better understanding of paper’s potential in terms of architecture, its material properties were researched on a micro, meso and macro level. This research of the possible applications of paper in architecture was informed by two main research questions: What is paper and to what extent can it be used in architecture? What is the most suitable way to use paper in emergency architecture? To answer the first research question, fundamental and material research on paper and paper products had to be conducted. The composition of the material, production methods and properties of paper were researched. Then paper products with the potential to be used in architecture were examined. The history of the development of paper and its influence on civilisation helped the author gain a better understanding of the nature of this material, which we encounter in our lives every day. Research on objects for everyday use, furniture, pavilions and architecture realised in the last 150 years allowed the author to distinguish various types of paper design and paper architecture. Analysis of realised buildings in which paper products were used as structural elements and parts of the building envelope resulted in a wide array of possible solutions. Structural systems, types of connections between the various elements, impregnation methods and the functionalities and lifespan of different types of buildings were systematised. The knowledge thus collected allowed the author to conduct a further exploration of paper architecture in the form of designs and prototypes. To answer the second research question, the analysed case studies were translated into designs and prototypes of emergency shelters. During the research-by-design, engineering and prototyping phases, more than a dozen prototypes were built. The prototypes differed in terms of structural systems, used materials, connections between structural elements, impregnation methods, functionality and types of building. The three versions of the Transportable Emergency Cardboard House project presented in the final chapter form the author’s final answer to the second research question. Paper will never replace traditional building materials such as timber, concrete, steel, glass or plastic. It can, however, complement them to a significant degree. &nbsp