Discrete Differential Geometry

This is the collection of extended abstracts for the 26 lectures and the open problem session at the fourth Oberwolfach workshop on Discrete Differential Geometry

Mathematical Surprises

This is open access book provides plenty of pleasant mathematical surprises. There are many fascinating results that do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics. This book presents a selection of these topics including the mathematical formalization of origami, construction with straightedge and compass (and other instruments), the five- and six-color theorems, a taste of Ramsey theory and little-known theorems proved by induction. Among the most surprising theorems are the Mohr-Mascheroni theorem that a compass alone can perform all the classical constructions with straightedge and compass, and Steiner's theorem that a straightedge alone is sufficient provided that a single circle is given. The highlight of the book is a detailed presentation of Gauss's purely algebraic proof that a regular heptadecagon (a regular polygon with seventeen sides) can be constructed with straightedge and compass. Although the mathematics used in the book is elementary (Euclidean and analytic geometry, algebra, trigonometry), students in secondary schools and colleges, teachers, and other interested readers will relish the opportunity to confront the challenge of understanding these surprising theorems

Mathematical Surprises

This is open access book provides plenty of pleasant mathematical surprises. There are many fascinating results that do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics. This book presents a selection of these topics including the mathematical formalization of origami, construction with straightedge and compass (and other instruments), the five- and six-color theorems, a taste of Ramsey theory and little-known theorems proved by induction. Among the most surprising theorems are the Mohr-Mascheroni theorem that a compass alone can perform all the classical constructions with straightedge and compass, and Steiner's theorem that a straightedge alone is sufficient provided that a single circle is given. The highlight of the book is a detailed presentation of Gauss's purely algebraic proof that a regular heptadecagon (a regular polygon with seventeen sides) can be constructed with straightedge and compass. Although the mathematics used in the book is elementary (Euclidean and analytic geometry, algebra, trigonometry), students in secondary schools and colleges, teachers, and other interested readers will relish the opportunity to confront the challenge of understanding these surprising theorems

Enabling New Functionally Embedded Mechanical Systems Via Cutting, Folding, and 3D Printing

Traditional design tools and fabrication methods implicitly prevent mechanical engineers from encapsulating full functionalities such as mobility, transformation, sensing and actuation in the early design concept prototyping stage. Therefore, designers are forced to design, fabricate and assemble individual parts similar to conventional manufacturing, and iteratively create additional functionalities. This results in relatively high design iteration times and complex assembly strategies

Origami fold as algebraic graph rewriting

AbstractWe formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewriting system (O,↬), where O is the set of abstract origamis and ↬ is a binary relation on O, that models fold. An abstract origami is a structure (Π,∽,≻), where Π is a set of faces constituting an origami, and ∽ and ≻ are binary relations on Π, each representing adjacency and superposition relations between the faces.We then address representation and transformation of abstract origamis and further reasoning about the construction for computational purposes. We present a labeled hypergraph of origami and define fold as algebraic graph transformation. The algebraic graph-theoretic formalism enables us to reason about origami in two separate domains of discourse, i.e. pure combinatorial domain where symbolic computation plays the main role and geometrical domain R×R. We detail the program language for the algebraic graph rewriting and graph rewriting algorithms for the fold, and show how fold is expressed by a set of graph rewrite rules

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 surfaces for kinetic architecture

This thesis departs from the conviction that spaces that can change their formal configuration through movement may endow buildings of bigger versatility. Through kinetic architecture may be possible to generate adaptable buildings able to respond to different functional solicitations in terms of the used spaces. The research proposes the exploration of rigidly folding origami surfaces as the means to materialize reconfigurable spaces through motion. This specific kind of tessellated surfaces are the result of the transformation of a flat element, without any special structural skill, into a self-supporting element through folds in the material, which gives them the aptitude to undertake various configurations depending on the crease pattern design and welldefined rules for folding according to rigid kinematics. The research follows a methodology based on multidisciplinary, practical experiments supported on digital tools for formal exploration and simulation. The developed experiments allow to propose a workflow, from concept to fabrication, of kinetic structures made through rigidly folding regular origami surfaces. The workflow is a step-by-step process that allows to take a logical path which passes through the main involved areas, namely origami geometry and parameterization, materials and digital fabrication and mechanisms and control. The investigation demonstrates that rigidly folding origami surfaces can be used as dynamic structures to materialize reconfigurable spaces at different scales and also that the use of pantographic systems as a mechanism associated to specific parts of the origami surface permits the achievement of synchronized motion and possibility of locking the structure at specific stages of the folding.A presente tese parte da convicção de que os espaços que são capazes de mudar a sua configuração formal através de movimento podem dotar os edifícios de maior versatilidade. Através da arquitectura cinética pode ser possível a geração de edifícios adaptáveis, capazes de responder a diferentes solicitações funcionais, em termos do espaço utilizado. Esta investigação propõe a exploração de superfícies de origami, dobráveis de forma rígida, como meio de materialização de espaços reconfiguráveis através de movimento. Este tipo de superfícies tesseladas são o resultado da transformação de um elemento plano, sem capacidade estrutural que, através de dobras no material, ganha propriedades de auto-suporte. Dependendo do padrão de dobragem e segundo regras de dobragem bem definidas de acordo com uma cinemática rígida, a superfície ganha a capacidade de assumir diferentes configurações. A investigação segue uma metodologia baseada em experiências práticas e multidisciplinares apoiada em ferramentas digitais para a exploração formal e simulação. Através das experiências desenvolvidas é proposto um processo de trabalho, desde a conceptualização à construção, de estruturas cinéticas baseadas em superfícies dobráveis de origami rígido de padrão regular. O processo de trabalho proposto corresponde a um procedimento passo-apasso que permite seguir um percurso lógico que atravessa as principais áreas envolvidas, nomeadamente geometria do origami e parametrização, materiais e fabricação digital e ainda mecanismos e controle. A dissertação demonstra que as superfícies de origami dobradas de forma rígida podem ser utilizadas como estruturas dinâmicas para materializar espaços reconfiguráveis a diferentes escalas. Demonstra ainda que a utilização de sistemas pantográficos como mecanismos associados a partes específicas da superfície permite atingir um movimento sincronizado e a possibilidade de bloquear o movimento em estados específicos da dobragem

One Tile to Rule Them All: Simulating Any Tile Assembly System with a Single Universal Tile

In the classical model of tile self-assembly, unit square tiles translate in the plane and attach edgewise to form large crystalline structures. This model of self-assembly has been shown to be capable of asymptotically optimal assembly of arbitrary shapes and, via information-theoretic arguments, increasingly complex shapes necessarily require increasing numbers of distinct types of tiles. We explore the possibility of complex and efficient assembly using systems consisting of a single tile. Our main result shows that any system of square tiles can be simulated using a system with a single tile that is permitted to flip and rotate. We also show that systems of single tiles restricted to translation only can simulate cellular automata for a limited number of steps given an appropriate seed assembly, and that any longer-running simulation must induce infinite assembly

Edge-unfolding almost-flat convex polyhedral terrains

Thesis (M. Eng.)--Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2013.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 97-98).In this thesis we consider the centuries-old question of edge-unfolding convex polyhedra, focusing specifically on edge-unfoldability of convex polyhedral terrain which are "almost at" in that they have very small height. We demonstrate how to determine whether cut-trees of such almost-at terrains unfold and prove that, in this context, any partial cut-tree which unfolds without overlap and "opens" at a root edge can be locally extended by a neighboring edge of this root edge. We show that, for certain (but not all) planar graphs G, there are cut-trees which unfold for all almost-at terrains whose planar projection is G. We also demonstrate a non-cut-tree-based method of unfolding which relies on "slice" operations to build an unfolding of a complicated terrain from a known unfolding of a simpler terrain. Finally, we describe several heuristics for generating cut-forests and provide some computational results of such heuristics on unfolding almost-at convex polyhedral terrains.by Yanping Chen.M.Eng

On the Nature of Students\u27 Digital Mathematical Performances

In this study I investigate the nature of digital mathematical performances (DMPs) produced by elementary school students (Grades 4-6). A DMP is a multimodal text/narrative (e.g., a video) in which one uses the performance arts to communicate mathematical ideas. I analyze twenty-two DMPs available at the Math + Science Performance Festival in 2008. Assuming a sociocultural/postmodern perspective with emphasis on multimodality, my focus is on the role of the arts and technology in shaping students’ mathematical communication and thinking. Methodologically, I employ qualitative case studies, along with video analysis. I conduct a descriptive analysis of each DMP using Boorstin’s (1990) categories of what makes good films, focusing on surprises, sense-making, emotions, and visceral sensations. I also conduct a cross-case analysis using Boorstin’s categories and the mathematical processes and strands of the Ontario Curriculum. The multimodal nature of DMP is one of its most significant pedagogic attributes. Mathematics is traditionally communicated through print-based texts, but the production of DMPs is an alternative that engages students in conceiving multimodal narratives. The playfulness offers scenarios for students’ collaboration, creativity, and imagination. By making DMPs available online, students share their ideas in a public and social environment, beyond the classrooms. Most of the DMPs only explore Geometry and offer opportunities to experience some surprises, sense-making, emotions, and visceral sensations. The lack of focus on other strands (e.g., Algebra) may be seen as a reflection on what (and how) students are (or not) learning in their classes. The production of conceptual DMPs is a rare event, although I acknowledge that I analyzed only DMPs of the first year of the Festival, that is, students did not have examples or references to produce their DMPs. Some DMPs potentially explore conceptual mathematical surprises, but they appear to have gaps in terms of sense-making. The use of the arts and technologies does not guarantee the mathematical conceptuality of DMPs. This study contributes to mathematics education with an exploratory discussion about how mathematical ideas can be (a) communicated and represented as multimodal texts at the elementary school level and (b) seen through a performance arts lens. The study also points out directions about the pedagogic components for conceiving conceptual DMPs in terms of the performance arts and the components of the Ontario Curriculum