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

Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^\circ, 360^\circ$\}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure

Fun with Fonts: Algorithmic Typography

Over the past decade, we have designed six typefaces based on mathematical theorems and open problems, specifically computational geometry. These typefaces expose the general public in a unique way to intriguing results and hard problems in hinged dissections, geometric tours, origami design, computer-aided glass design, physical simulation, and protein folding. In particular, most of these typefaces include puzzle fonts, where reading the intended message requires solving a series of puzzles which illustrate the challenge of the underlying algorithmic problem.Comment: 14 pages, 12 figures. Revised paper with new glass cane font. Original version in Proceedings of the 7th International Conference on Fun with Algorithm

Flat Folding an Unassigned Single-Vertex Complex (Combinatorially Embedded Planar Graph with Specified Edge Lengths) without Flat Angles

A foundational result in origami mathematics is Kawasaki and Justin's simple, efficient characterization of flat foldability for unassigned single-vertex crease patterns (where each crease can fold mountain or valley) on flat material. This result was later generalized to cones of material, where the angles glued at the single vertex may not sum to $360^\circ$. Here we generalize these results to when the material forms a complex (instead of a manifold), and thus the angles are glued at the single vertex in the structure of an arbitrary planar graph (instead of a cycle). Like the earlier characterizations, we require all creases to fold mountain or valley, not remain unfolded flat; otherwise, the problem is known to be NP-complete (weakly for flat material and strongly for complexes). Equivalently, we efficiently characterize which combinatorially embedded planar graphs with prescribed edge lengths can fold flat, when all angles must be mountain or valley (not unfolded flat). Our algorithm runs in $O(n \log^3 n)$ time, improving on the previous best algorithm of $O(n^2 \log n)$.Comment: 17 pages, 8 figures, to appear in Proceedings of the 38th International Symposium on Computational Geometr

Mechanisms and states of self-stress of planar trusses using graphic statics, part II: Applications and extensions

This paper extends the overview (Mitchell et al. [11]) relating graphic statics and reciprocal diagrams to linear algebra-based matrix structural analysis. Focus is placed on infinitesimal mechanisms, both in-plane (linkage) and out-of-plane (polyhedral Airy stress functions). Each self-stress in the original diagram corresponds to an out-of-plane polyhedral mechanism. Decomposition into sub-polyhedra leads to a basis set of reciprocal figures which may then be linearly combined. This leads to an intuitively-appealing approach to the identification of states of self-stress for use in structural design, and to a natural “structural algebra” for use in structural optimisation. A 90° rotation of the sub-reciprocal generated by any sub-polyhedron leads to the displacement diagram of an in-plane mechanism. Any self-stress in the original thus corresponds to an in-plane mechanism of the reciprocal, summarised by the equation s = M* (where s is the number of states of self-stress in one figure, and M* is the number of in-plane mechanisms, including rigid body rotation, in the other). Since states of self-stress correspond to out-of-plane polyhedral mechanisms, this leads to a form of “conservation of mechanisms” under reciprocity. It is also shown how external forces may be treated via a triple-layer Airy stress function, consisting of a structural layer, a load layer, and a layer formed by coordinate vectors of the structural perimeter

Morphing Contact Representations of Graphs

We consider the problem of morphing between contact representations of a plane graph. In a contact representation of a plane graph, vertices are realized by internally disjoint elements from a family of connected geometric objects. Two such elements touch if and only if their corresponding vertices are adjacent. These touchings also induce the same embedding as in the graph. In a morph between two contact representations we insist that at each time step (continuously throughout the morph) we have a contact representation of the same type. We focus on the case when the geometric objects are triangles that are the lower-right half of axis-parallel rectangles. Such RT-representations exist for every plane graph and right triangles are one of the simplest families of shapes supporting this property. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs. We study piecewise linear morphs, where each step is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. We provide a polynomial-time algorithm that decides whether there is a piecewise linear morph between two RT-representations of a plane triangulation, and, if so, computes a morph with a quadratic number of linear morphs. As a direct consequence, we obtain that for 4-connected plane triangulations there is a morph between every pair of RT-representations where the "top-most" triangle in both representations corresponds to the same vertex. This shows that the realization space of such RT-representations of any 4-connected plane triangulation forms a connected set