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.

The Milli-Motein: A self-folding chain of programmable matter with a one centimeter module pitch

The Milli-Motein (Millimeter-Scale Motorized Protein) is ca chain of programmable matter with a 1 cm pitch. It can fold itself into digitized approximations of arbitrary three-dimensional shapes. The small size of the Milli-Motein segments is enabled by the use of our new electropermanent wobble stepper motors, described in this paper, and by a highly integrated electronic and mechanical design. The chain is an interlocked series of connected motor rotors and stators, wrapped with a continuous flex circuit to provide communications, control, and power transmission capabilities. The Milli-Motein uses off-the-shelf electronic components and fasteners, and custom parts fabricated by conventional and electric discharge machining, assembled with screws, glue, and solder using tweezers under a microscope. We perform shape reconfiguration experiments using a four-segment Milli-Motein. It can switch from a straight line to a prescribed shape in 5 seconds, consuming 2.6 W power during reconfiguration. It can hold its shape indefinitely without power. During reconfiguration, a segment can lift the weight of one but not two segments as a horizontal cantilever.United States. Defense Advanced Research Projects Agency. Programmable Matter ProgramUnited States. Defense Advanced Research Projects Agency. Maximum Mobility and Manipulation (M3) ProgramUnited States. Army Research Office (Grant W911NF-08-1-0254)United States. Army Research Office (Grant W911NF-11-1-0096)Massachusetts Institute of Technology. Center for Bits and Atom

Unfolding some classes of polycubes

An unfolding of a polyhedron is a cutting along its surface such that the surface remains connected and it can be flattened to the plane without any overlap. An edge- unfolding is a restricted kind of unfolding, we are only allowed to cut along the edges of the faces of the polyhedron. A polycube is a special case of orthogonal polyhedron formed by glueing several unit cubes together face-to-face. In the case of polycubes, the edges of all cubes are available for cuts in edge-unfolding. We focus on one-layer polycubes and present several algorithms to unfold some classes of them. We show that it is possible to edge-unfold any one-layer polycube with cubic holes, thin horizontal holes and separable rectangular holes. The question of edge-unfolding general one-layer polycubes remains open. We also briefly study some classes of multi-layer polycubes. 1Rozklad mnohostěnu je tvořen řezy jeho povrchu takovými, že rozřezaný povrch je možné rozložit do roviny, aniž by vznikl překryv. Hranový rozklad je omezený typ roz- kladu, ve kterém je povolené řezy vést jen po hranách mnohostěnu. Kostičkový mno- hostěn je speciální druh mnohostěnu, který je tvořen jednotkovými krychlemi slepenými k sobě celými stěnami. V případě kostičkových mnohostěnů můžeme v hranovém roz- kladu řezat po hranách všech jednotkových krychlí. V této práci se zabýváme zejména jednovrstvými kostičkovými mnohostěny a popíšeme několik algoritmů pro rozklad růz- ných speciálních tříd. Ukážeme, že je možné hranově rozložit jednovrstvé krychličkové mnohostěny s krychlovými dírami, tenkými horizontálními dírami a oddělitelnými ob- délníkovými dírami. Otázka hranového rozkladu obecných jednovrstvých krychličkových zůstává otevřena. Také se krátce zabýváme rozklady některých tříd vícevrstvých krych- ličkových mnohostěnů. 1Department of Applied MathematicsKatedra aplikované matematikyMatematicko-fyzikální fakultaFaculty of Mathematics and Physic

Tile-based Pattern Design with Topology Control

International audiencePatterns with desired aesthetic appearances and physical structures are ubiquitous.However, such patterns are challenging to produce - manual authoring requires significant expertise and efforts while automatic computation lacks sufficient flexibility and user control.We propose a method that automatically synthesizes vector patterns with visual appearance and topological structures designated by users via input exemplars and output conditions.The input can be an existing vector graphics design or a new one manually drawn by the user through our interactive interface.Our system decomposes the input pattern into constituent components (tiles) and overall arrangement (tiling).The tile sets are general and flexible enough to represent a variety of patterns, and can produce different outputs with user specified conditions such as size, shape, and topological properties for physical manufacturing

Growing machines

Thesis (Ph. D.)--Massachusetts Institute of Technology, School of Architecture and Planning, Program in Media Arts and Sciences, 2004.Includes bibliographical references.construction is developed in three dimensions. It is similarly shown that right-angled tetrahedrons, when folded from an edge-connected string, can generate any three dimensional structure where the primitive pixel (or voxel) is a rhombic hexahedron. This construction also suggests a concept of 3D completeness for assembly, somewhat analogous to the concept of Turing completeness in computation. In combination, these pieces of work suggest that a manufacturing system based on four tiles, with seven states per tile, is capable of self-replication of arbitrary 3D structure by copying, then folding, bit strings of those tiles where the desired structure is encoded in the tile sequence.Biological systems are replete with examples of high complexity structures that have "self assembled," or more accurately, programmatically assembled from many smaller, simpler components. By comparison, the fabrication systems engineered by humans are typically top down, or subtractive, processes where systems of limited complexity are carved from bulk materials. Self-assembly to date has resembled crystallization more than it has the programmatic assembly of complex or useful structures--these systems are information limited. This thesis explores the programming of self-assembling systems by the introduction of small amounts of state to the sub-units of the assembly. A six-state, kinematic, conformational latching component is presented that is capable of self-replicating bit strings of two shape differentiated versions of the same component where the two variants represent the 0 and 1 bits. Individual units do not assemble until a string is introduced to the assembly environment to be copied. Electro-mechanical state machine emulators were constructed. Operating on an air table, the units demonstrated logic limited aggregation, or error-preventing assembly, as well as autonomous self-replication of bit strings. A new construction was developed that demonstrates that any two dimensional shape composed of square pixels can be deterministically folded from a linear string of vertex-connected square tiles. This non-intersecting series of folds implies a 'resolution' limit of four tiles per pixel. It is shown that four types of tiles, patterned magnetically, is sufficient to construct any shape given sequential folding. The construction was implemented to fold the letters 'M I T' from sequences of the 4 tile types. An analogousSaul Thomas Griffith.Ph.D

DNA Tile Self-Assembly for 3D-Surfaces: Towards Genus Identification

We introduce a new DNA tile self-assembly model: the Surface Flexible Tile Assembly Model (SFTAM), where 2D tiles are placed on host 3D surfaces made of axis-parallel unit cubes glued together by their faces, called polycubes. The bonds are flexible, so that the assembly can bind on the edges of the polycube. We are interested in the study of SFTAM self-assemblies on 3D surfaces which are not always embeddable in the Euclidean plane, in order to compare their different behaviors and to compute the topological properties of the host surfaces. We focus on a family of polycubes called order-1 cuboids. Order-0 cuboids are polycubes that have six rectangular faces, and order-1 cuboids are made from two order-0 cuboids by substracting one from the other. Thus, order-1 cuboids can be of genus 0 or of genus 1 (then they contain a tunnel). We are interested in the genus of these structures, and we present a SFTAM tile assembly system that determines the genus of a given order-1 cuboid. The SFTAM tile assembly system which we design, contains a specific set Y of tile types with the following properties. If the assembly is made on a host order-1 cuboid C of genus 0, no tile of Y appears in any producible assembly, but if C has genus 1, every terminal assembly contains at least one tile of Y. Thus, for order-1 cuboids our system is able to distinguish the host surfaces according to their genus, by the tiles used in the assembly. This system is specific to order-1 cuboids but we can expect the techniques we use to be generalizable to other families of shapes

Shape Design and Optimization for 3D Printing

In recent years, the 3D printing technology has become increasingly popular, with wide-spread uses in rapid prototyping, design, art, education, medical applications, food and fashion industries. It enables distributed manufacturing, allowing users to easily produce customized 3D objects in office or at home. The investment in 3D printing technology continues to drive down the cost of 3D printers, making them more affordable to consumers. As 3D printing becomes more available, it also demands better computer algorithms to assist users in quickly and easily generating 3D content for printing. Creating 3D content often requires considerably more efforts and skills than creating 2D content. In this work, I will study several aspects of 3D shape design and optimization for 3D printing. I start by discussing my work in geometric puzzle design, which is a popular application of 3D printing in recreational math and art. Given user-provided input figures, the goal is to compute the minimum (or best) set of geometric shapes that can satisfy the given constraints (such as dissection constraints). The puzzle design also has to consider feasibility, such as avoiding interlocking pieces. I present two optimization-based algorithms to automatically generate customized 3D geometric puzzles, which can be directly printed for users to enjoy. They are also great tools for geometry education. Next, I discuss shape optimization for printing functional tools and parts. Although current 3D modeling software allows a novice user to easily design 3D shapes, the resulting shapes are not guaranteed to meet required physical strength. For example, a poorly designed stool may easily collapse when a person sits on the stool; a poorly designed wrench may easily break under force. I study new algorithms to help users strengthen functional shapes in order to meet specific physical properties. The algorithm uses an optimization-based framework — it performs geometric shape deformation and structural optimization iteratively to minimize mechanical stresses in the presence of forces assuming typical use scenarios. Physically-based simulation is performed at run-time to evaluate the functional properties of the shape (e.g., mechanical stresses based on finite element methods), and the optimizer makes use of this information to improve the shape. Experimental results show that my algorithm can successfully optimize various 3D shapes, such as chairs, tables, utility tools, to withstand higher forces, while preserving the original shape as much as possible. To improve the efficiency of physics simulation for general shapes, I also introduce a novel, SPH-based sampling algorithm, which can provide better tetrahedralization for use in the physics simulator. My new modeling algorithm can greatly reduce the design time, allowing users to quickly generate functional shapes that meet required physical standards

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

Complementing the shape group method : assessing chirality

The Shape Group method is a powerful tool in the analysis of the shape of molecules, and in the correlation of molecular shape features to molecular properties in Quantitative Shape-Activity Relationship (QShAR) studies. However, the main disadvantage inherent in the method is that mirror image molecules are considered to be "exactly" similar. As such, the method requires a complementary chirality measure to allow for complete analysis where chirality is involved. In this work, two methods of creating chirality measures to complement the Shape Group method are presented. The first is based upon the assigning of handedness values to each array point of the computer file that contains specific property information and uses the parallels between a lattice animal inscribed in a Jordan curve, and the array points inscribed in an isodensity contour. Each array point can then be treated as a face-labelled cube, which is often a chiral object that can have an assigned handedness value. Grouping of these handedness values allows for the creation of chirality measures. In the second method, the Shape Group method is applied to electron density representations created by subtracting one fragmentary electron density from others and analysing the shape similarities of the resultant difference densities. With both methods, chirality information that is already embedded within the shape descriptions of electron density representations is emphasized. The Shape Group method and the developed chirality measures are then used to simply correlate the shape and chirality of the stereogenic carbon of molecules to optical rotation and rotational strengths of various classes of molecules