Computability and Tiling Problems
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Computational Design Framework 3D Graphic Statics
Computational Design Framework for 3D Graphic Statics 3D Graphic for Computational Design Framework Computational Design Framework for 3D Graphic Statics Juney Lee Juney Lee Juney ETH Zurich • PhD Dissertation No. 25526 Diss. ETH No. 25526 DOI: 10.3929/ethz-b-000331210 Computational Design Framework for 3D Graphic Statics A thesis submitted to attain the degree of Doctor of Sciences of ETH Zurich (Dr. sc. ETH Zurich) presented by Juney Lee 2015 ITA Architecture & Technology Fellow Supervisor Prof. Dr. Philippe Block Technical supervisor Dr. Tom Van Mele Co-advisors Hon. D.Sc. William F. Baker Prof. Allan McRobie PhD defended on October 10th, 2018 Degree confirmed at the Department Conference on December 5th, 2018 Printed in June, 2019 For my parents who made me, for Dahmi who raised me, and for Seung-Jin who completed me. Acknowledgements I am forever indebted to the Block Research Group, which is truly greater than the sum of its diverse and talented individuals. The camaraderie, respect and support that every member of the group has for one another were paramount to the completion of this dissertation. I sincerely thank the current and former members of the group who accompanied me through this journey from close and afar. I will cherish the friendships I have made within the group for the rest of my life. I am tremendously thankful to the two leaders of the Block Research Group, Prof. Dr. Philippe Block and Dr. Tom Van Mele. This dissertation would not have been possible without my advisor Prof. Block and his relentless enthusiasm, creative vision and inspiring mentorship. -
Bottom-Up Self-Assembly Based on DNA Nanotechnology
nanomaterials Review Bottom-Up Self-Assembly Based on DNA Nanotechnology 1, 1, 1 1 1,2,3, Xuehui Yan y, Shujing Huang y, Yong Wang , Yuanyuan Tang and Ye Tian * 1 College of Engineering and Applied Sciences, State Key Laboratory of Analytical Chemistry for Life Science, Nanjing University, Nanjing 210023, China; [email protected] (X.Y.); [email protected] (S.H.); [email protected] (Y.W.); [email protected] (Y.T.) 2 Shenzhen Research Institute of Nanjing University, Shenzhen 518000, China 3 Chemistry and Biomedicine Innovation Center, Nanjing University, Nanjing 210023, China * Correspondence: [email protected] These authors contributed equally to this work. y Received: 9 September 2020; Accepted: 12 October 2020; Published: 16 October 2020 Abstract: Manipulating materials at the atomic scale is one of the goals of the development of chemistry and materials science, as it provides the possibility to customize material properties; however, it still remains a huge challenge. Using DNA self-assembly, materials can be controlled at the nano scale to achieve atomic- or nano-scaled fabrication. The programmability and addressability of DNA molecules can be applied to realize the self-assembly of materials from the bottom-up, which is called DNA nanotechnology. DNA nanotechnology does not focus on the biological functions of DNA molecules, but combines them into motifs, and then assembles these motifs to form ordered two-dimensional (2D) or three-dimensional (3D) lattices. These lattices can serve as general templates to regulate the assembly of guest materials. In this review, we introduce three typical DNA self-assembly strategies in this field and highlight the significant progress of each. -
Rational Design of DNA Nanoarchitectures Udo Feldkamp* and Christof M
Reviews U. Feldkamp and C. M. Niemeyer DOI: 10.1002/anie.200502358 Nanoscience Rational Design of DNA Nanoarchitectures Udo Feldkamp* and Christof M. Niemeyer* Keywords: DNA · materials science · nanostructures · self-assembly · supramolecular chemistry Angewandte Chemie 1856 www.angewandte.org 2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2006, 45, 1856 – 1876 Angewandte DNANanoarchitectures Chemie DNA has many physical and chemical properties that make it a From the Contents powerful material for molecular constructions at the nanometer length scale. In particular, its ability to form duplexes and other 1. Introduction 1857 secondary structures through predictable nucleotide-sequence- 2. General Considerations of DNA- directed hybridization allows for the design of programmable Sequence Design 1858 structural motifs which can self-assemble to form large supra- molecular arrays, scaffolds, and even mechanical and logical 3. One-Dimensional DNA Strands for nanodevices. Despite the large variety of structural motifs used Assembly and Immobilization of Non- Nucleic Acid Compounds 1859 as building blocks in the programmed assembly of supra- molecular DNA nanoarchitectures, the various modules share 4. Design and Assembly of DNA Motifs 1860 underlying principles in terms of the design of their hierarchical configuration and the implemented nucleotide sequences. This 5. Three-Dimensional Structures from DNA 1866 Review is intended to provide an overview of this fascinating and rapidly growing field of research from the structural design 6. Applications of DNA Nanoarchitectures 1868 point of view. 7. Conclusions and Perspectives 1872 1. Introduction combination of distinct ssDNA and dsDNA elements within an artificial DNA motif allows structural building blocks to be Biomolecules, such as proteins and nucleic acids, possess designed with tailored flexibility and rigidity. -
How to Construct Convex Polyominoes on DNA Wang Tiles ?
How to construct convex polyominoes on DNA Wang tiles ? F. De Carli ∗ A. Frosini ∗ S. Rinaldi ∗ L. Vuillon † 1 Introduction The goal of this article is to present a method to construct various classes of convex polyominoes using DNA Wang tiles. The construction is based on some results of [4], where the authors present a coding of convex polyominoes in terms of two dimensional languages, by means of tiling systems [12]. Adopting the same formalism we used in [4], and applying an algorithm by L. De Prophetis and S. Varricchio [11], we are able to transform the tiles of the tiling systems for convex polyominoes to labelled Wang tiles. The last step of the construction is based on a conversion from labelled Wang tiles into DNA Wang tiles [1], which gives an effective way to construct nano structures with convex polyomino shapes. The possibility of constructing polyominoes by means of DNA Wang tiles seems interesting since polyominoes are simple and important discrete structures that appear in several problems related to theoretical computer science and discrete mathematics. In the half century, since Solomon Golomb used the term in his seminal article [13], the study of polyominoes has proved a fertile topic of research. By this period in the mid-1950s, it was clearly a timely notion in discrete models, as the increasingly influential work of Neville Temperley, on problems drawn from statistical mechanics and molecular dynamics [16], and of John Hammersely, dealing with percolation [14], bear witness. More recent years have seen the treatment of numerous related problems, such as the problem of covering a polyomino by rectangles [5] or problems of tiling regions by polyominoes [2, 6]. -
Efficient Methods
Efficient Methods for Tile-Based Synthesis and Computational Photography DISSERTATION zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften an der Universitat¨ Konstanz Fachbereich Informatik & Informationswissenschaft vorgelegt von Johannes Peter Kopf Tag der mundlichen¨ Prufung:¨ 27. Oktober 2008 1. Referent: Prof. Dr. Oliver Deussen 2. Referent: Prof. Dr. Dietmar Saupe 3. Referent: Prof. Dr. Hans-Peter Seidel Abstract This thesis presents contributions to two major topics in computer graphics. The first part de- scribes new algorithms for tile-based synthesis of blue noise point sets and solid textures; the second part describes systems for capturing, viewing, and manipulating outdoor photographs. Well distributed point sets play an important role in computer graphics, as well as many other fields, since they lie in the very foundation of any sampling technique. Many previous re- searchers have pointed out that point sets with a blue noise Fourier spectrum, i.e. the points are distributed both evenly and randomly, are desirable in many applications. In this thesis, we introduce a novel technique for rapidly generating such point sets. Through the use of Wang tiles, our technique deterministically generates infinite non-periodic patterns. Any local area may be consistently regenerated as needed. The points in each tile form a progressive sequence, enabling matching arbitrary spatially varying point densities. Recursion allows our technique to adaptively subdivide tiles where high density is required, and makes it possible to zoom into point sets by an arbitrary amount, while maintaining a constant apparent density. The technique is extremely fast (point generation is in the order of several millions of points per second) and has a compact memory footprint. -
Convex Polytopes and Tilings with Few Flag Orbits
Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals. -
An Introduction to Tile-Based Self-Assembly and a Survey of Recent Results
Nat Comput DOI 10.1007/s11047-013-9379-4 An introduction to tile-based self-assembly and a survey of recent results Matthew J. Patitz Ó Springer Science+Business Media Dordrecht 2013 Abstract We first give an introduction to the field of tile- state, spontaneously and without external guidance coa- based self-assembly, focusing primarily on theoretical lesce to form more complex structures. The process is models and their algorithmic nature. We start with a guided by only local interactions between the components, description of Winfree’s abstract Tile Assembly Model which typically follow a basic set of rules. Despite the (aTAM) and survey a series of results in that model, dis- seemingly simplistic nature of self-assembly, its power can cussing topics such as the shapes which can be built and the be harnessed to form structures of incredible complexity computations which can be performed, among many others. and intricacy. In fact, self-assembling systems abound in Next, we introduce the more experimentally realistic kinetic nature, resulting in everything from the delicate crystalline Tile Assembly Model (kTAM) and provide an overview of structure of snowflakes to many of the structurally and kTAM results, focusing especially on the kTAM’s ability to functionally varied components of biological systems. model errors and several results targeted at preventing and Beyond the purely mathematically interesting properties correcting errors. We then describe the 2-Handed Assembly of self-assembling systems, such systems have been rec- Model (2HAM), which allows entire assemblies to combine ognized as an excellent template for the fabrication of with each other in pairs (as opposed to the restriction of artificial structures on the nanoscale. -
Wang Tile and Aperiodic Sets of Tiles
Wang Tile and Aperiodic Sets of Tiles Chao Yang Guangzhou Discrete Mathematics Seminar, 2017 Room 416, School of Mathematics Sun Yat-Sen University, Guangzhou, China Hilbert’s Entscheidungsproblem The Entscheidungsproblem (German for "decision problem") was posed by David Hilbert in 1928. He asked for an algorithm to solve the following problem. I Input: A statement of a first-order logic (possibly with a finite number of axioms), I Output: "Yes" if the statement is universally valid (or equivalently the statement is provable from the axioms using the rules of logic), and "No" otherwise. Church-Turing Theorem Theorem (1936, Church and Turing, independently) The Entscheidungsproblem is undecidable. Wang Tile In order to study the decidability of a fragment of first-order logic, the statements with the form (8x)(9y)(8z)P(x; y; z), Hao Wang introduced the Wang Tile in 1961. Wang’s Domino Problem Given a set of Wang tiles, is it possible to tile the infinite plane with them? Hao Wang Hao Wang (Ó, 1921-1995), Chinese American philosopher, logician, mathematician. Undecidability and Aperiodic Tiling Conjecture (1961, Wang) If a finite set of Wang tiles can tile the plane, then it can tile the plane periodically. If Wang’s conjecture is true, there exists an algorithm to decide whether a given finite set of Wang tiles can tile the plane. (By Konig’s infinity lemma) Domino problem is undecidable Wang’s student, Robert Berger, gave a negative answer to the Domino Problem, by reduction from the Halting Problem. It was also Berger who coined the term "Wang Tiles". -
Tiling with Penalties and Isoperimetry with Density
Rose-Hulman Undergraduate Mathematics Journal Volume 13 Issue 1 Article 6 Tiling with Penalties and Isoperimetry with Density Yifei Li Berea College, [email protected] Michael Mara Williams College, [email protected] Isamar Rosa Plata University of Puerto Rico, [email protected] Elena Wikner Williams College, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Li, Yifei; Mara, Michael; Plata, Isamar Rosa; and Wikner, Elena (2012) "Tiling with Penalties and Isoperimetry with Density," Rose-Hulman Undergraduate Mathematics Journal: Vol. 13 : Iss. 1 , Article 6. Available at: https://scholar.rose-hulman.edu/rhumj/vol13/iss1/6 Rose- Hulman Undergraduate Mathematics Journal Tiling with Penalties and Isoperimetry with Density Yifei Lia Michael Marab Isamar Rosa Platac Elena Wiknerd Volume 13, No. 1, Spring 2012 aDepartment of Mathematics and Computer Science, Berea College, Berea, KY 40404 [email protected] bDepartment of Mathematics and Statistics, Williams College, Williamstown, MA 01267 [email protected] cDepartment of Mathematical Sciences, University of Puerto Rico at Sponsored by Mayagez, Mayagez, PR 00680 [email protected] dDepartment of Mathematics and Statistics, Williams College, Rose-Hulman Institute of Technology Williamstown, MA 01267 [email protected] Department of Mathematics Terre Haute, IN 47803 Email: [email protected] http://www.rose-hulman.edu/mathjournal Rose-Hulman Undergraduate Mathematics Journal Volume 13, No. 1, Spring 2012 Tiling with Penalties and Isoperimetry with Density Yifei Li Michael Mara Isamar Rosa Plata Elena Wikner Abstract. We prove optimality of tilings of the flat torus by regular hexagons, squares, and equilateral triangles when minimizing weighted combinations of perime- ter and number of vertices. -
University of Southampton Research Repository Eprints Soton
University of Southampton Research Repository ePrints Soton Copyright © and Moral Rights for this thesis are retained by the author and/or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder/s. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders. When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given e.g. AUTHOR (year of submission) "Full thesis title", University of Southampton, name of the University School or Department, PhD Thesis, pagination http://eprints.soton.ac.uk UNIVERSITY OF SOUTHAMPTON Faculty of Engineering, Science and Mathematics School of Chemistry Synthesis of oligonucleotide analogues for use in DNA nanostructures By Adeline Durand A thesis submitted for the degree of Doctor of Philosophy. March 2010 UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF ENGINEERING, SCIENCE & MATHEMATICS SCHOOL OF CHEMISTRY Doctor of Philosophy SYNTHESIS OF OLIGONUCLEOTIDE ANALOGUES FOR USE IN DNA NANOSTRUCTURES by Adeline Durand Thanks to its ability to form duplexes through selective base-pair recognition, DNA is a unique material for orderly self-assembled construction at the nanoscale. To develop a nanotechnology platform on a grid of addressable molecular building blocks using DNA node structures, DNA complexes need to be fixed onto surfaces. To fulfil this requirement on lipid membranes, phosphoramidites monomers modified with a cholesterol moiety and a spacer unit were synthesised. -
Troisième Cycle De La Physique Mathematical
TROISIÈME CYCLE DE LA PHYSIQUE EN SUISSE ROMANDE MATHEMATICAL DIFFRACTION THEORY IN EUCLIDIAN SPACES Michael BAAKE Fakultät für Mathematik, Universität Bielefeld D – 33501 Bielefeld Hôte du Laboratoire de Cristallographie EPFL Lausanne SEMESTRE D’HIVER 2004-2005 TROISIÈME CYCLE DE LA PHYSIQUE EN SUISSE ROMANDE UNIVERSITÉS DE FRIBOURG - GENÈVE - NEUCHÂTEL & ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE ************** Archives - Polycopiés EPFL Cubotron 1015 Lausanne http://cristallo.epfl.ch/3cycle/courses/Baake-2004.pdf MATHEMATICAL DIFFRACTION THEORY IN EUCLIDEAN SPACES: AN INTRODUCTORY SURVEY MICHAEL BAAKE Abstract. Mathematical diffraction theory is concerned with the Fourier transform of the autocorrelation of translation bounded complex measures. While the latter are meant to encapsulate the relevant order of various forms of matter, the corresponding diffraction measures describe the outcome of kinematic diffraction experiments, as obtained from X-ray or neutron scattering in the far field (or Fraunhofer) picture. In this introductory article, the mathematical approach to diffraction is summarized, with special emphasis on simple derivations of results. Apart from (fully) periodic order, also aperiodic order is discussed, both in terms of model sets (perfect order) and various stochastic extensions (lattice gases and random tilings). Keywords: Diffraction Theory, Lattice Systems, Quasicrystals, Disorder Introduction The diffraction theory of crystals is a subject with a long history, and one can safely say that it is well understood [31, 21]. Even though the advent of quasicrystals, with their sharp diffraction images with perfect non-crystallographic symmetry, seemed to question the general understanding, the diffraction theory of perfect quasicrystals, in terms of the cut and project method, is also rather well understood by now, see [36, 37] and references therein. -
Search Methods for Tile Sets in Patterned DNA Self-Assembly$
Search methods for tile sets in patterned DNA self-assemblyI Mika G¨o¨os1, Tuomo Lempi¨ainen2, Eugen Czeizler3,∗, Pekka Orponen3 Department of Information and Computer Science and Helsinki Institute for Information Technology HIIT Aalto University, Finland Abstract The Pattern self-Assembly Tile set Synthesis (PATS) problem, which arises in the theory of structured DNA self-assembly, is to determine a set of coloured tiles that, starting from a bordering seed structure, self-assembles to a given rectangular colour pattern. The task of finding minimum-size tile sets is known to be NP-hard. We explore several complete and incomplete search techniques for finding minimal, or at least small, tile sets and also assess the reliability of the solutions obtained according to the kinetic Tile Assembly Model. Keywords: DNA self-assembly, tilings, Tile Assembly Model, pattern assembly, tile set synthesis, reliable self-assembly 1. Introduction Algorithmic assembly of nucleic acids (DNA and RNA) has advanced extensively in the past 30 years, from a seminal idea to the current designs and experimental implementations of complex nanostructures and nanodevices with dynamic, programmable evolution and machinelike properties. Recent developments in the field include fundamental constructions such as in vitro complex 3D pattern formation and functionalisation [3, 4], robotic designs such as mobile arms, walkers, motors [5, 6], computational primitives [7, 8], and also applications to in vivo biosensors [9] and potential drug delivery mechanisms and therapeutics [10]. Self-assembly of nucleic acids can be seen both as a form of structural nanotechnology and as a model of computation. As a computational model, one first encodes the input of a computational problem into an algorithmically designed (DNA) pattern or shape.