Computational Design Framework 3D Graphic Statics
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Final Poster
Associating Finite Groups with Dessins d’Enfants Luis Baeza, Edwin Baeza, Conner Lawrence, and Chenkai Wang Abstract Platonic Solids Rotation Group Dn: Regular Convex Polygon Approach Each finite, connected planar graph has an automorphism group G;such Following Magot and Zvonkin, reduce to easier cases using “hypermaps” permutations can be extended to automorphisms of the Riemann sphere φ : P1(C) P1(C), then composing β = φ f where S 2(R) P1(C). In 1984, Alexander Grothendieck, inspired by a result of f : 1( ) ! 1( )isaBely˘ımapasafunctionofeither◦ zn or ' P C P C Gennadi˘ıBely˘ıfrom 1979, constructed a finite, connected planar graph 4 zn/(zn +1)! 2 such that Aut(f ) Z or Aut(f ) D ,respectively. ' n ' n ∆β via certain rational functions β(z)=p(z)/q(z)bylookingatthe inverse image of the interval from 0 to 1. The automorphisms of such a Hypermaps: Rotation Group Zn graph can be identified with the Galois group Aut(β)oftheassociated 1 1 rational function β : P (C) P (C). In this project, we investigate how Rigid Rotations of the Platonic Solids I Wheel/Pyramids (J1, J2) ! w 3 (w +8) restrictive Grothendieck’s concept of a Dessin d’Enfant is in generating all n 2 I φ(w)= 1 1 z +1 64 (w 1) automorphisms of planar graphs. We discuss the rigid rotations of the We have an action : PSL2(C) P (C) P (C). β(z)= : v = n + n, e =2 n, f =2 − n ◦ ⇥ 2 !n 2 4 zn · Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and I Zn = r r =1 and Dn = r, s s = r =(sr) =1 are the rigid I Cupola (J3, J4, J5) dodecahedron), the Archimedean solids, the Catalan solids, and the rotations of the regular convex polygons,with 4w 4(w 2 20w +105)3 I φ(w)= − ⌦ ↵ ⌦ 1 ↵ Rotation Group A4: Tetrahedron 3 2 Johnson solids via explicit Bely˘ımaps. -
On a Remarkable Cube of Pyrite, Carrying Crys- Tallized Gold and Galena of Unusual Habit
ON A REMARKABLE CUBE OF PYRITE, CARRYING CRYS- TALLIZED GOLD AND GALENA OF UNUSUAL HABIT By JOSEPH E. POGUE Assistant Curator, Division of Mineralogy, U. S. National Museum With One Plate The intergrowth or interpenetration of two or more minerals, especially if these be well crystallized, often shows a certain mutual crystallographic control in the arrangement of the individuals, sug- gestive of interacting molecular forces. Occasionally a crystal upon nearly completing its growth exerts what may be termed "surface affinit}'," in that it seems to attract molecules of composition differ- ent from its own and causes these to crystallize in positions bearing definite crystallographic relations to the host crystal, as evidenced, for example, by the regular arrangement of marcasite on calcite, chalcopyrite on galena, quartz on fluorite, and so on. Of special interest, not only because exhibiting the features mentioned above, but also on account of the unusual development of the individuals and the great beauty of the specimen, is a large cube of pyrite, studded with crystals of native gold and partly covered by plates of galena, acquired some years ago by the U. S. National Museum. This cube measures about 2 inches (51 mm.) along its edge, and is prominently striated, as is often the case with pyrite. It contains something more than 130 crystals of gold attached to its surface, has about one-fourth of its area covered with galena, and upon one face shows an imperfect crystal of chalcopyrite. The specimen came into the possession of the National Museum in 1906 and was ob- tained from the Snettisham District, near Juneau, Southeast Alaska. -
Magneto-Luminescence Correlation in the Textbook Dysprosium(III
Magnetochemistry 2016, 2, 41; doi:10.3390/magnetochemistry2040041 S1 of S3 Supplementary Materials: Magneto‐Luminescence Correlation in the Textbook Dysprosium(III) Nitrate Single‐Ion Magnet Ekaterina Mamontova, Jérôme Long, Rute A. S. Ferreira, Alexandre M. P. Botas, Dominique Luneau, Yannick Guari, Luis D. Carlos and Joulia Larionova Table S1. Crystal and structure refinement data for compound 1. Compound 1 Formula DyH12N3O15 Formula weight 456.60 Temperature/K 293 Crystal system Triclinic Space group P‐1 a/Å 6.7429(8) b/Å 9.1094(9) c/Å 11.6502(11) α/° 70.369(9) β/° 88.714(9) γ/° 69.113(11) Volume/Å3 625.86(13) Z 2 Dc/g∙cm−3 2.4228 μ(Mo‐Kα)/mm−1 6.054 F(000) 414.2 Crystal size/mm 0.1 × 0.1 × 0.1 Crystal type Colourless plates range 3.25 to 29.29 −9 ≤ h ≤ 9 Index ranges −12 ≤ k ≤ 11 −14 ≤ l ≤ 14 Reflections collected 5351 Independent reflections 2879 (Rint = 0.0424) Data/parameters 2879/171 R1 = 0.0369 Final R indices [I > 2σ(I)]a,b wR2 = 0.0765 R1 = 0.0443 Final R indices (all data)a,b wR2 = 0.0848 Largest diff. peak and hole 1.18 and −1.64 eÅ−3 a b 2222 2 RFFF1 / ; wR2 w F F/ w F oc o oc o Magnetochemistry 2016, 2, 41 S2 of S3 Table S2. SHAPE analysis for compound 1. DP EPY OBPY PPR PAPR JBCCU JPCSAPR JMBIC JATDI JSPC SDD TD HD 33.818 23.117 17.902 9.556 14.372 11.933 4.385 8.791 17.518 2.204 6.675 6.077 10.278 DP, decagon; EPY, ennegonal pyramid; OBPY, octagonal pyramid; PPR, pentagonal prism; PAPR, pentagonal antiprism; JBCCU, bicapped cube; JPCSPAR, bicapped square antiprism; JMBIC, metabidiminished icosahedron; JATDI, augmented tridiminished icosahedron; JSPC, spherocorona; SDD, staggered dodecahedron; TD, tetradecahedron; HD, hexadecahedron. -
Volumes of Polyhedra in Hyperbolic and Spherical Spaces
Volumes of polyhedra in hyperbolic and spherical spaces Alexander Mednykh Sobolev Institute of Mathematics Novosibirsk State University Russia Toronto 19 October 2011 Alexander Mednykh (NSU) Volumes of polyhedra 19October2011 1/34 Introduction The calculation of the volume of a polyhedron in 3-dimensional space E 3, H3, or S3 is a very old and difficult problem. The first known result in this direction belongs to Tartaglia (1499-1557) who found a formula for the volume of Euclidean tetrahedron. Now this formula is known as Cayley-Menger determinant. More precisely, let be an Euclidean tetrahedron with edge lengths dij , 1 i < j 4. Then V = Vol(T ) is given by ≤ ≤ 01 1 1 1 2 2 2 1 0 d12 d13 d14 2 2 2 2 288V = 1 d 0 d d . 21 23 24 1 d 2 d 2 0 d 2 31 32 34 1 d 2 d 2 d 2 0 41 42 43 Note that V is a root of quadratic equation whose coefficients are integer polynomials in dij , 1 i < j 4. ≤ ≤ Alexander Mednykh (NSU) Volumes of polyhedra 19October2011 2/34 Introduction Surprisely, but the result can be generalized on any Euclidean polyhedron in the following way. Theorem 1 (I. Kh. Sabitov, 1996) Let P be an Euclidean polyhedron. Then V = Vol(P) is a root of an even degree algebraic equation whose coefficients are integer polynomials in edge lengths of P depending on combinatorial type of P only. Example P1 P2 (All edge lengths are taken to be 1) Polyhedra P1 and P2 are of the same combinatorial type. -
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. -
Article Published Under an ACS Authorchoice License, Which Permits Copying and Redistribution of the Article Or Any Adaptations for Non-Commercial Purposes
This is an open access article published under an ACS AuthorChoice License, which permits copying and redistribution of the article or any adaptations for non-commercial purposes. Article www.acsnano.org Standardizing Size- and Shape-Controlled Synthesis of Monodisperse Magnetite (Fe3O4) Nanocrystals by Identifying and Exploiting Effects of Organic Impurities † † † ‡ † † † § Liang Qiao, Zheng Fu, Ji Li, , John Ghosen, Ming Zeng, John Stebbins, Paras N. Prasad, † and Mark T. Swihart*, † § Department of Chemical and Biological Engineering and Institute for Lasers, Photonics, Biophotonics, University at Buffalo (SUNY), Buffalo, New York 14260, United States ‡ MIIT Key Laboratory of Critical Materials Technology for New Energy Conversion and Storage, School of Chemistry and Chemical Engineering, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China *S Supporting Information ABSTRACT: Magnetite (Fe3O4) nanocrystals (MNCs) are among the most-studied magnetic nanomaterials, and many reports of solution-phase synthesis of monodisperse MNCs have been published. However, lack of reproducibility of MNC synthesis is a persistent problem, and the keys to producing monodisperse MNCs remain elusive. Here, we define and explore synthesis parameters in this system thoroughly to reveal their effects on the product MNCs. We demonstrate the essential role of benzaldehyde and benzyl benzoate produced by oxidation of benzyl ether, the solvent typically used for MNC synthesis, in producing mono- disperse MNCs. This insight allowed us to develop stable formulas for producing monodisperse MNCs and propose a model to rationalize MNC size and shape evolution. Solvent polarity controls the MNC size, while short ligands shift the morphology from octahedral to cubic. We demonstrate preparation of specific assemblies with these MNCs. -
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. -
Petrie Schemes
Canad. J. Math. Vol. 57 (4), 2005 pp. 844–870 Petrie Schemes Gordon Williams Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grunbaum–Dress¨ polyhedra, pos- sess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. 1 Introduction Historically, polyhedra have been conceived of either as closed surfaces (usually topo- logical spheres) made up of planar polygons joined edge to edge or as solids enclosed by such a surface. In recent times, mathematicians have considered polyhedra to be convex polytopes, simplicial spheres, or combinatorial structures such as abstract polytopes or incidence complexes. A Petrie polygon of a polyhedron is a sequence of edges of the polyhedron where any two consecutive elements of the sequence have a vertex and face in common, but no three consecutive edges share a commonface. For the regular polyhedra, the Petrie polygons form the equatorial skew polygons. Petrie polygons may be defined analogously for polytopes as well. Petrie polygons have been very useful in the study of polyhedra and polytopes, especially regular polytopes. -
Motions and Stresses of Projected Polyhedra
Motions and Stresses of Projected Polyhedra by Walter Whiteley* R&urn& Topologie Structuraie #7, 1982 Abstract Structural Topology #7,1982 L’utilisation de mouvements infinitesimaux de structures a panneaux permet Using infinitesimal motions of panel structures, a new proof is given for Clerk d’apporter une nouvelle preuve au theoreme de Clerk Maxwell affirmant que la Maxwell’s theorem that the projection of an oriented polyhedron from 3-space projection d’un polyedre de I’espace a 3 dimensions donne un diagramme plane gives a plane diagram of lines and points which forms a stressed bar and joint de lignes et de points qui forme une charpente contrainte a barres et a joints. framework. The methods extend to prove a simple converse for frameworks Les methodes tendent a prouver une reciproque simple pour les charpentes a with planar graphs - and a general converse for other polyhedra under appropriate graphes planaires - et une reciproque g&&ale pour ,les autres polyedres soumis conditions on the stress. The method of proof also yields a correspondence bet- a des conditions appropriees de contraintes. La methode utilisee pour la preuve ween the form of the stress on a bar (tension/compression) and the form of the pro&it aussi une correspondance entre la forme de la contrainte sur une bar dihedral angle (concave/convex). (tension/compression) et la forme de I’angle diedrique (concave/convexe). Les resultats ont une applicatioin potentielle a la fois sur I’etude des charpentes The results have potential application both to the study of frameworks and to et sur I’analyse de la scene (la reconnaissance d’images de polyedres). -
Computability and Tiling Problems
Computability and Tiling Problems Mark Richard Carney University of Leeds School of Mathematics Submitted in accordance with the requirements for the degree of Doctor of Philosophy October 2019 Intellectual Property Statement The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copy- right material and that no quotation from the thesis may be pub- lished without proper acknowledgement. The right of Mark Richard Carney to be identified as Author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. c October 2019 The University of Leeds and Mark Richard Carney. i Abstract In this thesis we will present and discuss various results pertaining to tiling problems and mathematical logic, specifically computability theory. We focus on Wang prototiles, as defined in [32]. We begin by studying Domino Problems, and do not restrict ourselves to the usual problems concerning finite sets of prototiles. We first consider two domino problems: whether a given set of prototiles S has total planar tilings, which we denote T ILE, or whether it has infinite connected but not necessarily total tilings, W T ILE (short for ‘weakly tile’). We show that both T ILE ≡m ILL ≡m W T ILE, and thereby both T ILE 1 and W T ILE are Σ1-complete. We also show that the opposite problems, :T ILE and SNT (short for ‘Strongly Not Tile’) are such that :T ILE ≡m W ELL ≡m 1 SNT and so both :T ILE and SNT are both Π1-complete. -
Math 311. Polyhedra Name: a Candel CSUN Math ¶ 1. a Polygon May Be
Math 311. Polyhedra A Candel Name: CSUN Math ¶ 1. A polygon may be described as a finite region of the plane enclosed by a finite number of segments, arranged in such a way that (a) exactly two segments meets at every vertex, and (b) it is possible to go from any segment to any other segment by crossing vertices of the polygon. A polyhedron (plural: polyhedra) is a three dimensional figure enclosed by a finite number of polygons arranged in such a way that (a) exactly two polygons meet (at any angle) at every edge. (b) it is possible to get from every polygon to every other polygon by crossing edges of the polyhedron. ¶ 2. The simplest class of polyhedra are the simple polyhedra: those that can be continuously deformed into spheres. Within this class we find prisms and pyramids. Prism Pyramid If you round out the corners and smooth out the edges, you can think of a polyhedra as the surface of a spherical body. An so you may also think of the polyhedra, with its faces, edges, and vertices, as determining a decomposition of the surface of the sphere. ¶ 3. A polyhedron is called convex if the segment that joins any two of its points lies entirely in the polyhedron. Put in another way, a convex polyhedron is a finite region of space enclosed by a finite number of planes. Convexity is a geometric property, it can be destroyed by deformations. It is important because a convex polyhe- dron is always a simple polyhedron. (a) Can you draw a simple polyhedron that is not a convex polyhedron? November 4, 2009 1 Math 311. -
15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm
15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P .