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Готові списки джерел за темами / Polyhedron of arrangements / Статті в журналах

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Статті в журналах з теми "Polyhedron of arrangements"

Автор: Grafiati

Опубліковано: 28 травня 2022

Оновлено: 31 липня 2025

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1

Yemets, Oleg A., Alexandra O. Yemets, and Ivan M. Polyakov. "Optimization on Arrangements: the Simplex Form of Polyhedron of Arrangements." Journal of Automation and Information Sciences 49, no. 12 (2017): 14–28. http://dx.doi.org/10.1615/jautomatinfscien.v49.i12.20.

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2

Duan, Jin-Wei, and Tao Deng. "Blueprints of the DNA Archimedean Polyhedra with Minimum Component Number." MATCH – Communications in Mathematical and in Computer Chemistry 92, no. 1 (2024): 5–22. http://dx.doi.org/10.46793/match.92-1.005d.

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DNA has emerged as a versatile material for constructing functional nanostructures with specific topological arrangements, making it highly desirable for synthesizing of DNA nanostructures using minimal components. In this study, we propose a novel approach to fabricate polyhedra using the fewest possible components and investigate the roles played by different components. Our results reveal that even-sided polygon components are composed of subunits distributed contiguously or alternately, while odd-sided polygon components are composed of subunits distributed alternately, which play a crucia

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3

Iemets, O. Ol, O. Ol Yemets’, and I. M. Polyakov. "Criterion of an Edge of a General Polyhedron of Arrangements." Cybernetics and Systems Analysis 54, no. 5 (2018): 796–805. http://dx.doi.org/10.1007/s10559-018-0081-5.

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4

Yemets, O. A., and O. A. Chernenko. "A Nonreducible System of Constraints of a Combinatorial Polyhedron in a Linear-Fractional Optimization Problem on Arrangements." Cybernetics and Systems Analysis 41, no. 2 (2005): 246–54. http://dx.doi.org/10.1007/s10559-005-0057-0.

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5

Babu, Chatla Naga, Paladugu Suresh, Natarajan Sampath, and Ganesan Prabusankar. "Cadmium coordination polymers based on flexible bis(imidazole) ligands: A rare example for doublet of doublet cadmium polyhedron arrangements." Journal of Molecular Structure 1075 (October 2014): 147–53. http://dx.doi.org/10.1016/j.molstruc.2014.06.074.

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6

Berg, Mark de, Dan Halperin, Mark Overmars, and Marc van Kreveld. "Sparse Arrangements and the Number of Views of Polyhedral Scenes." International Journal of Computational Geometry & Applications 07, no. 03 (1997): 175–95. http://dx.doi.org/10.1142/s0218195997000120.

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In this paper we study several instances of the problem of determining the maximum number of topologically distinct two-dimensional images that three-dimensional scenes can induce. To bound this number, we investigate arrangements of curves and of surfaces that have a certain sparseness property. Given a collection of n algebraic surface patches of constant maximum degree in 3-space with the property that any vertical line stabs at most k of them, we show that the maximum combinatorial complexity of the entire arrangement that they induce is Θ(n2 k). We extend this result to collections of hyp

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7

Kabešová, Mária, Zlatica Kožíšková, and Michal Dunaj-Jurčo. "The crystal and molecular structure of the bis(thiocyanato)-bis(4-methylpyridine)copper(II) complex at 180 K." Collection of Czechoslovak Chemical Communications 55, no. 5 (1990): 1184–92. http://dx.doi.org/10.1135/cccc19901184.

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At 180 K, [Cu(4-Mepy)2(NCS)2] crystallizes in the monoclinic system, space group P21, Z = 6, with the elementary cell parameters a = 0.963(1), b = 2.653(2), c = 0.984(2) nm, β = 106.40(0.07)°. The central atom possesses the tetragonal bipyramidal coordination; the heterocyclic ligands are bonded in the equatorial plane of the coordination polyhedron in the trans positions, the thiocyanate ligands are bridge ones and are coordinated in both the equatorial and axial positions of the coordination polyhedron. The crystal structure involves three symmetrically independent coordination polyhedra dif

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8

Siidra, Oleg I., Diana O. Nekrasova, Dmitry O. Charkin, et al. "Anhydrous alkali copper sulfates – a promising playground for new Cu²⁺ oxide complexes: new Rb-analogues of fumarolic minerals." Mineralogical Magazine 85, no. 6 (2021): 831–45. http://dx.doi.org/10.1180/mgm.2021.73.

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AbstractWe report the crystal structures of eight new synthetic multinary Rb–Cu sulfates representing four new structure types: δ-Rb2Cu(SO4)2, γ-RbNaCu(SO4)2, γ-RbKCu(SO4)2, Rb2Cu2(SO4)3, Rb2Cu2(SO4)3(H2O), β-Rb2Cu(SO4)Cl2, β-Rb4Cu4O2(SO4)4⋅(Cu0.83Rb0.17Cl) and Rb2Cu5O(SO4)5. The determination of their crystal structures significantly expands the family of anhydrous copper sulfates. Some of the anhydrous rubidium copper sulfates obtained turned out to be isostructural to known compounds and minerals. Rb2Cu5O(SO4)5 is isostructural to cesiodymite, CsKCu5O(SO4)5 and cryptochalcite, K2Cu5O(SO4)5.

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9

Boulware, Naomi, Naihuan Jing, and Kailash C. Misra. "On Smith normal forms of q-Varchenko matrices." Algebra and Discrete Mathematics 34, no. 2 (2022): 187–222. http://dx.doi.org/10.12958/adm2006.

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In this paper, we investigate q-Varchenko matrices for some hyperplane arrangements with symmetry in two andthree dimensions, and prove that they have a Smith normal formover Z[q]. In particular, we examine the hyperplane arrangement forthe regular n-gon in the plane and the dihedral model in the spaceand Platonic polyhedra. In each case, we prove that the q-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over Z[q] and realize their congruent transformation matrices over Z[q] as well.

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10

Xu, Kai, David Hutchins, and Kunshan Gao. "Coccolith arrangement follows Eulerian mathematics in the coccolithophore Emiliania huxleyi." PeerJ 6 (April 9, 2018): e4608. http://dx.doi.org/10.7717/peerj.4608.

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Background The globally abundant coccolithophore, Emiliania huxleyi, plays an important ecological role in oceanic carbon biogeochemistry by forming a cellular covering of plate-like CaCO3 crystals (coccoliths) and fixing CO2. It is unknown how the cells arrange different-sized coccoliths to maintain full coverage, as the cell surface area of the cell changes during daily cycle. Methods We used Euler’s polyhedron formula and CaGe simulation software, validated with the geometries of coccoliths, to analyze and simulate the coccolith topology of the coccosphere and to explore the arrangement mec

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11

Miyazaki, Yuzuru, Naoki Igawa та Kunio Yubuta. "Incommensurately modulated crystal structure of α′ (O′3)-type sodium cobalt oxide Na x CoO2 (x ∼ 0.78)". Acta Crystallographica Section B Structural Science, Crystal Engineering and Materials 77, № 3 (2021): 371–77. http://dx.doi.org/10.1107/s205252062100370x.

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A single-phase sample of α′ (O′3)-type layered sodium cobalt oxide Na x CoO2 (x ∼ 0.78) was prepared and its incommensurately modulated crystal structure was analyzed using the (3+1)-dimensional superspace approach to the powder neutron diffraction data. The crystal structure of the cobaltate is accurately described based on the superspace group C2/m(α0γ)00, wherein the positions of Na atoms are most significantly modulated in the monoclinic a direction to form an ordered arrangement. Such a displacive modulation causes a quasi-periodic shift of Na atoms from the centers of the NaO6 polyhedra

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12

Му, Цюань, Борис Александрович Каргин, and Евгения Геннадьевна Каблукова. "Computer-aided construction of three-dimensional convex bodies of arbitrary shapes." Вычислительные технологии, no. 2 (May 11, 2022): 54–61. http://dx.doi.org/10.25743/ict.2022.27.2.005.

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Представлена компьютерная реализация алгоритмов, предназначенных для численного моделирования трехмерных выпуклых тел со случайным числом вершин. Построение выпуклых многогранников является классической проблемой вычислительной геометрии. Интерес к таким моделям возникает в целом ряде задач физики, биологии, медицины. В данной работе модели построения выпуклых тел ориентированы на приложения, связанные с решением задач расчета характеристик рассеяния оптического излучения ледяными кристаллами перистых облаков. Алгоритмы построения выпуклых многогранников были положены в основу программы Convex

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13

Sadre-Marandi, Farrah, and Praachi Das. "Extension of Caspar-Klug theory to higher order pentagonal polyhedra." Computational and Mathematical Biophysics 6, no. 1 (2018): 1–13. http://dx.doi.org/10.1515/cmb-2018-0001.

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Abstract Many viral capsids follow an icosahedral fullerene-like structure, creating a caged polyhedral arrangement built entirely from hexagons and pentagons. Viral capsids consist of capsid proteins,which group into clusters of six (hexamers) or five (pentamers). Although the number of hexamers per capsid varies depending on the capsid size, Caspar-Klug Theory dictates there are exactly twelve pentamers needed to form a closed capsid.However, for a significant number of viruses, including viruses of the Papovaviridae family, the theory doesn’t apply. The anomaly of the Caspar-Klug Theory has

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14

Stoeber, Miriam, Pascale Schellenberger, C. Alistair Siebert, Cedric Leyrat, Ari Helenius, and Kay Grünewald. "Model for the architecture of caveolae based on a flexible, net-like assembly of Cavin1 and Caveolin discs." Proceedings of the National Academy of Sciences 113, no. 50 (2016): E8069—E8078. http://dx.doi.org/10.1073/pnas.1616838113.

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Caveolae are invaginated plasma membrane domains involved in mechanosensing, signaling, endocytosis, and membrane homeostasis. Oligomers of membrane-embedded caveolins and peripherally attached cavins form the caveolar coat whose structure has remained elusive. Here, purified Cavin1 60S complexes were analyzed structurally in solution and after liposome reconstitution by electron cryotomography. Cavin1 adopted a flexible, net-like protein mesh able to form polyhedral lattices on phosphatidylserine-containing vesicles. Mutating the two coiled-coil domains in Cavin1 revealed that they mediate di

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15

Panov, D., and A. Petrunin. "Ramification conjecture and Hirzebruch’s property of line arrangements." Compositio Mathematica 152, no. 12 (2016): 2443–60. http://dx.doi.org/10.1112/s0010437x16007648.

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The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on $\mathbb{C}\text{P}^{2}$ with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a $\text{CAT}[0]$ ramification and prove this in several cases. In the latter case we prove that the ramification is $\text{CAT}[0]$ if the metric on $\mathbb{C}\text{P}^{2}$ is non-negatively c

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16

Brow, Richard K., David R. Tallant, and Gary L. Turner. "Polyhedral Arrangements in Lanthanum Aluminoborate Glasses." Journal of the American Ceramic Society 80, no. 5 (2005): 1239–44. http://dx.doi.org/10.1111/j.1151-2916.1997.tb02970.x.

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17

Zhou, Wenjie, Yuanwei Li, Kwanghwi Je, et al. "Space-tiled colloidal crystals from DNA-forced shape-complementary polyhedra pairing." Science 383, no. 6680 (2024): 312–19. http://dx.doi.org/10.1126/science.adj1021.

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Generating space-filling arrangements of most discrete polyhedra nanostructures of the same shape is not possible. However, if the appropriate individual building blocks are selected (e.g., cubes), or multiple shapes of the appropriate dimensions are matched (e.g., octahedra and tetrahedra) and their pairing interactions are subsequently forced, space-filled architectures may be possible. With flexible molecular ligands (polyethylene glycol–modified DNA), the shape of a polyhedral nanoparticle can be deliberately altered and used to realize geometries that favor space tessellation. In this wor

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18

Podlaha, Jaroslav, and Jana Podlahová. "Crystal structure of dicadmium(II) ethylenediphosphinetetraacetate octahydrate." Collection of Czechoslovak Chemical Communications 56, no. 7 (1991): 1454–60. http://dx.doi.org/10.1135/cccc19911454.

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The crystal structure of the title compound containing a phosphorus analogue of EDTA as the ligand was determined from single-crystal X-ray diffraction data and refined to R = 0.028 for 1 920 observed reflections. The crystals are monocyclic, space group P21/c,a = 8.6169(8), b = 16.0539(9), c = 8.2504(6)Å, β = 103.458(7)° and Z = 2. The structure is completely different from that of the nitrogen prototype, [CdEDTA]2-. It consists of double-stranded polymeric layers which are composed of metal coordination polyhedra linked by μ6-hexadentate ligands; the layers lie parallel to the bc plane and a

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19

Hudelson, Matthew, Barbara Logan Mooney, and Aurora E. Clark. "Determining polyhedral arrangements of atoms using PageRank." Journal of Mathematical Chemistry 50, no. 9 (2012): 2342–50. http://dx.doi.org/10.1007/s10910-012-0033-7.

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20

Bao, Wen-Xiu, Zi-Qi Zhou, Hong-Xin Tang, Rui-Biao Fu, Zu-Ju Ma, and Xin-Tao Wu. "KPb₃(o-C₅H₄NCOO)₂Cl₅: a brand-new stable lead chloride with good comprehensive nonlinear optical performance." Inorganic Chemistry Frontiers 9, no. 8 (2022): 1830–35. http://dx.doi.org/10.1039/d2qi00171c.

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KPb3(o-C5H4NCOO)2Cl5 featuring a π-conjugated nicotinate group is simultaneously connected by a [PbO2Cl] trigonal pyramid and a distorted [PbNCl3] polyhedron into an ordered arrangement, exhibits good comprehensive NLO performance.

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21

Grinev, Nikolai. "Multi-criteria comparison of 220 kV switchgears which have different circuits of electrical connections." E3S Web of Conferences 461 (2023): 01015. http://dx.doi.org/10.1051/e3sconf/202346101015.

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The paper compares different open 220 kV switchgears in the context of a new complex solution offered by the author. The solution is a polyhedron circuit topology switchgear with novel scheme and arrangement approaches. Comparison is viewed here as a multi-criteria task with the following criteria: reliability, cost-effectiveness, maintenance activities, environment friendliness, compactness. To assess reliability, an unconventional calculation model is offered. The characteristics of open switchgears with a polyhedron circuit topology are better or at least not worse than the best analogs in

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22

Kabešová, Mária, and Zlatica Kožíšková. "Bis(1,10-phenantroline)copper(II) Complexes. Refinement of the Crystal Structure of the Dithiocyanato-bis(1,10-phenanthroline)copper(II) Complex." Collection of Czechoslovak Chemical Communications 57, no. 6 (1992): 1269–77. http://dx.doi.org/10.1135/cccc19921269.

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[Cu(NCS)2(phen)2] possesses the orthorhombic structure Pbcn at room temperature; the elementary cell parameters are a = 1.3029(21), b = 0.9900(15), c = 1.7256(23) nm, Z = 4. The central atom has the pseudooctahedral CuN6 coordination. The thiocyanate ligands are coordinated by the nitrogen atom in the equatorial plane of the coordination polyhedron. Each phenanthroline ligand in the coordination polyhedron coordinates one equatorial and one axial position. All the coordinating ligands possess the cis arrangement in the coordination sphere. The experimental results are correlated with published

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23

Govender, Nicolin, and Patrick Pizette. "A study on the effect of grain morphology on shear strength in granular materials." EPJ Web of Conferences 249 (2021): 06013. http://dx.doi.org/10.1051/epjconf/202124906013.

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The Discrete Element Method (DEM) has been successfully used to further understand GM behaviour where experimental means are not possible or limited. However, the vast majority of DEM publications use simplified spheres with rolling friction to account for particle shape, with a few using clumped spheres and super quadratics to better capture grain geometric detail. In this study, we compare the shear strength of packed polyhedral assemblies to spheres with rolling resistance to account for shape. Spheres were found to have the highest shear resistance as the limited rolling friction model cou

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24

Grigoriev, D. "Randomized Complexity Lower Bound for Arrangements and Polyhedra." Discrete & Computational Geometry 21, no. 3 (1999): 329–44. http://dx.doi.org/10.1007/pl00009425.

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25

Woidy, Patrick, and Florian Kraus. "Crystal structure of Cs2[Th(NO3)6]." Acta Crystallographica Section E Structure Reports Online 70, no. 8 (2014): 98–100. http://dx.doi.org/10.1107/s1600536814015876.

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Dicaesium hexanitratothorate(IV), Cs2[Th(NO3)6], was synthesized in the form of colourless crystals by reaction of thorium nitrate and caesium nitrate in aqueous solution. The Th atom is located on an inversion centre and is coordinated by six chelating nitrate anions. The resulting ThO12coordination polyhedron is best described as a slightly distorted icosahedron. The Cs atom also has a coordination number of 12, but its coordination polyhedron is considerably more distorted. The crystal packing can be derived from an hexagonal dense packing (hcp) of idealized spherical CsO12and ThO12units. T

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26

Lech, Andrew T., Christopher L. Turner, Reza Mohammadi, Sarah H. Tolbert, and Richard B. Kaner. "Structure of superhard tungsten tetraboride: A missing link between MB2 and MB12 higher borides." Proceedings of the National Academy of Sciences 112, no. 11 (2015): 3223–28. http://dx.doi.org/10.1073/pnas.1415018112.

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Superhard metals are of interest as possible replacements with enhanced properties over the metal carbides commonly used in cutting, drilling, and wear-resistant tooling. Of the superhard metals, the highest boride of tungsten—often referred to as WB4 and sometimes as W1–xB3—is one of the most promising candidates. The structure of this boride, however, has never been fully resolved, despite the fact that it was discovered in 1961—a fact that severely limits our understanding of its structure–property relationships and has generated increasing controversy in the literature. Here, we present a

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27

Grinev, Nikolai. "Polyhedron circuit topology switchgear as a novel complex solution." E3S Web of Conferences 461 (2023): 01014. http://dx.doi.org/10.1051/e3sconf/202346101014.

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The paper offers and analyzes a new complex solution – open switchgear with polyhedron circuit topology, with novel circuit and arrangement approaches including new electrical equipment. Decreasing numbers of disconnectors, better structural characteristics, elimination of vulnerabilities such as sections and busbar systems are the key characteristics of this solution. Due to a new circuit topology and equipment components, the open switchgear on the whole becomes more reliable, flexible, and less costly. Examples of arrangement solutions are given. We study the data on the 220 kB switchgear c

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28

Hershberger, J. E., and J. S. Snoeyink. "Erased arrangements of lines and convex decompositions of polyhedra." Computational Geometry 9, no. 3 (1998): 129–43. http://dx.doi.org/10.1016/s0925-7721(97)00024-2.

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29

Wang, Xing, Mingjun Xia, and R. K. Li. "Two potential self-activated orthoborates Cd4NdO(BO3)3 and Ca3Nd3(BO3)5: growth, crystal structures and optical properties." New Journal of Chemistry 40, no. 3 (2016): 2057–62. http://dx.doi.org/10.1039/c5nj02710a.

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30

Cuntz, Michael, Sophia Elia, and Jean-Philippe Labbé. "Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids." Annals of Combinatorics 26, no. 1 (2021): 1–85. http://dx.doi.org/10.1007/s00026-021-00555-2.

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AbstractA catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and also the weak order through the poset of regions. The weak order is known to be a congruence normal lattice, and congruence normality of lattices of regions of simplicial arrangements can be determined using polyhedral cones called shards. In this article, we update Grünbaum’s catalogue by providing normals realizing all known simplicial arrangements with up to 37 lines and key invariants. Then we add structure to this catalogue

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31

Flanders, D. J., D. J. Rawlins, P. J. Shaw, and C. W. Lloyd. "Computer-aided 3-D reconstruction of interphase epidermal cells of Datura stramonium reveals assembly." Development 106, no. 3 (1989): 531–41. http://dx.doi.org/10.1242/dev.106.3.531.

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From immunofluorescence microscopy it has been cortical microtubules form whole-cell arrays. This has clearly seen in cylindrical hairs where the existence testifies to the continuity of the array around the cell. It is not, however, clear how microtubules pack polyhedral cells with multiple, angled facets. In problem, elongated and isodiametric cells in the stramonium L. were subjected to anti-tubulin avoiding distortion by cellulase treatment and air- sections were then deblurred by computer, the digitized, reconstructed and then rotated in order to arrangement of microtubules along the anti

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32

Inoue, Kazutoshi, Kazuaki Kawahara, Mitsuhiro Saito, Motoko Kotani, and Yuichi Ikuhara. "3D arrangement of atomic polyhedra in tilt grain boundaries." Acta Materialia 202 (January 2021): 266–76. http://dx.doi.org/10.1016/j.actamat.2020.10.017.

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33

Fukunaga, Toshiharu, Keiji Itoh, Kazuhiro Mori, and Masaaki Sugiyama. "Reverse Monte Carlo Modeling of Atomic Configuration for Amorphous Materials." Solid State Phenomena 127 (September 2007): 51–56. http://dx.doi.org/10.4028/www.scientific.net/ssp.127.51.

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Reverse Monte Carlo (RMC) modeling, based on diffraction data, was applied to various kinds of amorphous materials to visualizing the three-dimensional atomic arrangement and to elucidate topological characteristics. For an as-grown amorphous carbon nanocoil, it could be clarified that graphene sheets are winding and the regular ABAB… stacking is lost and the configuration gradually changes to the hexagonal network with great regularity through heat treatment. Voronoi analysis of the RMC model could characterize the atomic configurations for NiZr2 and CuZr2 metallic glasses. The Zr environment

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34

Xu, Zhi Gang, Zi Xiang Li, Jin Wang, Tao Tao Liu, and Chun Po Sun. "Research on Situated Reconstitution of Non-Manifold Polyhedral from Function-to-Form Mapping." Advanced Materials Research 774-776 (September 2013): 1766–69. http://dx.doi.org/10.4028/www.scientific.net/amr.774-776.1766.

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Situated reconstitution is proposed in this paper to cope with un-manifold polyhedral from the function to form mapping processes. Specifically the situated reconstitution is not simply re-arrangement of geometric elements, mainly face entities in the un-manifold polyhedral, rather it needs a complex geometric/algebraic reasoning to reconstitute a manifold object, novel mathematical as well as AI techniques are introduced in this paper, several tools are developed to testify the proposed situated reconstitution in the “top-down” design automation systems.

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35

Liu, Yanting, Jun Zhou, Lu Zhou, et al. "Self-assembled structures of polyhedral gold nanocrystals: shape-directive arrangement and structure-dependent plasmonic enhanced characteristics." RSC Advances 6, no. 62 (2016): 57320–26. http://dx.doi.org/10.1039/c6ra12868h.

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Self-assembly structures of different types of polyhedral nanocrystals through drop casting method and their plasmonic enhancement characteristics and SERS performances due to the nano-antenna effect.

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36

Uchiyama, Tamotsu, Taiji Nakamura, Miyabi Hiyama, and Takako Kudo. "Theoretical Study of Si/C Equally Mixed Dodecahedrane Analogues." Molecules 28, no. 6 (2023): 2769. http://dx.doi.org/10.3390/molecules28062769.

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To gain insight into the effect of Si/C arrangement on the molecular framework of strained polyhedral compounds, dodecahedrane analogues containing equal numbers of carbon and silicon (Si/C equally mixed dodecahedrane analogues) were investigated using the ab initio molecular orbital method. There are 1648 isomers for which the Si/C arrangement on the molecular framework is different. Based on the geometry optimization of all the isomers as well as the carbon and silicon analogues, the characteristics of the structure, relative energy, and strain energy of the Si/C equally mixed compounds are

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37

Болотников, Алексей Игоревич, and Alexey Igorevich Bolotnikov. "On a Matching Arrangement of a Graph and $LP$-Orientations of a Matching Polyhedron." Matematicheskie Zametki 114, no. 2 (2023): 312–15. http://dx.doi.org/10.4213/mzm14007.

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38

Bolotnikov, A. I. "On a Matching Arrangement of a Graph and $$LP$$-Orientations of a Matching Polyhedron." Mathematical Notes 114, no. 1-2 (2023): 271–74. http://dx.doi.org/10.1134/s0001434623070283.

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39

Guibas, Leonidas J., Dan Halperin, Hirohisa Hirukawa, Jean-Claude Latombe, and Randall H. Wilson. "Polyhedral Assembly Partitioning Using Maximally Covered Cells in Arrangements of Convex Polytopes." International Journal of Computational Geometry & Applications 08, no. 02 (1998): 179–99. http://dx.doi.org/10.1142/s0218195998000102.

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We study the following problem: Given a collection A of polyhedral parts in 3D, determine whether there exists a subset S of the parts that can be moved as a rigid body by infinitesimal translation and rotation, without colliding with the rest of the parts, A\S. A negative result implies that the object whose constituent parts are the collection A cannot be taken apart with two hands. A positive result, together with the list of movable parts in S and a direction of motion for S, can be used by an assembly sequence planner (it does not, however, give the complete specification of an assembly o

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40

Makovicky, Emil, and Dan Topa. "Crankshaft chains of LEP cations as a solution to the cation size misfit." Acta Crystallographica Section A Foundations and Advances 70, a1 (2014): C1708. http://dx.doi.org/10.1107/s2053273314082916.

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In Pb-As and Pb-Sb sulfosalts, columns of large coordination prisms of Pb combine with intervening slabs of As- or/and Sb coordination polyhedra. The LEP of these elements dictate the configurations resembling SnS (especially for As) or inflated PbS (especially for Sb). The size misfit of cation polyhedra forces the short As-S and Sb-S bonds to form crankshaft chains. In the homologues with thicker slabs, parallel crankshaft chains and incorporation of Pb in the slabs compensate for the misfit. In the low homologues, N = 3 for sartorite series and N = 4 for andorites, the misfit becomes critic

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41

Serezhkin, Viktor N., Ekaterina F. Rogaleva, Anton V. Savchenkov, Denis V. Pushkin, and Larisa B. Serezhkina. "Aspects of the topology of actinide atom substructures in crystal structures and the concept of antiliquid." Acta Crystallographica Section A Foundations and Advances 75, no. 2 (2019): 370–78. http://dx.doi.org/10.1107/s2053273318018326.

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Using the parameters of Voronoi–Dirichlet (VD) polyhedra the authors have verified the maximum space-filling principle in substructures constructed of actinide atoms (from thorium to einsteinium) in all crystal structures from the Inorganic Crystal Structure Database (ICSD) and Cambridge Structural Database (CSD). It is shown that most of the actinide atoms in such substructures are surrounded by 14 or 12 neighboring atoms. It was discovered that U substructures with greater than or equal to 20 crystallographically independent U atoms in the unit cell feature 15-faceted VD polyhedra as the mos

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42

Hawthorne, F. C., and M. A. Cooper. "The crystal structure of chalcoalumite: mechanisms of Jahn-Teller-driven distortion in [6]Cu2+-containing oxysalts." Mineralogical Magazine 77, no. 7 (2013): 2901–12. http://dx.doi.org/10.1180/minmag.2013.077.7.02.

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AbstractThe crystal structure of chalcoalumite, ideally Cu2+Al4(SO4)(OH)12(H2O)3, monoclinic, P21/n, Z = 4:a 10.228(3), b 8.929(3), c 17.098(6) Å, β 95.800(11)°, V 1553.6(1.5) Å3, has been refined to R1 = 3.08% for 4,022 unique observed (4σ) reflections collected on a Bruker D8 three-circle diffractometer equipped with a rotating-anode generator, multilayer optics and an APEX-II CCD detector. In the structure of chalcoalumite, there is one S site, tetrahedrally coordinated by four O anions, with <S–O> = 1.472 Å. There are four Al sites with site-scattering values in accord with occupancy

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43

Hu, Mingqiang, Chengbing Ma, Huimin Wen, Honghua Cui та Changneng Chen. "Heptacarbonylbis(μ-propane-1,3-dithiolato)triiron(I,II)(2Fe—Fe)". Acta Crystallographica Section E Structure Reports Online 70, № 4 (2014): m124. http://dx.doi.org/10.1107/s1600536814004619.

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The trinuclear title compound, [Fe3(C3H6S2)2(CO)7], is a mixed-valent FeI/FeIIcomplex and crystallizes with two molecules of similar configuration in the asymmetric unit. The three Fe atoms in each molecule display a bent arrangement [Fe—Fe—Fe = 156.22 (4) and 157.06 (3)°]. Both outer FeIatoms are six-coordinated in a distorted ocahedral coordination geometry defined by the bridging FeIIatom, three carbonyl C atoms and two bridging S atoms. The coordination number of the central FeIIatom is seven and includes bonding to the two outer FeIatoms, four bridging S atoms and one carbonyl C atom. The

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44

Wyckoff, Kira, Jadon Bienz, and Ram Seshadri. "Relating Crystallographic Shear to Rate Performance in Wadsely-Roth Materials." ECS Meeting Abstracts MA2023-01, no. 2 (2023): 506. http://dx.doi.org/10.1149/ma2023-012506mtgabs.

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Wadsley-Roth shear structures represent a promising and high performance class of high-voltage Nb- and W-based anode materials that intrinsically avoid SEI formation, are capable of mulitelectron redox, and suppress large lattice expansion. Currently, there is huge momentum to design faster charging batteries. Wadsley-Roth materials, typically described by their block size, have been found to display outstanding charge storage at fast rates. Within this class of materials, however, there is significant diversity in terms of block size and block arrangement. It is understood that these material

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45

Cressey, G., B. A. Cressey, and F. J. Wicks. "Polyhedral serpentine: a spherical analogue of polygonal serpentine?" Mineralogical Magazine 72, no. 6 (2008): 1229–42. http://dx.doi.org/10.1180/minmag.2008.072.6.1229.

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AbstractVugs in late hydrothermal veins in the serpentinite at Gew-graze, Lizard, Cornwall, UK, contain serpentine spheres ≤0.7 mm in diameter composed of a crystallographically controlled radial array of well crystallized lizardite-1Tcrystals. Examinations with optical and scanning electron microscopy reveal that the spheres actually have polyhedral morphology. The polyhedral facets at the sphere surface are the (0001) terminations of individual single crystals of lizardite. Each lizardite crystal is a hexagonal prism and tapers inwards to the core. The angle from prism axis to prism axis is

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46

Kusaka, Katsuhiro, Kenji Hagiya, Masaaki Ohmasa, and Kazuaki Iishi. "Variation of large-scale regularity in modulated structures of Ca2CoSi2O7 studied by a simulation method." Acta Crystallographica Section B Structural Science 60, no. 4 (2004): 369–74. http://dx.doi.org/10.1107/s0108768104011735.

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The period of two-dimensional modulation in Ca2CoSi2O7 varies with temperature (q = 0.285–1/3). The change in the modulated structure with the variation of q has been clarified by the construction of the structures using various q values and the modulation amplitudes determined at 293 K [q = 0.2913 (1), tetragonal]. The features of the modulated structures are characterized by the formation of CaO6 polyhedra and the variable distribution of bundles along the c-axis, composed of four arrays of CaO6 polyhedra and an array of CoO4 tetrahedra. The formation of octagonal arrangements of the bundles

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47

Murad, N. A. "Histological study of the adrenal gland in the adult Iraqi local male rabbits." Al-Qadisiyah Journal of Veterinary Medicine Sciences 11, no. 3 (2012): 24. http://dx.doi.org/10.29079/vol11iss3art211.

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The present study was designed to provide basic data about the histological features of theadrenal gland in the adult Iraqi local male rabbits. The adrenal glands were paired, lie close thecraniomedial border of the kidney, together they weigh about 0.59 g ± 0.38. Each adrenal glandwas covered by a dense connective tissue capsule and consists of two layers: adrenal cortex andadrenal medulla. The adrenal cortex made up of columnar or polyhedral cells, and can be dividedin to three layers according to arrangement of its cells were: The zona glomerulosa immediatelyinside the capsule, consisting o

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48

Avis, David, and Komei Fukuda. "A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra." Discrete & Computational Geometry 8, no. 3 (1992): 295–313. http://dx.doi.org/10.1007/bf02293050.

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49

Mahoney, John F., B. E. Kalensher, Julius Perel, and Terry D. Lee. "Model for charged cluster stability based on polyhedral arrangement of point charges." International Journal of Mass Spectrometry and Ion Processes 83, no. 3 (1988): 231–44. http://dx.doi.org/10.1016/0168-1176(88)80030-2.

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50

Dauter, Zbigniew, and Mariusz Jaskolski. "Multiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedra." Acta Crystallographica Section A Foundations and Advances 76, no. 5 (2020): 580–83. http://dx.doi.org/10.1107/s2053273320007093.

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The famous Euler's rule for three-dimensional polyhedra, F − E + V = 2 (F, E and V are the numbers of faces, edges and vertices, respectively), when extended to many tested cases of space-filling polyhedra such as the asymmetric unit (ASU), takes the form Fn − En + Vn = 1, where Fn, En and Vn enumerate the corresponding elements, normalized by their multiplicity, i.e. by the number of times they are repeated by the space-group symmetry. This modified formula holds for the ASUs of all 230 space groups and 17 two-dimensional planar groups as specified in the International Tables for Crystallogra

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