Polyhedra

The article discusses the handwritten revisions and drawn additions by Albrecht Dürer in his own copy of the treatise on geometry, Underweysung der Messung (1525). Situating Dürer’s interest in mathematics within the scholarly milieu of Renaissance Nuremberg, the article addresses the shifts in style and content that Dürer proscribes and offers new perspectives on the artist’s relation to his own late work. The article concludes by displaying several drawings bound into the edition by Dürer that illustrate variations on perspectival apparatuses. These lesser known drawings illuminate the evolution of Dürer’s conceptualization of the dynamics between artist and subject, as well as the tools used to facilitate this interaction.

Emergence of a pantheon -- cognitive or otherwise
Pantheons as patterns of cognitive N-foldness
Pantheon dynamics in a globalized semi-secular civilization?
Mathematical theology enabling the quest for a meta-pattern?
Complex equations forming "pantheons" of mathematical experience?
Prime number and curvature implications for global governance?
Polyhedra suggestive of arrays of requisite variety of pantheons in 3D
Configuring the 64 subjects of mathematics as a 64-edged drilled truncated cube
Exploring potential dynamics within a pantheon?
Engendering and navigating pantheons -- "angelic" and "demonic"?
Pantheon as a psychosocial "O-ring" -- speculatively understood?

Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin elimination method (a version of Gaussian elimination for inequalities) is discussed in some detail. I also included some material on projective spaces, projective maps and polar duality w.r.t. a nondegenerate quadric in order to define a suitable notion of ``projective polyhedron'' based on cones. To the best of our knowl...

Methodology
Commentary on patterns of N-foldness
Polyhedra of secondary value for memorable mapping?
Constructible polyhedra in the light of constructible polygons?
Memorability of 17 Sustainable Development Goals with 169 tasks
Relevance of polyhedral mapping of patterns of complexity with symbolic appeal
Strategic viability of global governance enabled by mappings on exotic polyhedra
Polyhedral "memory palaces": an ordering pattern for sustainable self-governance?
Memorability implied by Euler's isomorphism of musical and polyhedral order?
Reframing forms of connectivity through the logic of oppositional geometry

1.Introduction The following exercise is based on experiments conducted in circular Origami. This type of paper folding allows for a completely different geometry than the square type since it lends itself very easily to the creation of shapes based on 30-60-90 degree angles. This allows for experimentation with shapes made up of equilateral triangles such as deltahedra. The results of this research were used in Annenberg sponsored activities conducted in a progressive middle school in Houston TX, as well as a workshop presented at the 1999 Bridges Conference in Winfield, KS. Not including preparatory and follow up work by the teacher, the activities in Houston were composed of two main parts, the collaborative construction of a three-yard-across, eighty-faced regular deltahedron (the EndoPentakis Icosi-dodecahedron) and the following exercise. The barn-raising was presented last year in Winfield, and the paper folding is the topic of this paper. Circular Origami introduces a whole ...

The "hidden-line problem" for computer-drawn polyhedra is the problem of determining which edges, or parts of edges, of a polyhedra are visible from a given vantage point. This is an important problem in computer graphics, and its fast solution is especially critical for on-line CRT display applications. The method presented here for solving this problem is believed to be faster than previously known methods. An edge classification scheme is described that eliminates at once most of the totally invisible edges. The remaining, potentially visible edges are then tested in paths, which eventually cover the whole polyhedra. These paths are synthesized in such a way as to minimize the number of calculations. Both the case of a cluster of polyhedra and the illumination problem in which a polyhedron is illuminated from a point source of light are treated as applications of the general algorithm. Several illustrative examples are included.

Eliciting provocative clues for psychosocial challenges
Tensegrity form-finding of relevance to integrative configuration of polarities
Psychosocial tensegrity as a necessarily mysterious collective blindspot?
Towards polarity containment -- psychosocial tensegrity in practice
Predetermination of tensegrity forms of relevance to integrative configuration of polarities
Matching sets of psychosocial polarities to tensegrities: case of the 10 Commandments?
Spherical tensegrity as "container" for polarization dynamics of global civilization?
Tensegrity torus as complementary framing of integrative psychosocial structure?
Matching sets of psychosocial polarities to tensegrities: case of Axes of Bias?
Matching sets of psychosocial polarities to tensegrities: case of Sustainable Development Goals?
Tensegrity as a key to psychosocial polarity reversal in practice?

When rolling a single die, the probability of generating any random number is 1/n, were n is the number of faces on the die.  For an icosahedron, numbered 1-20, the probability of generating any particular number is 1/20, or a 5% chance.

For a cube, numbered 1-6, the probability of generating any particular random number is 1/6, or a 16.67% chance.

When rolling multiple dice, the probability of generating any particular sum follows a normal distribution, rather than a linear distribution, with the shape of the bell curve becoming more defined as the number of dice rolls increases.

In this paper, I will explain why the character generation algorithm of Dungeons & Dragons suggests rolling four six-sided dice, and removing the lowest roll, for each attribute.

My presentation in Kanazawa aims to bring a renewed interest and a new nexus of activities around William Huff's work, especially as an exercise in transformational thinking in the field of architectural education. After his retirement, professor Huff donated a beautiful collection of his students' work to the HfG-Archiv in Ulm, Germany, which inspired us at the Institute of Art and Design in Vienna to conduct a semester assignment for 450 students around the topic of Parquet Deformations (P.D.). The open search for strategies to transfer the spatiotemporal idea of a planar P.D. into 3D led to fascinating results. Hereby we discovered a field of formal research that opened the students' eyes for two and three-dimensional relationships and made them enthusiastic and sensitive for spatial transformations. This is a brief report on how we conducted the exercise, as a fruitful exercise for the basic design education of young architects and designers. William Huff, experimenting with his students on parquets consisting of one geometrical shape only, discovered a way to transform, slowly and subtly, one geometrical shape into another and set up strict rules for the assignment: the parquet had to be made up with one geometrical shape only (no overlaps, no gaps). The allowed lattices were square, rhombic (the rhombic has to be a 60°-120° rhombus) or regularly hexagonal. Parquet Deformations (P.D.s), designs, produced for over three decades in a curriculum of basic design studies, are rooted in two analytical disciplines: monohedral tiling from geometry; continuous deformation from biological morphology (after D'Arcy Thompson). The HfG-archive also hosts intricate 3D-models made under Huff's guidance. Especially the "trisections of the cube" and William Huff's comment

The article addresses the genesis and visualization of the capstone image to Kepler's polyhedral hypothesis of the planetary intervals from his first major work, Mysterium Cosmographicum (1596). The contention is that the famous Tabula III was directed less by Kepler than it was an initiative spearheaded by Georg Gruppenbach, the printer of Mysterium, and Kepler's mentor Michael Mäistlin, who sought to produce a marketable broadsheet that would appeal to the contemporary German fashion for illustrations of polyhedral geometry. More generally, the article seeks to redefine the key role played by the printing workshop and the decorative arts in the theory's inception and ultimate graphic manifestation.

Abstract
Over 90% of man-made solids, and naturally occurring solids, are crystalline. Most solids form with a regular arrangement of their particles and the particles pack in the most efficient manner, in cubic lattice structures. This geometry of inorganic matter-crystals, minerals, and metals-is based on the √ 2 and √ 3. However, biological life is based on a different geometry. In this paper I take a stab at describing the difference between inorganic and organic life from a purely geometric perspective.

Torus interconnect -- as used in supercomputers
Cubic organization of I Ching trigrams -- an eightfold way
Pattern language and polyhedral mapping
Higher dimensionality, polyhedral packing and transformation
Brain organization, cognition, comprehension -- and music
Oppositional logic and its requisite polyhedral geometry
Reflexivity in multi-loop thinking and higher order learning
Toroidal constraint -- nuclear fusion as metaphor of cognitive fusion
Supercomputers, hypercomputing and superquestions?
Framing the space for conscious creativity?
Dancing cognitively inside the box -- and beyond

Patterns, their visualization and their comprehension
Possible cognitive implications of 5-foldness
Tracing complex feedback loops
Requisite fifth-order thinking?
Trajectories to our future -- and their recursive implication?
Clues to systemic implications of 5-fold cognitive organization?
Cognitive glass ceilings impeding integrative comprehension
Beyond the 4-fold cognitive modality: integrating the 5-fold, 6-fold and 7-fold?
Eliciting systemic answers from a 5-fold web of meaning?
Pattern spiralling challenge of biomimetics
Polyhedral possibilities of interrelating patterns comprehensibly: a "cognitive gearbox"?
Speculation regarding a requisite "sixth discipline" and more

This paper introduces a unified geometric 2D building unit comprising a selfassembling or folding system of 3D structures. In one manifestation of the model, the building unit comprises eight equilateral triangles, one of 11 known nets of the octahedron, serving as the common denominator of several series of both known and unknown polyhedra. Employing a magnetic attraction-rejection pattern, complex polyhedral structures are formed quickly and easily without any mathematical knowledge or even glue. The result is an accessible world of polyhedra available as a folding game by which both simple and complex multidimensional polyhedral structures are explaine

Polyhedral insights from the travelling salesman problem of operations research
Travelling consumer problem in terms of polyhedral combinatorics
Symmetrical polyhedra as a key to memorable organization of thematic sites
Polyhedral implications for "cognitive governance" understood in "global" terms?
Transformation: changing cognitive gear and pattern shape shifting
Web navigation of a global configuration?
Implications of closest packing of polyhedra for wizdome design?
Memorable pattern language for configuring sets of websites?

Table of the important parameters of a non-uniform tetradecahedron, having 2 congruent regular hexagonal faces, 12 congruent trapezoidal faces & 18 vertices lying on a spherical surface with a certain radius, derived by the author by applying "HCR's Theory of Polygon" to calculate solid angle subtended by each regular hexagonal & trapezoidal face & their normal distances from the center of non-uniform tetradecahedron, inscribed radius, circumscribed radius, mean radius, surface area & volume. These formulas are very useful in the analysis, designing & modeling of various non-uniform polyhedra.

Two new three-dimensional frameworks with zeolite-like channels were prepared in the presence of 1,6-diaminohexane. Cu1.5(H3N–(CH2)6–NH3)0.5[C6H2(COO)4]·5H2O (1) crystallizes in the triclinic space group Pequation image with a = 772.56(7), b = 1110.36(7), c = 1111.98(8) pm, α = 98.720(7)°, β = 108.246(9)°, and γ = 95.559(7)°. Cu2(H3N–(CH2)6–NH3)0.5(OH)[C6H2(COO)4]·3H2O (2) crystallizes in the monoclinic space group P2/c with a = 1159.34(11), b = 1059.44(7), c = 1582.2(2) pm, and β = 106.130(11)°. The Cu2+ coordination polyhedra are connected by [C6H2(COO)4]4– anions to yield three-dimensional frameworks with wide centrosymmetric channel-like voids. Complex 1 reveals voids extending along [100] with diagonals of 900 pm and 300 pm, whereas in complex 2 the diagonal of the nearly rectangular crossection of the channels extending parallel to [001] is 900 pm. The negative excess charges of the frameworks are compensated by [H3N–(CH2)6–NH3]2+ cations, which occupy the voids along with water molecules. The [H3N–(CH2)6–NH3]2+ cations are not connected to Cu2+ and have served as templates.

Combining artistic creativity with highly simplified complex geometry, this workshop allows participants, teachers, parents, and children from age 7 and above, to experience first hand the folding movement of two-dimensional nets into three-dimensional polyhedral structures. Participants will become acquainted with the folding and unfolding processes of a broad range of complex polyhedral structures, in a simple, easy way. The building unit comprises eight equilateral triangles, one of 11 known nets of the octahedron, and its underlying mechanism allows not only for the formation of many known polyhedral (deltahedral) structures but also that of many unknown deltahedral formations. Using this model, complex polyhedral structures may be built quickly and easily in a matter of seconds devoid of the need of glue or scissors, angle calculations, or mathematical formulae. The result is an accessible world of polyhedrons available as a new self-assembling geometric model and a new folding game governed by a simple pedagogical technique by which both simple and complex multidimensional polyhedral structures may be formulated.

Colourless single crystals of BaC4O4 have been obtained from aqueous solution at 80 °C. BaC4O4 is stable in air up to 490 °C. BaCO3 is formed by further increase of temperature. BaC4O4 crystallizes in the tetragonal space group I4/mcm (nr. 140) with a=635.95(5), c=1240.77(13) pm, Z=4. Ba2+ is coordinated by eight oxygen atoms of the squarate dianions; Ba—O 276.1(1) pm. The coordination polyhedron is a distorted, square anti-prism. The squarate dianions occupy crystallographic mirror planes and posses approximately 4/mmm symmetry; C—C 145.7(5) and 146.5(5) pm, C—O 125.9(3) pm. A layer-like separation between the cations and anions exists with respect to the [001] direction.

Basic design education, as an introduction of design principles to novice students, has two-way of teaching which are design thinking and professional training, since Bauhaus. Initiated in 2009, the Computation-based Basic Design Studio creates a common ground through discussions between students, academics, and professionals from various backgrounds. In this paper, the implementation of parallel courses named Computation-based Basic Design Studio and Design Geometry is discussed upon final assignment of the first semester-New Year's Lanterns. The given assignment structured as a cyclic process through constant feedback between geometric relations, material performance, and, joinery details to achieve novel outcomes that exceed the preliminarily set structural criteria. In relation to individual processes and outcomes of the final assignment, observed tendencies developed by students', at the end of their first-term in design education, will be discussed as final remarks.

A well-characterized kaolinite has been hydrated in order to test the hypothesis that platey kaolinite will roll upon hydration. Kaolinite hydrates are prepared by repeated intercalation of kaolinite with potassium acetate and subsequent washing with water. On hydration, kaolinite plates roll along the major crystallographic directions to form tubes identical to proper tubular halloysite. Most tubes are elongated along the b crystallographic axis, while some are elongated along the a axis. Overall, the tubes exhibit a range of crystallinity. Well-ordered examples show a 2-layer structure, while poorly ordered tubes show little o1 no 3-dimensional order. Cross-sectional views of the formed tubes show both smoothly curved layers and planar faces. These characteristics of the experimentally formed tubes are shared by natural halloysites. Therefore, it is proposed that planar kaolinite can transform to tubular halloysite.

Analizaremos las características de los cinco poliedros regulares convexos que Platón describe en el Timeo y esclareceremos los siguientes problemas: primero, mientras los poliedros regulares son cinco, los elementos naturales son solo cuatro; segundo, la transformación de unos elementos en otros; tercero, las proporciones que rigen las mezclas de los elementos, y cuarto, las consecuencias de la propuesta platónica en dos problemas científico-filosóficos de ese momento: la causa del movimiento y la existencia o no del vacío.

Introduzione Il Libellus de impletione loci Il presente lavoro ha come obiettivo principale quello di fornire l'edizione critica del Libellus de impletione loci, scritto dal matematico messinese Francesco Maurolico (Messina, 1494 Messina, 1575).

In this paper we present an algorithm for detecting and repairing defects in the boundary of a polyhedron. These defects, usually caused by problems in CAD software, consist of small gaps bounded by edges that are incident to only one polyhedron face. The algorithm uses a partial curve matching technique for matching parts of the defects, and an optimal triangulation of 3-D polygons for resolving the unmatched parts. It is also shown that finding a consistent set of partial curve matches with maximum score, a subproblem which is related to our repairing process, is NP-hard. Experimental results on several polyhedra are presented.

Weaving carpets of triangles into spheres and simpler polyhedra
Eliciting memorable spherically symmetrical semiregular polyhedra
Illustrative application to symbolic reconciliation
Spherical tiling and spherical polyhedra
Use of force directed layout to elicit memorable polyhedra
Potential significance of memorable irregularity?

Order in psychosocial space -- and strategic design implications
Values: Cannon-balls? Wealth symbols? Ball-games? Strange attractors?
Use of distinctive geometries as mapping surfaces for value-goals
Polyhedral mapping reconciling value-goals and their antitheses in the light of ball-games
Global mapping of 17 value-goals alone
Global mapping of 17 value-goals interacting with their 17 problematic opponents
Unusually memorable mapping possibilities of a tetrahedral geodesic spheres
Interplay of value-goals -- integrating opposing "hemispheres of the global brain"
Global mapping in 4D of interactions of competing value-goals?
Integrative comprehension of disparate global value-goals through sphere-packing
Requisite 12-fold organization implied by the 17-fold articulation of value goals -- as its "stellations"?
Unexplored cognitive implications of 17 Sustainable Development Goals -- through "magic squares"?

Monoclinic single crystals of Co[(C6H10)(NH3)2][C6H2(COO)4] · 2H2O have been prepared in aqueous solution at 80 °C. Space group C2/c (no. 15), a = 1065.92(8), b = 1568.97(9), c = 1140.88(9) pm, β = 90.101(6)°, V = 1.9080(2) nm3, Z = 4. Co2+, which is situated on a twofold crystallographic axis, is coordinated in a moderately distorted tetrahedral fashion by four oxygen atoms stemming from the pyromellitate anions (Co-O 197.87(12) and 200.64(12) pm). A three-dimensionally connected coordination polymer is made up by Co2+ and C6H2(COO)44- featuring channel-like voids, which accomodate water molecules and (C6H10)(NH3)22+ cations compensating for the negative excess charge of the three-dimensional framework. Thermogravimetric analysis in air showed that the dehydrated compound was stable between 198 and 361 °C. Further decomposition yielded CoO. Zn[(C6H12)(NH3)2][C6H2(COO)4]·1/2H2O (2) was prepared analogously to 1 employing 1, 6-diaminohexane. Space group P21/n (no. 14), a = 1087.78(8), b = 1515.18(11), c = 1162.21(10) pm, β=96.249(7)°, V = 1.9042(3) nm3, Z = 4. Zn2+ is coordinated tetrahedrally like Co2+ by oxygen atoms of the pyromellitate anions (Zn—O 195.0(4) - 197.8(4)). The connection of Zn2+ with the anions leads similar to 1 to a three-dimensional framework with voids accomodating (C6H12)(NH3)22+-cations and water molecules. 2 was stable anhydrously between approx. 120 and 340 °C, the further decomposition was completed at 700 °C yielding ZnO.

Analizaremos las características de los cinco poliedros regulares convexos que Platón describe en el Timeo y esclareceremos los siguientes problemas: primero, mientras los poliedros regulares son cinco, los elementos naturales son solo cuatro; segundo, la transformación de unos elementos en otros; tercero, las proporciones que rigen las mezclas de los elementos, y cuarto, las consecuencias de la propuesta platónica en dos problemas científico filosóficos de ese momento: la causa del movimiento y la existencia o no del vacío.

Élève de Peter Behrens et d’Auguste Perret, l’architecte Alfred Neumann (1900-1968) est universellement connu par la notion de systèmes de compactage spatial en tant qu’arrangements tridimensionnels d’unités géométriques, conduisant à une morphologie d’agrégations modulaires. Il enseignera au Technion Israel Institute of Technology, Faculté d’Architecture (Haïfa, Israël). Ses principaux projets (avec Zvi Hecker, Eldar Sharon) incluent l’hôtel de ville de Bat Yam (1959-1963), ou les appartements Dubiner à Ramat Gan (1961-1964) en Israël. Par la suite, Alfred Neumann enseignera comme Professeur invité à l’École d’Architecture de l’Université Laval à Québec (QC, CA) en 1962-63; il deviendra Professeur à l’Université Laval (1965-1968) et directeur du Programme de Maitrise. Zvi Hecker, son élève, y sera nommé Professeur adjoint (1969-1972). Suite au legs de sa bibliothèque personnelle, les professeurs Georges Teyssot et Samuel Bernier-Lavigne ont mis sur pied un projet de valorisation de son œuvre. Celui-ci se décline en trois parties : la réalisation d’un meuble (inspiré de son œuvre) pour y accueillir ses livres ; un catalogue illustré comprenant des textes d’Alfred Neumann, d’Eva-Marie Neumann, de Georges Teyssot et de Samuel Bernier-Lavigne ; et une exposition présentant les livres rares de la collection.

All the important parameters of a non-uniform tetradecahedron, having 2 congruent regular hexagonal faces, 12 congruent trapezoidal faces & 18 vertices lying on a spherical surface with a certain radius, have been derived by the author by applying "HCR's Theory of Polygon" to calculate solid angle subtended by each regular hexagonal & trapezoidal face & their normal distances from the center of non-uniform tetradecahedron, inscribed radius, circumscribed radius, mean radius, surface area & volume. These formulas are very useful in the analysis, designing & modeling of various non-uniform polyhedra.

We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in R³. In particular, we prove that the maximum number of facets of the Minkowski sum of two convex polyhedra with m and n facets respectively is bounded from above by f(m, n) = 4mn − 9m − 9n + 26. Given two positive integers m and n, we describe how to construct two convex polyhedra with m and n facets respectively, such that the number of facets of their Minkowski sum is exactly f(m, n). We generalize the construction to yield a lower bound on the maximum complexity of Minkowski sums of many convex polyhedra in R3. That is, given k positive integers m1, m2,..., mk, we describe how to construct k convex polyhedra with corresponding number of facets, such that the number of facets of their Minkowski sum is � 1≤i<j≤k (2mi − 5)(2mj − 5) + � � k � 2 + 1≤i≤k mi. We also provide a conservative upper bound for the general case. Snapshots of several preconstructed convex polyhedra, the Minkows...

The following exercise is based on experiments conducted in circular Origami. This type of paper folding allows for a completely different geometry than the square type since it lends itself very easily to the creation of shapes based on 30-60-90 degree angles. This allows for experimentation with shapes made up of equilateral triangles such as deltahedra. The results of this research were used in Annenberg sponsored activities conducted in a progressive middle school in Houston TX, as well as a workshop presented at the 1999 Bridges Conference in Winfield, KS. Not including preparatory and follow up work by the teacher, the activities in Houston were composed of two main parts, the collaborative construction of a three-yard-across, eighty-faced regular deltahedron (the Endo-Pentakis Icosi-dodecahedron) and the following exercise. The barn-raising was presented last year in Winfield, and the paper folding is the topic of this paper.

A k-orbit map is a map with k flag-orbits under the action of its automorphism group. We give a basic theory of k-orbit maps and classify them up to k 4. "Hurwitz-like" upper bounds for the cardinality of the automorphism groups of 2orbit and 3-orbit maps on surfaces are given. Furthermore, we consider effects of operations like medial and truncation on k-orbit maps and use them in classifying 2-orbit and 3-orbit maps on surfaces of small genus.

Black-brown needle-shaped single crystals of [Co 2 (en) 4 (O 2)(OH)][C 4 O 4 ] 1.5 4H 2 O (en = ethylenediamine) have been prepared in aqueous solution at room temperature. Space group P (no.2) with a = 800.20(8), b = 1225.48(7), c = 1403.84(9) pm, alpha= 100.282(5), beta= 94.515(7), gamma= 95.596(6)°. The Co 3+ cations [Co(1), Co(2)] are coordinated in an octahedral manner by four nitrogen atoms stemming from the ethylenediamine molecules and two oxygen atoms each from a hydroxo group and a peroxo group, respectively. Both Co 3+ coordination polyhedra are connected by a common corner and by the peroxo group leading to the binuclear [(en) 2 Co(O 2)(OH)Co(en) 2 ]3+ cation. The squarate dianions, not bonded to Co 3+ , and the [(en) 2 Co(O 2)(OH)Co(en) 2 ] 3+ cations are linked by hydrogen bonds forming a three-dimensional supramolecular network containing water molecules. Magnetic measurements reveal a diamagnetic behaviour indicating a low-spin electron configuration of Co 3+. The UV-Vis spectra shows two LMCT bands (pi*(O2 2-) ->d (Co 3+)) at 274 and 368 nm and the d-d transition (1A1g-> 1T1g) at 542 nm. Thermoanalytical investigations in air show that the compound is stable up to 120 °C. Subsequent decomposition processes to cobalt oxide are finished at 460 °C.

Combining artistic creativity with highly simplified complex geometry, this workshop allows participants, teachers, parents, and children from age 7 and above, to experience first hand the folding movement of two-dimensional nets into three-dimensional polyhedral structures. Participants will become acquainted with the folding and unfolding processes of a broad range of complex polyhedral structures, in a simple, easy way. The building unit comprises eight equilateral triangles, one of 11 known nets of the octahedron, and its underlying mechanism allows not only for the formation of many known polyhedral (deltahedral) structures but also that of many unknown deltahedral formations. Using this model, complex polyhedral structures may be built quickly and easily in a matter of seconds devoid of the need of glue or scissors, angle calculations, or mathematical formulae. The result is an accessible world of polyhedrons available as a new self-assembling geometric model and a new folding...

The role of geometry in human movement is not as evident as the presence of geometric configurations in art, architecture and design. In this paper, I summarize how my research on geometric configurations transitioned, from theoretical explorations and computational design, to built objects and movement practices. The geometric properties of the icosahedron, one of the five regular polyhedra, are applied to interpret the proportions of the human body, guiding and inspiring movements leading to mindfulness. Introduction: The Geometry of Movement Galileo Galilei poetically defined the universe as “written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures” [1]. Mathematics and geometry are essential in the understanding of the physical world, not only as foundation of science, but also as components in the human-made world and the built environment. Geometry is not only a means in physical sciences but also guides and inspires arti...

Representing a design graphically can make certain properties of a design obvious that otherwise would require tedious reasoning. This holds for symmetries, resolvability, and subdesign inclusion in appropriate cases. We describe how a graphical representation can be achieved in many cases and show several examples, in particular visualizations of the biplanes on up to 16 vertices and subdesigns of the large Witt design.

Resumo Este artigo mostra o desenvolvimento de um ambiente web para a construção de poliedros de Arquimedes em Realidade Aumentada (RA) e Realidade Virtual (RV). Neste ambiente foram utilizadas as transformações geométricas de translação e rotação com a estrutura de hierarquias de páginas HTML, sem o uso das coorde-nadas de cada vértice do poliedro. O ambiente desenvolvido pode ser usado em sala de aula para a visualização dos poliedros em Realidade Aumentada, com a possibilidade de manipulações das representações gráficas pelos alunos no am-biente criado em Realidade Virtual. Outros estudos que podem ser desenvolvidos com os poliedros modelados são de áreas, volumes e a relação de Euler. Outro conteúdo importante que pode ser desenvolvido é o de seções planas, pois sete poliedros de Arquimedes são obtidos através de seções planas dos poliedros de Platão. Com este trabalho, torna-se possível o desenvolvimento de materiais di-dáticos com uma tecnologia simples, gratuita e com grande contribuição para a melhoria do ensino da Geometria e outras áreas que utilizam objetos 3D. Palavras-chave: Realidade Aumentada; Realidade Virtual; Visualização gráfica.
Abstract This paper shows the development of a web environment for the construction of Archimedes polyhedra in Augmented Reality (AR) and Virtual Reality (VR). In this environment we used the geometric transformations of translation and rotation with the structure of hierarchies of HTML pages, without the use of the coordinates of each vertex of the polyhedron. The developed environment can be used in classroom to visualize the polyhedra in Augmented Reality, with the possibility of manipulations of the graphical representations by students in the environment created in Virtual Reality. Other studies that can be developed with the polyhedra modeled are areas, volumes and the relation of Euler. Another important content that can be developed is truncation, because seven Archimedes polyhedra are obtained through truncation of Plato's polyhedrons. With this work, it becomes possible to develop didactic materials with a simple technology, free and with great contribution to improvement of the teaching of Geometry and other areas that use 3D objects.

Coronavirus representations compared with a global pattern of nuclear explosions
Designing a rotating global pattern of "erupting spikes" in 3D
Number of equidistant "spikes" in a global spherical array?
Options for representation in 3D of a "mushroom-spike"
Distribution of "mushroom-spikes" around a sphere
Biomimicry and global polyhedral organization
Polyhedral organization of coronavirus?
Global "spiky-prickly" indications of a "pattern that connects"
Global deployment of "proactive" technology
Psychosocial "global implication" of a "pattern that connects"?
Global spiking of knowledge and ignorance
Global connectivity as reinforced through symbolic configuration?

Questions
Interrelating the questionable faces of Facebook via polyhedral animation
Role of Facebook AI: superficial connectivity vs systemic insight?
Polyhedra versus checklists in the coherent organization of categories
Virtual reality of Facebook -- how it really "works"?

Explanation as interplay of projection and "conjection"?
Surrogates of "conjection" as an unrecognized cognitive process?
Reframing NATO and The Pentagon?
Simple possibilities of exploring the complexity of NATO otherwise
Polyhedral duality as a means of engaging otherwise with dualism: NATO's shadow?
Comprehension of NATO as a four-dimensional dynamic entity?
Implications of stellation of the NATO star symbol?
Comprehending NATO through biomimicry?
NATO as a dynamic configuration of coats of arms and feedback cycles?
Classification of "projections" versus Classification of "conjections"?

Triclinic single crystals of Cu4(H3N–(CH2)9–NH3)(OH)2[C6H2(COO)4]2•5H2O were prepared in aqueous solution at 80 °C in the presence of 1,9-diaminononane. Space group P-1 (no. 2) with a = 1057.5(2), b = 1166.0(2), c = 1576.7(2) pm, α = 106.080(10)°, β = 90.73(2)° and γ = 94.050(10)°. The four crystallographic independent Cu2+ ions are surrounded by five oxygen atoms each with Cu–O distances between 191.4(3) and 231.7(4) pm. The connection between the Cu2+ coordination polyhedra and the [C6H2(COO)4]4– anions yields three-dimensional framework with negative excess charge and wide centrosymmetric channel-like voids. These voids extend parallel to [001] with the diagonal of the nearly rectangular cross-section of approximately 900 pm. The channels of the framework accommodate [H3N–(CH2)9–NH3]2+ cations and water molecules, which are not connected to Cu2+. The nonane-1,9-diammonium cations adopt a partial gauche conformation. Thermoanalytical measurements in air show a loss of water of crystallization starting at 90 °C and finishing at approx. 170 °C. The dehydrated compound is stable up to 260 °C followed by an exothermic decomposition yielding copper oxide.

netic resonance imaging) apparata. These cross sections, In this paper we present a new technique for piecewise-linear hereafter called slices, are the basis for interpolating the surface reconstruction from a series of parallel polygonal cross boundary surface of the organ. The interpolated object sections. This is an important problem in medical imaging, can then be displayed in graphics applications, or (more surface reconstruction from topographic data, and other applirecently) even manufactured by an NC (numerically concations. We reduce the problem, as in most previous works, to trolled) or an RP (rapid prototyping) machine. Another a series of problems of piecewise-linear interpolation between motivation for this problem is the nondestructive digitizaeach pair of successive slices. Our algorithm uses a partial curve tion of objects: after an object is scanned by an echomatching technique for matching parts of the contours, an graphic or an X-ray apparatus, the obtained slices are used optimal triangulation of 3-D polygons for resolving the unfor the reconstruction of the original object. Yet another matched parts, and a minimum spanning tree heuristic for interpolating between nonsimply connected regions. Unlike motivation is the reconstruction of a three-dimensional previous attempts at solving this problem, our algorithm seems model of a terrain from topographic elevation contours. to handle successfully in practice any kind of data. It allows Many solutions were suggested for the pure raster intermultiple contours in each slice, with any hierarchy of contour polation. These usually handle two raster images, where nesting, and avoids the introduction of counterintuitive bridges each pixel is either white or black, or assigned a gray level between contours, proposed in some earlier papers to handle taken from a fixed range. The interpolation produces one interpolation between multiply connected regions. Experimenor more intermediate raster images, which smoothly and tal results on various complex examples, involving actual medilocally turn the first image into the second one. Then, the cal imaging data, are presented and show the good and robust bounding surface is detected using other methods, such as performance of our algorithm.

Global strategic plan for coronavirus pandemic?
Relevance of doughnut model to a pandemic?
Torus as a key to strategic knowledge organization?
Strategic implications of the helical core of the coronavirus?
Pineapple model of global governance?
Formalizing a pineapple model of governance through zonagons and zonohedrification?
Zomes as a key to appropriate organizational and knowledge architecture?
Perception and uncertainty in global governance -- "one model fits all"?
Nine-fold configuration of planetary boundaries?
Playful transformation between forms of global governance -- "organizing"?

We consider the problem of finding a strip separating between two polygons, whose intersection with a third (convex) polygon is of maximum area. We present an optimal linear-time algorithm for computing the optimum strip. When the third polygon is not convex, the running time of the algorithm is quadratic in the size of the input. The application in mind is the piecewise-linear surface interpolation in simple branching cases, where the sought volume branches from one contour in one slice into two contours in the other slice.