Convex Segmentochora
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Systematic Narcissism
Systematic Narcissism Wendy W. Fok University of Houston Systematic Narcissism is a hybridization through the notion of the plane- tary grid system and the symmetry of narcissism. The installation is based on the obsession of the physical reflection of the object/subject relation- ship, created by the idea of man-made versus planetary projected planes (grids) which humans have made onto the earth. ie: Pierre Charles L’Enfant plan 1791 of the golden section projected onto Washington DC. The fostered form is found based on a triacontahedron primitive and devel- oped according to the primitive angles. Every angle has an increment of 0.25m and is based on a system of derivatives. As indicated, the geometry of the world (or the globe) is based off the planetary grid system, which, according to Bethe Hagens and William S. Becker is a hexakis icosahedron grid system. The formal mathematical aspects of the installation are cre- ated by the offset of that geometric form. The base of the planetary grid and the narcissus reflection of the modular pieces intersect the geometry of the interfacing modular that structures the installation BACKGROUND: Poetics: Narcissism is a fixation with oneself. The Greek myth of Narcissus depicted a prince who sat by a reflective pond for days, self-absorbed by his own beauty. Mathematics: In 1978, Professors William Becker and Bethe Hagens extended the Russian model, inspired by Buckminster Fuller’s geodesic dome, into a grid based on the rhombic triacontahedron, the dual of the icosidodeca- heron Archimedean solid. The triacontahedron has 30 diamond-shaped faces, and possesses the combined vertices of the icosahedron and the dodecahedron. -
Geometric Shapes and Cut Sections
GEOMETRIC SHAPES AND CUT SECTIONS MODEL NO: EG.GEO • Wooden and Perspex Material • Long Life and Sturdy • Solid and Hollow sections • Clear Labeling • Small and Large sizes • Magnetic Fractions also Available Object Drawing Models Interpenetration of Solids Models Dissected and Non-Dissected • Two Cylinders of Different Diameters Cube Penetrating Each Other at Right Angles Cone • Two Cylinders of Different Diameters with Sphere Oblique Penetration Cylinder • A Cylinder into a Cone with Oblique Rectangular Prism Penetration Rectangular Pyramid • A Cone & Cylinder with Penetration at Triangular Prism Right Angles Triangular Pyramid • Two Cones at Right Angles Square Prism • Two Square Prisms of Different Sizes at Square Pyramid Right Angles Pentagonal Prism • Two Square Prisms at Oblique Angles Pentagonal Pyramid • Sphere & Cone at Centre Hexagonal Prism • Sphere & Cylinder Eccentric Hexagonal Pyramid • Sphere &Cylinder at Centre Octagonal Prism Octagonal Pyramid Decagonal Prism Decagonal Pyramid Development of Surface for all Shapes *Oblique Polyhedrons for Above Available Available TEACHING AIDS FOR ENGINEERING DRAWING Set of Cut Sectioned Geometric Shapes Set of Solid Wooden Prism and Pyramids Set of Oblique Polyhedrons Set of Geometric Shapes with Development of Surface Paper Nets * Mechmatics reserves the right to modify its Product without notice * In the interest of continued improvements to our products, we reserve the right to change specifications without prior notification. Common Examples of Loci of Point with Pencil tracing position and Theoretical Diagrams * In the interest of continued improvements to our products, we reserve the right to change specifications without prior notification. DEVELOPMENT OF SURFACES MODEL ED.DS A development is the unfold / unrolled flat / plane figure of a 3-D object. -
The Continuum of Space Architecture: from Earth to Orbit
42nd International Conference on Environmental Systems AIAA 2012-3575 15 - 19 July 2012, San Diego, California The Continuum of Space Architecture: From Earth to Orbit Marc M. Cohen1 Marc M. Cohen Architect P.C. – Astrotecture™, Palo Alto, CA, 94306 Space architects and engineers alike tend to see spacecraft and space habitat design as an entirely new departure, disconnected from the Earth. However, at least for Space Architecture, there is a continuum of development since the earliest formalizations of terrestrial architecture. Moving out from 1-G, Space Architecture enables the continuum from 1-G to other gravity regimes. The history of Architecture on Earth involves finding new ways to resist Gravity with non-orthogonal structures. Space Architecture represents a new milestone in this progression, in which gravity is reduced or altogether absent from the habitable environment. I. Introduction EOMETRY is Truth2. Gravity is the constant.3 Gravity G is the constant – perhaps the only constant – in the evolution of life on Earth and the human response to the Earth’s environment.4 The Continuum of Architecture arises from geometry in building as a primary human response to gravity. It leads to the development of fundamental components of construction on Earth: Column, Wall, Floor, and Roof. According to the theoretician Abbe Laugier, the column developed from trees; the column engendered the wall, as shown in FIGURE 1 his famous illustration of “The Primitive Hut.” The column aligns with the human bipedal posture, where the spine, pelvis, and legs are the gravity- resisting structure. Caryatids are the highly literal interpretation of this phenomenon of standing to resist gravity, shown in FIGURE 2. -
Composite Concave Cupolae As Geometric and Architectural Forms
Journal for Geometry and Graphics Volume 19 (2015), No. 1, 79–91. Composite Concave Cupolae as Geometric and Architectural Forms Slobodan Mišić1, Marija Obradović1, Gordana Ðukanović2 1Faculty of Civil Engineering, University of Belgrade Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia emails: [email protected], [email protected] 2Faculty of Forestry, University of Belgrade, Serbia email: [email protected] Abstract. In this paper, the geometry of concave cupolae has been the starting point for the generation of composite polyhedral structures, usable as formative patterns for architectural purposes. Obtained by linking paper folding geometry with the geometry of polyhedra, concave cupolae are polyhedra that follow the method of generating cupolae (Johnson’s solids: J3, J4 and J5); but we removed the convexity criterion and omitted squares in the lateral surface. Instead of alter- nating triangles and squares there are now two or more paired series of equilateral triangles. The criterion of face regularity is respected, as well as the criterion of multiple axial symmetry. The distribution of the triangles is based on strictly determined and mathematically defined parameters, which allows the creation of such structures in a way that qualifies them as an autonomous group of polyhedra — concave cupolae of sorts II, IV, VI (2N). If we want to see these structures as polyhedral surfaces (not as solids) connecting the concept of the cupola (dome) in the architectural sense with the geometrical meaning of (concave) cupola, we re- move the faces of the base polygons. Thus we get a deltahedral structure — a shell made entirely from equilateral triangles, which is advantageous for the purpose of prefabrication. -
The Volumes of a Compact Hyperbolic Antiprism
The volumes of a compact hyperbolic antiprism Vuong Huu Bao joint work with Nikolay Abrosimov Novosibirsk State University G2R2 August 6-18, Novosibirsk, 2018 Vuong Huu Bao (NSU) The volumes of a compact hyperbolic antiprism Aug. 6-18,2018 1/14 Introduction Calculating volumes of polyhedra is a classical problem, that has been well known since Euclid and remains relevant nowadays. This is partly due to the fact that the volume of a fundamental polyhedron is one of the main geometrical invariants for a 3-dimensional manifold. Every 3-manifold can be presented by a fundamental polyhedron. That means we can pair-wise identify the faces of some polyhedron to obtain a 3-manifold. Thus the volume of 3-manifold is the volume of prescribed fundamental polyhedron. Theorem (Thurston, Jørgensen) The volumes of hyperbolic 3-dimensional hyperbolic manifolds form a closed non-discrete set on the real line. This set is well ordered. There are only finitely many manifolds with a given volume. Vuong Huu Bao (NSU) The volumes of a compact hyperbolic antiprism Aug. 6-18,2018 2/14 Introduction 1835, Lobachevsky and 1982, Milnor computed the volume of an ideal hyperbolic tetrahedron in terms of Lobachevsky function. 1993, Vinberg computed the volume of hyperbolic tetrahedron with at least one vertex at infinity. 1907, Gaetano Sforza; 1999, Yano, Cho ; 2005 Murakami; 2005 Derevnin, Mednykh gave different formulae for general hyperbolic tetrahedron. 2009, N. Abrosimov, M. Godoy and A. Mednykh found the volumes of spherical octahedron with mmm or 2 m-symmetry. | 2013, N. Abrosimov and G. Baigonakova, found the volume of hyperbolic octahedron with mmm-symmetry. -
Uniform Polychora
BRIDGES Mathematical Connections in Art, Music, and Science Uniform Polychora Jonathan Bowers 11448 Lori Ln Tyler, TX 75709 E-mail: [email protected] Abstract Like polyhedra, polychora are beautiful aesthetic structures - with one difference - polychora are four dimensional. Although they are beyond human comprehension to visualize, one can look at various projections or cross sections which are three dimensional and usually very intricate, these make outstanding pieces of art both in model form or in computer graphics. Polygons and polyhedra have been known since ancient times, but little study has gone into the next dimension - until recently. Definitions A polychoron is basically a four dimensional "polyhedron" in the same since that a polyhedron is a three dimensional "polygon". To be more precise - a polychoron is a 4-dimensional "solid" bounded by cells with the following criteria: 1) each cell is adjacent to only one other cell for each face, 2) no subset of cells fits criteria 1, 3) no two adjacent cells are corealmic. If criteria 1 fails, then the figure is degenerate. The word "polychoron" was invented by George Olshevsky with the following construction: poly = many and choron = rooms or cells. A polytope (polyhedron, polychoron, etc.) is uniform if it is vertex transitive and it's facets are uniform (a uniform polygon is a regular polygon). Degenerate figures can also be uniform under the same conditions. A vertex figure is the figure representing the shape and "solid" angle of the vertices, ex: the vertex figure of a cube is a triangle with edge length of the square root of 2. -
E8 Physics and Quasicrystals Icosidodecahedron and Rhombic Triacontahedron Vixra 1301.0150 Frank Dodd (Tony) Smith Jr
E8 Physics and Quasicrystals Icosidodecahedron and Rhombic Triacontahedron vixra 1301.0150 Frank Dodd (Tony) Smith Jr. - 2013 The E8 Physics Model (viXra 1108.0027) is based on the Lie Algebra E8. 240 E8 vertices = 112 D8 vertices + 128 D8 half-spinors where D8 is the bivector Lie Algebra of the Real Clifford Algebra Cl(16) = Cl(8)xCl(8). 112 D8 vertices = (24 D4 + 24 D4) = 48 vertices from the D4xD4 subalgebra of D8 plus 64 = 8x8 vertices from the coset space D8 / D4xD4. 128 D8 half-spinor vertices = 64 ++half-half-spinors + 64 --half-half-spinors. An 8-dim Octonionic Spacetime comes from the Cl(8) factors of Cl(16) and a 4+4 = 8-dim Kaluza-Klein M4 x CP2 Spacetime emerges due to the freezing out of a preferred Quaternionic Subspace. Interpreting World-Lines as Strings leads to 26-dim Bosonic String Theory in which 10 dimensions reduce to 4-dim CP2 and a 6-dim Conformal Spacetime from which 4-dim M4 Physical Spacetime emerges. Although the high-dimensional E8 structures are fundamental to the E8 Physics Model it may be useful to see the structures from the point of view of the familiar 3-dim Space where we live. To do that, start by looking the the E8 Root Vector lattice. Algebraically, an E8 lattice corresponds to an Octonion Integral Domain. There are 7 Independent E8 Lattice Octonion Integral Domains corresponding to the 7 Octonion Imaginaries, as described by H. S. M. Coxeter in "Integral Cayley Numbers" (Duke Math. J. 13 (1946) 561-578 and in "Regular and Semi-Regular Polytopes III" (Math. -
Archimedean Solids
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat). -
The View from Six Dimensions
Walls and Bridges The view from Six Dimensiosn Wendy Y. Krieger [email protected] ∗ January, Abstract Walls divide, bridges unite. This idea is applied to devising a vocabulary suited for the study of higher dimensions. Points are connected, solids divided. In higher dimensions, there are many more products and concepts visible. The four polytope products (prism, tegum, pyramid and comb), lacing and semiate figures, laminates are all discussed. Many of these become distinct in four to six dimensions. Walls and Bridges Consider a knife. Its main action is to divide solids into pieces. This is done by a sweeping action, although the presence of solid materials might make the sweep a little less graceful. What might a knife look like in four dimensions. A knife would sweep a three-dimensional space, and thus the blade is two-dimensional. The purpose of the knife is to divide, and therefore its dimension is fixed by what it divides. Walls divide, bridges unite. When things are thought about in the higher dimensions, the dividing or uniting nature of it is more important than its innate dimensionality. A six-dimensional blade has four dimensions, since its sweep must make five dimensions. There are many idioms that suggest the role of an edge or line is to divide. This most often hap pens when the referent dimension is the two-dimensional ground, but the edge of a knife makes for a three-dimensional referent. A line in the sand, a deadline, and to the edge, all suggest boundaries of two-dimensional areas, where the line or edge divides. -
Systematics of Atomic Orbital Hybridization of Coordination Polyhedra: Role of F Orbitals
molecules Article Systematics of Atomic Orbital Hybridization of Coordination Polyhedra: Role of f Orbitals R. Bruce King Department of Chemistry, University of Georgia, Athens, GA 30602, USA; [email protected] Academic Editor: Vito Lippolis Received: 4 June 2020; Accepted: 29 June 2020; Published: 8 July 2020 Abstract: The combination of atomic orbitals to form hybrid orbitals of special symmetries can be related to the individual orbital polynomials. Using this approach, 8-orbital cubic hybridization can be shown to be sp3d3f requiring an f orbital, and 12-orbital hexagonal prismatic hybridization can be shown to be sp3d5f2g requiring a g orbital. The twists to convert a cube to a square antiprism and a hexagonal prism to a hexagonal antiprism eliminate the need for the highest nodality orbitals in the resulting hybrids. A trigonal twist of an Oh octahedron into a D3h trigonal prism can involve a gradual change of the pair of d orbitals in the corresponding sp3d2 hybrids. A similar trigonal twist of an Oh cuboctahedron into a D3h anticuboctahedron can likewise involve a gradual change in the three f orbitals in the corresponding sp3d5f3 hybrids. Keywords: coordination polyhedra; hybridization; atomic orbitals; f-block elements 1. Introduction In a series of papers in the 1990s, the author focused on the most favorable coordination polyhedra for sp3dn hybrids, such as those found in transition metal complexes. Such studies included an investigation of distortions from ideal symmetries in relatively symmetrical systems with molecular orbital degeneracies [1] In the ensuing quarter century, interest in actinide chemistry has generated an increasing interest in the involvement of f orbitals in coordination chemistry [2–7]. -
A Congruence Problem for Polyhedra
A congruence problem for polyhedra Alexander Borisov, Mark Dickinson, Stuart Hastings April 18, 2007 Abstract It is well known that to determine a triangle up to congruence requires 3 measurements: three sides, two sides and the included angle, or one side and two angles. We consider various generalizations of this fact to two and three dimensions. In particular we consider the following question: given a convex polyhedron P , how many measurements are required to determine P up to congruence? We show that in general the answer is that the number of measurements required is equal to the number of edges of the polyhedron. However, for many polyhedra fewer measurements suffice; in the case of the cube we show that nine carefully chosen measurements are enough. We also prove a number of analogous results for planar polygons. In particular we describe a variety of quadrilaterals, including all rhombi and all rectangles, that can be determined up to congruence with only four measurements, and we prove the existence of n-gons requiring only n measurements. Finally, we show that one cannot do better: for any ordered set of n distinct points in the plane one needs at least n measurements to determine this set up to congruence. An appendix by David Allwright shows that the set of twelve face-diagonals of the cube fails to determine the cube up to conjugacy. Allwright gives a classification of all hexahedra with all face- diagonals of equal length. 1 Introduction We discuss a class of problems about the congruence or similarity of three dimensional polyhedra. -
Year 6 – Wednesday 24Th June 2020 – Maths
1 Year 6 – Wednesday 24th June 2020 – Maths Can I identify 3D shapes that have pairs of parallel or perpendicular edges? Parallel – edges that have the same distance continuously between them – parallel edges never meet. Perpendicular – when two edges or faces meet and create a 90o angle. 1 4. 7. 10. 2 5. 8. 11. 3. 6. 9. 12. C1 – Using the shapes above: 1. Name and sort the shapes into: a. Pyramids b. Prisms 2. Draw a table to identify how many faces, edges and vertices each shape has. 3. Write your own geometric definition: a. Prism b. pyramid 4. Which shape is the odd one out? Explain why. C2 – Using the shapes above; 1. Which of the shapes have pairs of parallel edges in: a. All their faces? b. More than one half of the faces? c. One face only? 2. The following shapes have pairs of perpendicular edges. Identify the faces they are in: 2 a. A cube b. A square based pyramid c. A triangular prism d. A cuboid 3. Which shape with straight edges has no perpendicular edges? 4. Which shape has perpendicular edges in the shape but not in any face? C3 – Using the above shapes: 1. How many faces have pairs of parallel edges in: a. A hexagonal pyramid? b. A decagonal (10-sided) based prism? c. A heptagonal based prism? d. Which shape has no face with parallel edges but has parallel edges in the shape? 2. How many faces have perpendicular edges in: a. A pentagonal pyramid b. A hexagonal pyramid c.