Instructions for Plato's Solids
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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. -
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). -
Shape Skeletons Creating Polyhedra with Straws
Shape Skeletons Creating Polyhedra with Straws Topics: 3-Dimensional Shapes, Regular Solids, Geometry Materials List Drinking straws or stir straws, cut in Use simple materials to investigate regular or advanced 3-dimensional shapes. half Fun to create, these shapes make wonderful showpieces and learning tools! Paperclips to use with the drinking Assembly straws or chenille 1. Choose which shape to construct. Note: the 4-sided tetrahedron, 8-sided stems to use with octahedron, and 20-sided icosahedron have triangular faces and will form sturdier the stir straws skeletal shapes. The 6-sided cube with square faces and the 12-sided Scissors dodecahedron with pentagonal faces will be less sturdy. See the Taking it Appropriate tool for Further section. cutting the wire in the chenille stems, Platonic Solids if used This activity can be used to teach: Common Core Math Tetrahedron Cube Octahedron Dodecahedron Icosahedron Standards: Angles and volume Polyhedron Faces Shape of Face Edges Vertices and measurement Tetrahedron 4 Triangles 6 4 (Measurement & Cube 6 Squares 12 8 Data, Grade 4, 5, 6, & Octahedron 8 Triangles 12 6 7; Grade 5, 3, 4, & 5) Dodecahedron 12 Pentagons 30 20 2-Dimensional and 3- Dimensional Shapes Icosahedron 20 Triangles 30 12 (Geometry, Grades 2- 12) 2. Use the table and images above to construct the selected shape by creating one or Problem Solving and more face shapes and then add straws or join shapes at each of the vertices: Reasoning a. For drinking straws and paperclips: Bend the (Mathematical paperclips so that the 2 loops form a “V” or “L” Practices Grades 2- shape as needed, widen the narrower loop and insert 12) one loop into the end of one straw half, and the other loop into another straw half. -
Rhombic Triacontahedron
Rhombic Triacontahedron Figure 1 Rhombic Triacontahedron. Vertex labels as used for the corresponding vertices of the 120 Polyhedron. COPYRIGHT 2007, Robert W. Gray Page 1 of 11 Encyclopedia Polyhedra: Last Revision: August 5, 2007 RHOMBIC TRIACONTAHEDRON Figure 2 Icosahedron (red) and Dodecahedron (yellow) define the rhombic Triacontahedron (green). Figure 3 “Long” (red) and “short” (yellow) face diagonals and vertices. COPYRIGHT 2007, Robert W. Gray Page 2 of 11 Encyclopedia Polyhedra: Last Revision: August 5, 2007 RHOMBIC TRIACONTAHEDRON Topology: Vertices = 32 Edges = 60 Faces = 30 diamonds Lengths: 15+ ϕ = 2 EL ≡ Edge length of rhombic Triacontahedron. ϕ + 2 EL = FDL ≅ 0.587 785 252 FDL 2ϕ 3 − ϕ = DVF ϕ 2 FDL ≡ Long face diagonal = DVF ≅ 1.236 067 977 DVF ϕ 2 = EL ≅ 1.701 301 617 EL 3 − ϕ 1 FDS ≡ Short face diagonal = FDL ≅ 0.618 033 989 FDL ϕ 2 = DVF ≅ 0.763 932 023 DVF ϕ 2 2 = EL ≅ 1.051 462 224 EL ϕ + 2 COPYRIGHT 2007, Robert W. Gray Page 3 of 11 Encyclopedia Polyhedra: Last Revision: August 5, 2007 RHOMBIC TRIACONTAHEDRON DFVL ≡ Center of face to vertex at the end of a long face diagonal 1 = FDL 2 1 = DVF ≅ 0.618 033 989 DVF ϕ 1 = EL ≅ 0.850 650 808 EL 3 − ϕ DFVS ≡ Center of face to vertex at the end of a short face diagonal 1 = FDL ≅ 0.309 016 994 FDL 2 ϕ 1 = DVF ≅ 0.381 966 011 DVF ϕ 2 1 = EL ≅ 0.525 731 112 EL ϕ + 2 ϕ + 2 DFE = FDL ≅ 0.293 892 626 FDL 4 ϕ ϕ + 2 = DVF ≅ 0.363 271 264 DVF 21()ϕ + 1 = EL 2 ϕ 3 − ϕ DVVL = FDL ≅ 0.951 056 516 FDL 2 = 3 − ϕ DVF ≅ 1.175 570 505 DVF = ϕ EL ≅ 1.618 033 988 EL COPYRIGHT 2007, Robert W. -
Can Every Face of a Polyhedron Have Many Sides ?
Can Every Face of a Polyhedron Have Many Sides ? Branko Grünbaum Dedicated to Joe Malkevitch, an old friend and colleague, who was always partial to polyhedra Abstract. The simple question of the title has many different answers, depending on the kinds of faces we are willing to consider, on the types of polyhedra we admit, and on the symmetries we require. Known results and open problems about this topic are presented. The main classes of objects considered here are the following, listed in increasing generality: Faces: convex n-gons, starshaped n-gons, simple n-gons –– for n ≥ 3. Polyhedra (in Euclidean 3-dimensional space): convex polyhedra, starshaped polyhedra, acoptic polyhedra, polyhedra with selfintersections. Symmetry properties of polyhedra P: Isohedron –– all faces of P in one orbit under the group of symmetries of P; monohedron –– all faces of P are mutually congru- ent; ekahedron –– all faces have of P the same number of sides (eka –– Sanskrit for "one"). If the number of sides is k, we shall use (k)-isohedron, (k)-monohedron, and (k)- ekahedron, as appropriate. We shall first describe the results that either can be found in the literature, or ob- tained by slight modifications of these. Then we shall show how two systematic ap- proaches can be used to obtain results that are better –– although in some cases less visu- ally attractive than the old ones. There are many possible combinations of these classes of faces, polyhedra and symmetries, but considerable reductions in their number are possible; we start with one of these, which is well known even if it is hard to give specific references for precisely the assertion of Theorem 1. -
Putting the Icosahedron Into the Octahedron
Forum Geometricorum Volume 17 (2017) 63–71. FORUM GEOM ISSN 1534-1178 Putting the Icosahedron into the Octahedron Paris Pamfilos Abstract. We compute the dimensions of a regular tetragonal pyramid, which allows a cut by a plane along a regular pentagon. In addition, we relate this construction to a simple construction of the icosahedron and make a conjecture on the impossibility to generalize such sections of regular pyramids. 1. Pentagonal sections The present discussion could be entitled Organizing calculations with Menelaos, and originates from a problem from the book of Sharygin [2, p. 147], in which it is required to construct a pentagonal section of a regular pyramid with quadrangular F J I D T L C K H E M a x A G B U V Figure 1. Pentagonal section of quadrangular pyramid basis. The basis of the pyramid is a square of side-length a and the pyramid is assumed to be regular, i.e. its apex F is located on the orthogonal at the center E of the square at a distance h = |EF| from it (See Figure 1). The exercise asks for the determination of the height h if we know that there is a section GHIJK of the pyramid by a plane which is a regular pentagon. The section is tacitly assumed to be symmetric with respect to the diagonal symmetry plane AF C of the pyramid and defined by a plane through the three points K, G, I. The first two, K and G, lying symmetric with respect to the diagonal AC of the square. -
Arxiv:1705.01294V1
Branes and Polytopes Luca Romano email address: [email protected] ABSTRACT We investigate the hierarchies of half-supersymmetric branes in maximal supergravity theories. By studying the action of the Weyl group of the U-duality group of maximal supergravities we discover a set of universal algebraic rules describing the number of independent 1/2-BPS p-branes, rank by rank, in any dimension. We show that these relations describe the symmetries of certain families of uniform polytopes. This induces a correspondence between half-supersymmetric branes and vertices of opportune uniform polytopes. We show that half-supersymmetric 0-, 1- and 2-branes are in correspondence with the vertices of the k21, 2k1 and 1k2 families of uniform polytopes, respectively, while 3-branes correspond to the vertices of the rectified version of the 2k1 family. For 4-branes and higher rank solutions we find a general behavior. The interpretation of half- supersymmetric solutions as vertices of uniform polytopes reveals some intriguing aspects. One of the most relevant is a triality relation between 0-, 1- and 2-branes. arXiv:1705.01294v1 [hep-th] 3 May 2017 Contents Introduction 2 1 Coxeter Group and Weyl Group 3 1.1 WeylGroup........................................ 6 2 Branes in E11 7 3 Algebraic Structures Behind Half-Supersymmetric Branes 12 4 Branes ad Polytopes 15 Conclusions 27 A Polytopes 30 B Petrie Polygons 30 1 Introduction Since their discovery branes gained a prominent role in the analysis of M-theories and du- alities [1]. One of the most important class of branes consists in Dirichlet branes, or D-branes. D-branes appear in string theory as boundary terms for open strings with mixed Dirichlet-Neumann boundary conditions and, due to their tension, scaling with a negative power of the string cou- pling constant, they are non-perturbative objects [2]. -
The Crystal Forms of Diamond and Their Significance
THE CRYSTAL FORMS OF DIAMOND AND THEIR SIGNIFICANCE BY SIR C. V. RAMAN AND S. RAMASESHAN (From the Department of Physics, Indian Institute of Science, Bangalore) Received for publication, June 4, 1946 CONTENTS 1. Introductory Statement. 2. General Descriptive Characters. 3~ Some Theoretical Considerations. 4. Geometric Preliminaries. 5. The Configuration of the Edges. 6. The Crystal Symmetry of Diamond. 7. Classification of the Crystal Forros. 8. The Haidinger Diamond. 9. The Triangular Twins. 10. Some Descriptive Notes. 11. The Allo- tropic Modifications of Diamond. 12. Summary. References. Plates. 1. ~NTRODUCTORY STATEMENT THE" crystallography of diamond presents problems of peculiar interest and difficulty. The material as found is usually in the form of complete crystals bounded on all sides by their natural faces, but strangely enough, these faces generally exhibit a marked curvature. The diamonds found in the State of Panna in Central India, for example, are invariably of this kind. Other diamondsJas for example a group of specimens recently acquired for our studies ffom Hyderabad (Deccan)--show both plane and curved faces in combination. Even those diamonds which at first sight seem to resemble the standard forms of geometric crystallography, such as the rhombic dodeca- hedron or the octahedron, are found on scrutiny to exhibit features which preclude such an identification. This is the case, for example, witb. the South African diamonds presented to us for the purpose of these studŸ by the De Beers Mining Corporation of Kimberley. From these facts it is evident that the crystallography of diamond stands in a class by itself apart from that of other substances and needs to be approached from a distinctive stand- point. -
Sink Or Float? Thought Problems in Math and Physics
AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 33 33 Sink or Float? Sink or Float: Thought Problems in Math and Physics is a collection of prob- lems drawn from mathematics and the real world. Its multiple-choice format Sink or Float? forces the reader to become actively involved in deciding upon the answer. Thought Problems in Math and Physics The book’s aim is to show just how much can be learned by using everyday Thought Problems in Math and Physics common sense. The problems are all concrete and understandable by nearly anyone, meaning that not only will students become caught up in some of Keith Kendig the questions, but professional mathematicians, too, will easily get hooked. The more than 250 questions cover a wide swath of classical math and physics. Each problem s solution, with explanation, appears in the answer section at the end of the book. A notable feature is the generous sprinkling of boxes appearing throughout the text. These contain historical asides or A tub is filled Will a solid little-known facts. The problems themselves can easily turn into serious with lubricated aluminum or tin debate-starters, and the book will find a natural home in the classroom. ball bearings. cube sink... ... or float? Keith Kendig AMSMAA / PRESS 4-Color Process 395 page • 3/4” • 50lb stock • finish size: 7” x 10” i i \AAARoot" – 2009/1/21 – 13:22 – page i – #1 i i Sink or Float? Thought Problems in Math and Physics i i i i i i \AAARoot" – 2011/5/26 – 11:22 – page ii – #2 i i c 2008 by The -
15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm
15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P . -
THE INVENTION of ATOMIST ICONOGRAPHY 1. Introductory
THE INVENTION OF ATOMIST ICONOGRAPHY Christoph Lüthy Center for Medieval and Renaissance Natural Philosophy University of Nijmegen1 1. Introductory Puzzlement For centuries now, particles of matter have invariably been depicted as globules. These glob- ules, representing very different entities of distant orders of magnitudes, have in turn be used as pictorial building blocks for a host of more complex structures such as atomic nuclei, mole- cules, crystals, gases, material surfaces, electrical currents or the double helixes of DNA. May- be it is because of the unsurpassable simplicity of the spherical form that the ubiquity of this type of representation appears to us so deceitfully self-explanatory. But in truth, the spherical shape of these various units of matter deserves nothing if not raised eyebrows. Fig. 1a: Giordano Bruno: De triplici minimo et mensura, Frankfurt, 1591. 1 Research for this contribution was made possible by a fellowship at the Max-Planck-Institut für Wissenschafts- geschichte (Berlin) and by the Netherlands Organization for Scientific Research (NWO), grant 200-22-295. This article is based on a 1997 lecture. Christoph Lüthy Fig. 1b: Robert Hooke, Micrographia, London, 1665. Fig. 1c: Christian Huygens: Traité de la lumière, Leyden, 1690. Fig. 1d: William Wollaston: Philosophical Transactions of the Royal Society, 1813. Fig. 1: How many theories can be illustrated by a single image? How is it to be explained that the same type of illustrations should have survived unperturbed the most profound conceptual changes in matter theory? One needn’t agree with the Kuhnian notion that revolutionary breaks dissect the conceptual evolution of science into incommensu- rable segments to feel that there is something puzzling about pictures that are capable of illus- 2 THE INVENTION OF ATOMIST ICONOGRAPHY trating diverging “world views” over a four-hundred year period.2 For the matter theories illustrated by the nearly identical images of fig. -
What Is a Polyhedron?
3. POLYHEDRA, GRAPHS AND SURFACES 3.1. From Polyhedra to Graphs What is a Polyhedron? Now that we’ve covered lots of geometry in two dimensions, let’s make things just a little more difficult. We’re going to consider geometric objects in three dimensions which can be made from two-dimensional pieces. For example, you can take six squares all the same size and glue them together to produce the shape which we call a cube. More generally, if you take a bunch of polygons and glue them together so that no side gets left unglued, then the resulting object is usually called a polyhedron.1 The corners of the polygons are called vertices, the sides of the polygons are called edges and the polygons themselves are called faces. So, for example, the cube has 8 vertices, 12 edges and 6 faces. Different people seem to define polyhedra in very slightly different ways. For our purposes, we will need to add one little extra condition — that the volume bound by a polyhedron “has no holes”. For example, consider the shape obtained by drilling a square hole straight through the centre of a cube. Even though the surface of such a shape can be constructed by gluing together polygons, we don’t consider this shape to be a polyhedron, because of the hole. We say that a polyhedron is convex if, for each plane which lies along a face, the polyhedron lies on one side of that plane. So, for example, the cube is a convex polyhedron while the more complicated spec- imen of a polyhedron pictured on the right is certainly not convex.