Polyhedra from Symmetric Placement of Regular Polygons
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On the Archimedean Or Semiregular Polyhedra
ON THE ARCHIMEDEAN OR SEMIREGULAR POLYHEDRA Mark B. Villarino Depto. de Matem´atica, Universidad de Costa Rica, 2060 San Jos´e, Costa Rica May 11, 2005 Abstract We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler’s polyhedral formula. Contents 1 Introduction 2 1.1 RegularPolyhedra .............................. 2 1.2 Archimedean/semiregular polyhedra . ..... 2 2 Proof techniques 3 2.1 Euclid’s proof for regular polyhedra . ..... 3 2.2 Euler’s polyhedral formula for regular polyhedra . ......... 4 2.3 ProofsofArchimedes’theorem. .. 4 3 Three lemmas 5 3.1 Lemma1.................................... 5 3.2 Lemma2.................................... 6 3.3 Lemma3.................................... 7 4 Topological Proof of Archimedes’ theorem 8 arXiv:math/0505488v1 [math.GT] 24 May 2005 4.1 Case1: fivefacesmeetatavertex: r=5. .. 8 4.1.1 At least one face is a triangle: p1 =3................ 8 4.1.2 All faces have at least four sides: p1 > 4 .............. 9 4.2 Case2: fourfacesmeetatavertex: r=4 . .. 10 4.2.1 At least one face is a triangle: p1 =3................ 10 4.2.2 All faces have at least four sides: p1 > 4 .............. 11 4.3 Case3: threefacesmeetatavertes: r=3 . ... 11 4.3.1 At least one face is a triangle: p1 =3................ 11 4.3.2 All faces have at least four sides and one exactly four sides: p1 =4 6 p2 6 p3. 12 4.3.3 All faces have at least five sides and one exactly five sides: p1 =5 6 p2 6 p3 13 1 5 Summary of our results 13 6 Final remarks 14 1 Introduction 1.1 Regular Polyhedra A polyhedron may be intuitively conceived as a “solid figure” bounded by plane faces and straight line edges so arranged that every edge joins exactly two (no more, no less) vertices and is a common side of two faces. -
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). -
Construction of Great Icosidodecahedron This Paper
Construction of Great IcosiDodecahedron This paper describes how to create the Great Icosidodecahedron analytically as a 3D object. This uniform polyhedron is particularly beautiful and can be constructed with 12 pentagrams and 20 equilateral triangles. You start with a pentagon and create the net for a Icosidodecahedron without the triangles (fig 1). Extend the sides of the pentagons to create pentagrams and fold the sides into the same planes as the icosidodecahedron and you get the first half of the polyhedron (fig 2). Now for the equilateral triangles. Pick every combination of two pentagrams where a single vertex shares a common point (fig 3). Create two equilateral triangles connecting the common vertex and each of the legs of the two pentagrams. Do this to for all pentagrams and you will come up with 20 distinct intersecting triangles. The final result is in figure 4. We can create the model using a second method. Based on the geometry of the existing model, create a single cup (fig 5). Add the adjoining pentagram star (fig 6). Now take this shape and rotate- duplicate for each plane of the pentagon icosidodecahedron and you get the model of figure 4. 1 Construction of Great IcosiDodecahedron fig1 The net for the Icosidodecahedron without the triangle faces. I start with a pentagon, triangulate it with an extra point in the center. 2 Construction of Great IcosiDodecahedron fig 2 Extend the sides of the pentagons to make pentagrams then fold the net into the same planes as the Icosidodecahedron. This makes 12 intersecting stars (pentagrams). fig 3 Now add equilateral triangles to each pair of pentagrams, which touch in only one vertex. -
The Truncated Icosahedron As an Inflatable Ball
https://doi.org/10.3311/PPar.12375 Creative Commons Attribution b |99 Periodica Polytechnica Architecture, 49(2), pp. 99–108, 2018 The Truncated Icosahedron as an Inflatable Ball Tibor Tarnai1, András Lengyel1* 1 Department of Structural Mechanics, Faculty of Civil Engineering, Budapest University of Technology and Economics, H-1521 Budapest, P.O.B. 91, Hungary * Corresponding author, e-mail: [email protected] Received: 09 April 2018, Accepted: 20 June 2018, Published online: 29 October 2018 Abstract In the late 1930s, an inflatable truncated icosahedral beach-ball was made such that its hexagonal faces were coloured with five different colours. This ball was an unnoticed invention. It appeared more than twenty years earlier than the first truncated icosahedral soccer ball. In connection with the colouring of this beach-ball, the present paper investigates the following problem: How many colourings of the dodecahedron with five colours exist such that all vertices of each face are coloured differently? The paper shows that four ways of colouring exist and refers to other colouring problems, pointing out a defect in the colouring of the original beach-ball. Keywords polyhedron, truncated icosahedron, compound of five tetrahedra, colouring of polyhedra, permutation, inflatable ball 1 Introduction Spherical forms play an important role in different fields – not even among the relics of the Romans who inherited of science and technology, and in different areas of every- many ball games from the Greeks. day life. For example, spherical domes are quite com- The Romans mainly used balls composed of equal mon in architecture, and spherical balls are used in most digonal panels, forming a regular hosohedron (Coxeter, 1973, ball games. -
Are Your Polyhedra the Same As My Polyhedra?
Are Your Polyhedra the Same as My Polyhedra? Branko Gr¨unbaum 1 Introduction “Polyhedron” means different things to different people. There is very little in common between the meaning of the word in topology and in geometry. But even if we confine attention to geometry of the 3-dimensional Euclidean space – as we shall do from now on – “polyhedron” can mean either a solid (as in “Platonic solids”, convex polyhedron, and other contexts), or a surface (such as the polyhedral models constructed from cardboard using “nets”, which were introduced by Albrecht D¨urer [17] in 1525, or, in a more mod- ern version, by Aleksandrov [1]), or the 1-dimensional complex consisting of points (“vertices”) and line-segments (“edges”) organized in a suitable way into polygons (“faces”) subject to certain restrictions (“skeletal polyhedra”, diagrams of which have been presented first by Luca Pacioli [44] in 1498 and attributed to Leonardo da Vinci). The last alternative is the least usual one – but it is close to what seems to be the most useful approach to the theory of general polyhedra. Indeed, it does not restrict faces to be planar, and it makes possible to retrieve the other characterizations in circumstances in which they reasonably apply: If the faces of a “surface” polyhedron are sim- ple polygons, in most cases the polyhedron is unambiguously determined by the boundary circuits of the faces. And if the polyhedron itself is without selfintersections, then the “solid” can be found from the faces. These reasons, as well as some others, seem to warrant the choice of our approach. -
An Enduring Error
Version June 5, 2008 Branko Grünbaum: An enduring error 1. Introduction. Mathematical truths are immutable, but mathematicians do make errors, especially when carrying out non-trivial enumerations. Some of the errors are "innocent" –– plain mis- takes that get corrected as soon as an independent enumeration is carried out. For example, Daublebsky [14] in 1895 found that there are precisely 228 types of configurations (123), that is, collections of 12 lines and 12 points, each incident with three of the others. In fact, as found by Gropp [19] in 1990, the correct number is 229. Another example is provided by the enumeration of the uniform tilings of the 3-dimensional space by Andreini [1] in 1905; he claimed that there are precisely 25 types. However, as shown [20] in 1994, the correct num- ber is 28. Andreini listed some tilings that should not have been included, and missed several others –– but again, these are simple errors easily corrected. Much more insidious are errors that arise by replacing enumeration of one kind of ob- ject by enumeration of some other objects –– only to disregard the logical and mathematical distinctions between the two enumerations. It is surprising how errors of this type escape detection for a long time, even though there is frequent mention of the results. One example is provided by the enumeration of 4-dimensional simple polytopes with 8 facets, by Brückner [7] in 1909. He replaces this enumeration by that of 3-dimensional "diagrams" that he inter- preted as Schlegel diagrams of convex 4-polytopes, and claimed that the enumeration of these objects is equivalent to that of the polytopes. -
New Perspectives on Polyhedral Molecules and Their Crystal Structures Santiago Alvarez, Jorge Echeverria
New Perspectives on Polyhedral Molecules and their Crystal Structures Santiago Alvarez, Jorge Echeverria To cite this version: Santiago Alvarez, Jorge Echeverria. New Perspectives on Polyhedral Molecules and their Crystal Structures. Journal of Physical Organic Chemistry, Wiley, 2010, 23 (11), pp.1080. 10.1002/poc.1735. hal-00589441 HAL Id: hal-00589441 https://hal.archives-ouvertes.fr/hal-00589441 Submitted on 29 Apr 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Journal of Physical Organic Chemistry New Perspectives on Polyhedral Molecules and their Crystal Structures For Peer Review Journal: Journal of Physical Organic Chemistry Manuscript ID: POC-09-0305.R1 Wiley - Manuscript type: Research Article Date Submitted by the 06-Apr-2010 Author: Complete List of Authors: Alvarez, Santiago; Universitat de Barcelona, Departament de Quimica Inorganica Echeverria, Jorge; Universitat de Barcelona, Departament de Quimica Inorganica continuous shape measures, stereochemistry, shape maps, Keywords: polyhedranes http://mc.manuscriptcentral.com/poc Page 1 of 20 Journal of Physical Organic Chemistry 1 2 3 4 5 6 7 8 9 10 New Perspectives on Polyhedral Molecules and their Crystal Structures 11 12 Santiago Alvarez, Jorge Echeverría 13 14 15 Departament de Química Inorgànica and Institut de Química Teòrica i Computacional, 16 Universitat de Barcelona, Martí i Franquès 1-11, 08028 Barcelona (Spain). -
Math.MG] 25 Sep 2012 Oisb Ue N Ops,O Ae Folding
Constructing the Cubus simus and the Dodecaedron simum via paper folding Urs Hartl and Klaudia Kwickert August 27, 2012 Abstract The archimedean solids Cubus simus (snub cube) and Dodecaedron simum (snub dodecahe- dron) cannot be constructed by ruler and compass. We explain that for general reasons their vertices can be constructed via paper folding on the faces of a cube, respectively dodecahedron, and we present explicit folding constructions. The construction of the Cubus simus is particularly elegant. We also review and prove the construction rules of the other Archimedean solids. Keywords: Archimedean solids Snub cube Snub dodecahedron Construction Paper folding · · · · Mathematics Subject Classification (2000): 51M15, (51M20) Contents 1 Construction of platonic and archimedean solids 1 2 The Cubus simus constructed out of a cube 4 3 The Dodecaedron simum constructed out of a dodecahedron 7 A Verifying the construction rules from figures 2, 3, and 4. 11 References 13 1 Construction of platonic and archimedean solids arXiv:1202.0172v2 [math.MG] 25 Sep 2012 The archimedean solids were first described by Archimedes in a now-lost work and were later redis- covered a few times. They are convex polyhedra with regular polygons as faces and many symmetries, such that their symmetry group acts transitively on their vertices [AW86, Cro97]. They can be built by placing their vertices on the faces of a suitable platonic solid, a tetrahedron, a cube, an octahedron, a dodecahedron, or an icosahedron. In this article we discuss the constructibility of the archimedean solids by ruler and compass, or paper folding. All the platonic solids can be constructed by ruler and compass. -
Polyhedral Forms Obtained by Combinig Lateral Sheet of CP II-10 and Truncated Dodecahedron
Polyhedral Forms Obtained by Combinig Lateral Sheet of CP II-10 and Truncated Dodecahedron Marija Đ. Obradovi ć The paper analyzes the possibility of obtaining polyhedral shapes formed Associate Professor, University of Belgrade, Faculty of Civil engineering, by combining polyhedral surfaces based on the segment surface of Belgrade elongated concave pyramids of the second sort (CeP II-10, type A and type B). In previous research, CP II type A and CP II type B were elaborated in Milena Stavri ć detail. In this paper we discuss further potential of these polyhedral Ass. Professor, Graz University of surfaces, on the example of combining them with Archimedean solid - Technology, Faculty of Architecture, Graz, Austria Truncated dodecahedron (U26). The faces of this solid consist of 12 decagons and 20 triangles. On the decagonal faces, decagonal polygons of Albert Wiltsche the CeP II segments ( CP II-10) can be added, which provides the new Ass. Professor, Graz University of polyhedral composite forms that are, furthermore, concave deltahedra. Technology, Faculty of Architecture, Graz, Austria There are considered possibilities of obtaining polyhedral shapes by combining sheet segments CP II-10-A, as well as of CP II-10-B with U26. Finally, a couple of new shape suggestions are given: compound polyhedra, obtained by intersection of paired composite concave polyhedra originated in the described manner. Keywords: concave polyhedra, composite polyhedra, truncated dodecahedron, CP II-10, augmentation, incavation 1. INTRODUCTION concerned in the paper arise is a regular polygon - decagon, we will seek possibilities of forming convex This article considers procedures for creating obtainable polyhedra which will also be deltahedra, consisting new polyhedral shapes - composite polyhedra, exclusively of equilateral triangles. -
About Weaving and Helical Holes
Bridges 2010: Mathematics, Music, Art, Architecture, Culture About Weaving and Helical Holes Rinus Roelofs Sculptor Lansinkweg 28 7553AL Hengelo The Netherlands E-mail: [email protected] www.rinusroelofs.nl Abstract Weaving is an invention of man, and maybe one of the most important inventions in the field of construction. There are many weave patterns but most weave patterns are based on just two different grids, the plain weave and the three way weave. The development of weaving is mainly connected to the possibilities of manufacturing. In this paper I want to investigate this ‘passing one thread over another’-concept from a mathematical-artistic point of view. 1. Introduction 1.1. Over and Under. In Leonardo’s painting on the ceiling of the Sala delle Asse in the Sforza Castle in Milan (Figure 1) we can see different weaving patterns: the knotted ropes and the interwoven branches. Both can only be realized with the aid of man. Nature doesn’t weave by itself, at least not in the way Leonardo shows us in his painting. When we look for a definition of weaving we can find the following: weaving is the textile art in which two distinct sets of yarns or threads, called the warp and the filling or weft, are interlaced with each other to form a fabric or cloth. To define the concept of weaving in a more general way we could say that a weaving is a line pattern in which for each pair of adjacent crossing points on the line(s) the position of that line changes from ‘over’ to ‘under’ or from ‘under’ to ‘over’ (Figure 2a,b). -
[ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated Into This Format from Data Given In
ENTRY POLYHEDRA [ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated into this format from data given in http://netlib.bell-labs.com/netlib tetrahedron The [tetrahedron] is a polyhedron with 4 vertices and 4 faces. The dual polyhedron is called tetrahedron. cube The [cube] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. hexahedron The [hexahedron] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. octahedron The [octahedron] is a polyhedron with 6 vertices and 8 faces. The dual polyhedron is called cube. dodecahedron The [dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called icosahedron. icosahedron The [icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called dodecahedron. small stellated dodecahedron The [small stellated dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called great dodecahedron. great dodecahedron The [great dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called small stellated dodecahedron. great stellated dodecahedron The [great stellated dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called great icosahedron. great icosahedron The [great icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called great stellated dodecahedron. truncated tetrahedron The [truncated tetrahedron] is a polyhedron with 12 vertices and 8 faces. The dual polyhedron is called triakis tetrahedron. cuboctahedron The [cuboctahedron] is a polyhedron with 12 vertices and 14 faces. The dual polyhedron is called rhombic dodecahedron.