The Maths Behind Polytopes
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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. -
Platonic Solids Generate Their Four-Dimensional Analogues
1 Platonic solids generate their four-dimensional analogues PIERRE-PHILIPPE DECHANT a;b;c* aInstitute for Particle Physics Phenomenology, Ogden Centre for Fundamental Physics, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, United Kingdom, bPhysics Department, Arizona State University, Tempe, AZ 85287-1604, United States, and cMathematics Department, University of York, Heslington, York, YO10 5GG, United Kingdom. E-mail: [email protected] Polytopes; Platonic Solids; 4-dimensional geometry; Clifford algebras; Spinors; Coxeter groups; Root systems; Quaternions; Representations; Symmetries; Trinities; McKay correspondence Abstract In this paper, we show how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone. Via the Cartan-Dieudonne´ theorem, the reflective symmetries of the arXiv:1307.6768v1 [math-ph] 25 Jul 2013 Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors gen- erating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arising from the Platonic Solids can thus in turn be interpreted as vertices in four- dimensional space, giving a simple construction of the 4D polytopes 16-cell, 24-cell, the F4 root system and the 600-cell. In particular, these polytopes have ‘mysterious’ symmetries, that are almost trivial when seen from the three-dimensional spinorial point of view. In fact, all these induced polytopes are also known to be root systems and thus generate rank-4 Coxeter PREPRINT: Acta Crystallographica Section A A Journal of the International Union of Crystallography 2 groups, which can be shown to be a general property of the spinor construction. -
Uniform Panoploid Tetracombs
Uniform Panoploid Tetracombs George Olshevsky TETRACOMB is a four-dimensional tessellation. In any tessellation, the honeycells, which are the n-dimensional polytopes that tessellate the space, Amust by definition adjoin precisely along their facets, that is, their ( n!1)- dimensional elements, so that each facet belongs to exactly two honeycells. In the case of tetracombs, the honeycells are four-dimensional polytopes, or polychora, and their facets are polyhedra. For a tessellation to be uniform, the honeycells must all be uniform polytopes, and the vertices must be transitive on the symmetry group of the tessellation. Loosely speaking, therefore, the vertices must be “surrounded all alike” by the honeycells that meet there. If a tessellation is such that every point of its space not on a boundary between honeycells lies in the interior of exactly one honeycell, then it is panoploid. If one or more points of the space not on a boundary between honeycells lie inside more than one honeycell, the tessellation is polyploid. Tessellations may also be constructed that have “holes,” that is, regions that lie inside none of the honeycells; such tessellations are called holeycombs. It is possible for a polyploid tessellation to also be a holeycomb, but not for a panoploid tessellation, which must fill the entire space exactly once. Polyploid tessellations are also called starcombs or star-tessellations. Holeycombs usually arise when (n!1)-dimensional tessellations are themselves permitted to be honeycells; these take up the otherwise free facets that bound the “holes,” so that all the facets continue to belong to two honeycells. In this essay, as per its title, we are concerned with just the uniform panoploid tetracombs. -
Platonic Solids Generate Their Four-Dimensional Analogues
This is a repository copy of Platonic solids generate their four-dimensional analogues. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/85590/ Version: Accepted Version Article: Dechant, Pierre-Philippe orcid.org/0000-0002-4694-4010 (2013) Platonic solids generate their four-dimensional analogues. Acta Crystallographica Section A : Foundations of Crystallography. pp. 592-602. ISSN 1600-5724 https://doi.org/10.1107/S0108767313021442 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ 1 Platonic solids generate their four-dimensional analogues PIERRE-PHILIPPE DECHANT a,b,c* aInstitute for Particle Physics Phenomenology, Ogden Centre for Fundamental Physics, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, United Kingdom, bPhysics Department, Arizona State University, Tempe, AZ 85287-1604, United States, and cMathematics Department, University of York, Heslington, York, YO10 5GG, United Kingdom. E-mail: [email protected] Polytopes; Platonic Solids; 4-dimensional geometry; Clifford algebras; Spinors; Coxeter groups; Root systems; Quaternions; Representations; Symmetries; Trinities; McKay correspondence Abstract In this paper, we show how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone. -
Convex Polytopes and Tilings with Few Flag Orbits
Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals. -
Coloring Uniform Honeycombs
Bridges 2009: Mathematics, Music, Art, Architecture, Culture Coloring Uniform Honeycombs Glenn R. Laigo, [email protected] Ma. Louise Antonette N. De las Peñas, [email protected] Mathematics Department, Ateneo de Manila University Loyola Heights, Quezon City, Philippines René P. Felix, [email protected] Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines Abstract In this paper, we discuss a method of arriving at colored three-dimensional uniform honeycombs. In particular, we present the construction of perfect and semi-perfect colorings of the truncated and bitruncated cubic honeycombs. If G is the symmetry group of an uncolored honeycomb, a coloring of the honeycomb is perfect if the group H consisting of elements that permute the colors of the given coloring is G. If H is such that [ G : H] = 2, we say that the coloring of the honeycomb is semi-perfect . Background In [7, 9, 12], a general framework has been presented for coloring planar patterns. Focus was given to the construction of perfect colorings of semi-regular tilings on the hyperbolic plane. In this work, we will extend the method of coloring two dimensional patterns to obtain colorings of three dimensional uniform honeycombs. There is limited literature on colorings of three-dimensional honeycombs. We see studies on colorings of polyhedra; for instance, in [17], a method of coloring shown is by cutting the polyhedra and laying it flat to produce a pattern on a two-dimensional plane. In this case, only the faces of the polyhedra are colored. In [6], enumeration problems on colored patterns on polyhedra are discussed and solutions are obtained by applying Burnside's counting theorem. -
Local Symmetry Preserving Operations on Polyhedra
Local Symmetry Preserving Operations on Polyhedra Pieter Goetschalckx Submitted to the Faculty of Sciences of Ghent University in fulfilment of the requirements for the degree of Doctor of Science: Mathematics. Supervisors prof. dr. dr. Kris Coolsaet dr. Nico Van Cleemput Chair prof. dr. Marnix Van Daele Examination Board prof. dr. Tomaž Pisanski prof. dr. Jan De Beule prof. dr. Tom De Medts dr. Carol T. Zamfirescu dr. Jan Goedgebeur © 2020 Pieter Goetschalckx Department of Applied Mathematics, Computer Science and Statistics Faculty of Sciences, Ghent University This work is licensed under a “CC BY 4.0” licence. https://creativecommons.org/licenses/by/4.0/deed.en In memory of John Horton Conway (1937–2020) Contents Acknowledgements 9 Dutch summary 13 Summary 17 List of publications 21 1 A brief history of operations on polyhedra 23 1 Platonic, Archimedean and Catalan solids . 23 2 Conway polyhedron notation . 31 3 The Goldberg-Coxeter construction . 32 3.1 Goldberg ....................... 32 3.2 Buckminster Fuller . 37 3.3 Caspar and Klug ................... 40 3.4 Coxeter ........................ 44 4 Other approaches ....................... 45 References ............................... 46 2 Embedded graphs, tilings and polyhedra 49 1 Combinatorial graphs .................... 49 2 Embedded graphs ....................... 51 3 Symmetry and isomorphisms . 55 4 Tilings .............................. 57 5 Polyhedra ............................ 59 6 Chamber systems ....................... 60 7 Connectivity .......................... 62 References -
The Classification of Archimedean 4-Polytopes
THEOREM OF THE DAY The Classification of Archimedean 4-Polytopes The Archimedean polytopes in four dimensions are: 1. the polytopes of Boole Stott–Wythoff type; 2. the prismatic polytopes; 3. Cartesian products of regular polygons; 4. Thorold Gosset’s snub-24-cell (1900); 5. the Grand Antiprism (Conway–Guy, 1965) Dwellersinatwo-dimensionalworldmightinvestigate theproperties of a Platonic solid, such as the octahedron, by examining the cross-sections pro- duced when a plane intersects it. Above, a plane intersection parallel with one of the faces produces a hexagonal section; a different intersection might reveal a square or a rectangle. Archimedean four-dimensional polytopes consist of copies of prisms and Archimedean solids meeting at vertices in identical configurations (more precisely, the symmetry group of the polytope acts transitively on its ver- tices). A drawing by Boole Stott is adapted, right, to show how a regular polytope, the 24-cell, consisting of 24 octahedra with six meeting at each of its at 24 vertices, may be ‘unfolded’ so that 3-dimensional sections through it may be visualised. Following pioneering work by Alicia Boole Stott, Willem Wythoff and Thorold Gosset in the early 1900s, work on polytopes was systematised by H.S.M. Coxeter in the 1940s. The classification of Archimedean 4-polytopes was completed by J.H. Conway and Michael Guy in the 1960s using computer search. Web link: johncarlosbaez.wordpress.com/2012/05/21/. Fine descriptions of Boole Stott’s work are www.ams.org/featurecolumn/archive/boole.html by Tony Phillips and www.sciencedirect.com/science/article/pii/S0315086007000973 by Irene Polo-Blanco. -
Wythoffian Skeletal Polyhedra
Wythoffian Skeletal Polyhedra by Abigail Williams B.S. in Mathematics, Bates College M.S. in Mathematics, Northeastern University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Dedication I would like to dedicate this dissertation to my Meme. She has always been my loudest cheerleader and has supported me in all that I have done. Thank you, Meme. ii Abstract of Dissertation Wythoff's construction can be used to generate new polyhedra from the symmetry groups of the regular polyhedra. In this dissertation we examine all polyhedra that can be generated through this construction from the 48 regular polyhedra. We also examine when the construction produces uniform polyhedra and then discuss other methods for finding uniform polyhedra. iii Acknowledgements I would like to start by thanking Professor Schulte for all of the guidance he has provided me over the last few years. He has given me interesting articles to read, provided invaluable commentary on this thesis, had many helpful and insightful discussions with me about my work, and invited me to wonderful conferences. I truly cannot thank him enough for all of his help. I am also very thankful to my committee members for their time and attention. Additionally, I want to thank my family and friends who, for years, have supported me and pretended to care everytime I start talking about math. Finally, I want to thank my husband, Keith. -
Represented by Quaternions
Turk J Phys 36 (2012) , 309 – 333. c TUB¨ ITAK˙ doi:10.3906/fiz-1109-11 Branching of the W (H4) polytopes and their dual polytopes under the coxeter groups W (A4) and W (H3) represented by quaternions Mehmet KOCA1,NazifeOzde¸¨ sKOCA2 and Mudhahir AL-AJMI3 Department of Physics, College of Science, Sultan Qaboos University P. O. Box 36, Al-Khoud 123, Muscat-SULTANATE OF OMAN e-mails: [email protected], [email protected], [email protected] Received: 11.09.2011 Abstract 4-dimensional H4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter- Weyl group W(H4), where the group elements and the vertices of the polytopes are represented by quater- nions. Projection of an arbitrary W(H4) orbit into three dimensions is made preserving the icosahedral subgroup W(H3) and the tetrahedral subgroup W(A3). The latter follows a branching under the Cox- eter group W(A4). The dual polytopes of the semi-regular and quasi-regular H4 polytopes have been constructed. Key Words: 4D polytopes, dual polytopes, coxeter groups, quaternions, W(H4) 1. Introduction It seems that there exists experimental evidence for the existence of the Coxeter-Weyl group W (E8). Radu Coldea et al. [1] have performed a neutron scattering experiment on CoNb 2 O6 (cobalt niobate), which describes the one dimensional quantum Ising chain. Their work have determined the masses of the five emerging particles; the first two are found to obey the relation m2 = τm1 . Their results could be attributed to the Zamolodchikov model [2] which describes the one-dimensional Ising model at critical temperature perturbed by an external magnetic field leading to eight spinless bosons with the mass relations π 7π 4π m1,m3 =2m1 cos 30 ,m4 =2m2 cos 30 ,m5 =2m2 cos 30 (1) m2 = τm1,m6=τm3,m7=τm4,m8=τm5, √ 1+ 5 where τ= 2 is the golden ratio. -
Naming Archimedean and Catalan Polyhedra and Tilings
-21- Naming Archimedean and Catalan Polyhedra and Tilings The book in which Archimedes enumerated the polyhedra that have regular faces and equivalent vertices is unfortunately lost; however, its contents were reconstructed by Kepler, from whom the tradi- tional names descend. In this chapter we explain these “Keplerian” names for the Archimedean and Catalan solids and extend them to the analogous tessellations of two- and three-dimensional Euclidean space. We shall describe the polyhedra in dual pairs indicated by the ar- rows, and at the same time give our abbreviations for them, starting with the five Platonic (or regular) ones: TC↔ OD↔ I Tetrahedron Cube Octahedron Dodecahedron Icosahedron Etymologically, the Greek stem “hedr-” is cognate with the Latin “sede-” and the English “seat,” so that, for instance, “dodecahedron” really means “twelve-seater.” (opposite page) The Archimedean solids and their duals can be nicely arranged so that their edges are mutually tangent, at their intersections, to a common sphere. 283 © 2016 by Taylor & Francis Group, LLC 284 21. Naming Archimedean and Catalan Polyhedra and Tilings Truncation and “Kis”ing These are followed by their “truncated” and “kis-” versions. Here, truncation means cutting off the corners in such a way that each regular n-gonal face is replaced by a regular 2n-gonal one. The dual operation is to erect a pyramid on each face, thus replacing a regular m-gon by m isoceles triangles. These give five Archimedean and five Catalan solids: truncated truncated truncated truncated truncated Tetrahedron Cube Octahedron Dodecahedron Icosahedron tT tC tO tD tI kT kC kO kD kI kisTetrahedron kisCube kisOctahedron kisDodecahedron kisIcosahedron The names used by Kepler for the Catalan ones were rather longer, namely, triakis tetrakis triakis pentakis triakis Downloaded by [University of Bergen Library] at 04:55 26 October 2016 tetrahedron hexahedron octahedron dodecahedron icosahedron and were usually printed as single words. -
John Horton Conway 2013 Book.Pdf
Princeton University Honors Faculty Members Receiving Emeritus Status May 2013 The biographical sketches were written by colleagues in the departments of those honored, except where noted. Copyright © 2013 by The Trustees of Princeton University 350509-13 Contents Faculty Members Receiving Emeritus Status Leonard Harvey Babby 1 Mark Robert Cohen 4 Martin C. Collcutt 6 John Horton Conway 10 Edward Charles Cox 14 Frederick Lewis Dryer 16 Thomas Jeffrey Espenshade 19 Jacques Robert Fresco 22 Charles Gordon Gross 24 András Peter Hámori 28 Marie-Hélène Huet 30 Morton Daniel Kostin 32 Heath W. Lowry 34 Richard Bryant Miles 36 Chiara Rosanna Nappi 39 Susan Naquin 42 Edward Nelson 44 John Abel Pinto 47 Albert Jordy Raboteau 49 François P. Rigolot 54 Daniel T. Rodgers 57 Gilbert Friedell Rozman 61 Peter Schäfer 64 José A. Scheinkman 68 Anne-Marie Slaughter 71 Robert Harry Socolow 74 Zoltán G. Soos 78 Eric Hector Vanmarcke 81 Maurizio Viroli 83 Frank Niels von Hippel 85 Andrew John Wiles 87 Michael George Wood 89 John Horton Conway John Conway is a mathematician whose interests run broad and deep, ranging from classical geometry to the 196,884-dimensional Monster group to infinity and beyond. Perhaps his greatest achievement (certainly his proudest achievement) is the invention of new system of numbers, the surreal numbers—a continuum of numbers that include not only real numbers (integers, fractions, and irrationals such as pi, which in his heyday he could recite from memory to more than 1,100 digits), but also the infinitesimal and the infinite numbers. When he discovered them in 1970, the surreals had John wandering around in a white-hot daydream for weeks.