Abstract Regular Polytopes
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Arxiv:Math/9906062V1 [Math.MG] 10 Jun 1999 Udo Udmna Eerh(Rn 96-01-00166)
Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices ∗ Michel DEZA CNRS and Ecole Normale Sup´erieure, Paris, France Mikhail SHTOGRIN Steklov Mathematical Institute, 117966 Moscow GSP-1, Russia Abstract We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic d-space and Coxeter’s regular hyperbolic honeycombs (with infinite or star-shaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice. In section 2 the last remaining 2-dimensional case is decided: for any odd m ≥ 7, star-honeycombs m m {m, 2 } are embeddable while { 2 ,m} are not (unique case of non-embedding for dimension 2). As a spherical analogue of those honeycombs, we enumerate, in section 3, 36 Riemann surfaces representing all nine regular polyhedra on the sphere. In section 4, non-embeddability of all remaining star-honeycombs (on 3-sphere and hyperbolic 4-space) is proved. In the last section 5, all cases of embedding for dimension d> 2 are identified. Besides hyper-simplices and hyper-octahedra, they are exactly those with bipartite skeleton: hyper-cubes, cubic lattices and 8, 2, 1 tilings of hyperbolic 3-, 4-, 5-space (only two, {4, 3, 5} and {4, 3, 3, 5}, of those 11 have compact both, facets and vertex figures). 1 Introduction arXiv:math/9906062v1 [math.MG] 10 Jun 1999 We say that given tiling (or honeycomb) T has a l1-graph and embeds up to scale λ into m-cube Hm (or, if the graph is infinite, into cubic lattice Zm ), if there exists a mapping f of the vertex-set of the skeleton graph of T into the vertex-set of Hm (or Zm) such that λdT (vi, vj)= ||f(vi), f(vj)||l1 = X |fk(vi) − fk(vj)| for all vertices vi, vj, 1≤k≤m ∗This work was supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach) and Russian fund of fundamental research (grant 96-01-00166). -
Petrie Schemes
Canad. J. Math. Vol. 57 (4), 2005 pp. 844–870 Petrie Schemes Gordon Williams Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grunbaum–Dress¨ polyhedra, pos- sess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. 1 Introduction Historically, polyhedra have been conceived of either as closed surfaces (usually topo- logical spheres) made up of planar polygons joined edge to edge or as solids enclosed by such a surface. In recent times, mathematicians have considered polyhedra to be convex polytopes, simplicial spheres, or combinatorial structures such as abstract polytopes or incidence complexes. A Petrie polygon of a polyhedron is a sequence of edges of the polyhedron where any two consecutive elements of the sequence have a vertex and face in common, but no three consecutive edges share a commonface. For the regular polyhedra, the Petrie polygons form the equatorial skew polygons. Petrie polygons may be defined analogously for polytopes as well. Petrie polygons have been very useful in the study of polyhedra and polytopes, especially regular polytopes. -
Four-Dimensional Regular Polytopes
faculty of science mathematics and applied and engineering mathematics Four-dimensional regular polytopes Bachelor’s Project Mathematics November 2020 Student: S.H.E. Kamps First supervisor: Prof.dr. J. Top Second assessor: P. Kiliçer, PhD Abstract Since Ancient times, Mathematicians have been interested in the study of convex, regular poly- hedra and their beautiful symmetries. These five polyhedra are also known as the Platonic Solids. In the 19th century, the four-dimensional analogues of the Platonic solids were described mathe- matically, adding one regular polytope to the collection with no analogue regular polyhedron. This thesis describes the six convex, regular polytopes in four-dimensional Euclidean space. The focus lies on deriving information about their cells, faces, edges and vertices. Besides that, the symmetry groups of the polytopes are touched upon. To better understand the notions of regularity and sym- metry in four dimensions, our journey begins in three-dimensional space. In this light, the thesis also works out the details of a proof of prof. dr. J. Top, showing there exist exactly five convex, regular polyhedra in three-dimensional space. Keywords: Regular convex 4-polytopes, Platonic solids, symmetry groups Acknowledgements I would like to thank prof. dr. J. Top for supervising this thesis online and adapting to the cir- cumstances of Covid-19. I also want to thank him for his patience, and all his useful comments in and outside my LATEX-file. Also many thanks to my second supervisor, dr. P. Kılıçer. Furthermore, I would like to thank Jeanne for all her hospitality and kindness to welcome me in her home during the process of writing this thesis. -
Regular Polyhedra Through Time
Fields Institute I. Hubard Polytopes, Maps and their Symmetries September 2011 Regular polyhedra through time The greeks were the first to study the symmetries of polyhedra. Euclid, in his Elements showed that there are only five regular solids (that can be seen in Figure 1). In this context, a polyhe- dron is regular if all its polygons are regular and equal, and you can find the same number of them at each vertex. Figure 1: Platonic Solids. It is until 1619 that Kepler finds other two regular polyhedra: the great dodecahedron and the great icosahedron (on Figure 2. To do so, he allows \false" vertices and intersection of the (convex) faces of the polyhedra at points that are not vertices of the polyhedron, just as the I. Hubard Polytopes, Maps and their Symmetries Page 1 Figure 2: Kepler polyhedra. 1619. pentagram allows intersection of edges at points that are not vertices of the polygon. In this way, the vertex-figure of these two polyhedra are pentagrams (see Figure 3). Figure 3: A regular convex pentagon and a pentagram, also regular! In 1809 Poinsot re-discover Kepler's polyhedra, and discovers its duals: the small stellated dodecahedron and the great stellated dodecahedron (that are shown in Figure 4). The faces of such duals are pentagrams, and are organized on a \convex" way around each vertex. Figure 4: The other two Kepler-Poinsot polyhedra. 1809. A couple of years later Cauchy showed that these are the only four regular \star" polyhedra. We note that the convex hull of the great dodecahedron, great icosahedron and small stellated dodecahedron is the icosahedron, while the convex hull of the great stellated dodecahedron is the dodecahedron. -
On Regular Polytope Numbers
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 1, Pages 65{75 S 0002-9939(02)06710-2 Article electronically published on June 12, 2002 ON REGULAR POLYTOPE NUMBERS HYUN KWANG KIM (Communicated by David E. Rohrlich) Abstract. Lagrange proved a theorem which states that every nonnegative integer can be written as a sum of four squares. This result can be generalized as the polygonal number theorem and the Hilbert-Waring problem. In this paper, we shall generalize Lagrange's sum of four squares theorem further. To each regular polytope V in a Euclidean space, we will associate a sequence of nonnegative integers which we shall call regular polytope numbers, and consider the problem of finding the order g(V )ofthesetofregularpolytope numbers associated to V . In 1770, Lagrange proved a theorem which states that every nonnegative integer can be written as a sum of four squares. There are two major generalizations of this beautiful result. One is a horizontal generalization due to Cauchy which is known as the polygonal number theorem, and the other is a higher dimensional generalization which is known as the Hilbert-Waring problem. A nonempty subset A of nonnegative integers is called a basis of order g if g is the minimum number with the property that every nonnegative integer can be written as a sum of g elements in A. Lagrange's sum of four squares can be restated as: the set n2 n =0; 1; 2;::: of nonnegative squares forms a basis of order 4. The polygonalf numbersj are sequencesg of nonnegative integers constructed geometrically from regular polygons. -
CONVEX and ABSTRACT POLYTOPES 1 Thursday, May 19, 2005 to Saturday, May, 21, 2005
CONVEX AND ABSTRACT POLYTOPES 1 Thursday, May 19, 2005 to Saturday, May, 21, 2005 MEALS Breakfast (Continental): 7:00 - 9:00 am, 2nd floor lounge, Corbett Hall, Friday & Saturday. Lunch (Buffet): 11:30 am - 1:30 pm, Donald Cameron Hall, Friday & Saturday. Dinner (Buffet): 5:30 - 7:30 pm, Donald Cameron Hall, Thursday & Friday. Coffee Breaks: As per daily schedule, 2nd floor lounge, Corbett Hall. **For other lighter meal options at the Banff Centre, there are two other options: Gooseberry’s Deli, located in the Sally Borden Building, and The Kiln Cafe, located beside Donald Cameron Hall. There are also plenty of restaurants and cafes in the town of Banff, a 10-15 minute walk from Corbett Hall.** MEETING ROOMS All lectures are held in Max Bell 159. Hours: 6 am - 12 midnight. LCD projector, overhead projectors and blackboards are available for presentations. Please note that the meeting space designated for BIRS is the lower level of Max Bell, Rooms 155-159. Please respect that all other space has been contracted to other Banff Centre guests, including any Food and Beverage in those areas. SCHEDULE Thursday 16:00 Check-in begins (Front Desk - Professional Development Centre - open 24 hours) 19:30 Informal gathering in 2nd floor lounge, Corbett Hall. Beverages and small assortment of snacks available in the lounge on a cash honour-system basis. Friday 9:15-10:05 C.Lee — Some Construction Techniques for Convex Polytopes. 10:15-10:35 J.Lawrence — Intersections of Convex Transversals. 10:45-11:15 Coffee Break, 2nd floor lounge, Corbett Hall. 11:15-11:35 R.Dawson — New Triangulations of the Sphere. -
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. -
3D Visualization Models of the Regular Polytopes in Four and Higher Dimensions
BRIDGES Mathematical Connections in Art, Music, and Science 3D Visualization Models of the Regular Polytopes in Four and Higher Dimensions Carlo H. Sequin Computer Science Division, EECS Department University of California, Berkeley, CA 94720 E-maih [email protected] Abstract This paper presents a tutorial review of the construction of all regular polytopes in spaces of all possible dimensions . .It focusses on how to make instructive, 3-dimensional, physical visualization models for the polytopes of dimensions 4 through 6, using solid free-form fabrication technology. 1. Introduction Polytope is a generalization of the terms in the sequence: point, segment, polygon, polyhedron ... [1]. Such a polytope is called regular, if all its elements (vertices, edges, faces, cells ... ) are indistinguishable, i.e., if there exists a group of spatial transformations (rotations, mirroring) that will bring the polytope into coverage with itself. Through these symmetry operations, it must be possible to transform any particular element of the polytope into any other chosen element of the same kind. In two dimensions, there exist infinitely many regular polygons; the first five are shown in Figure 1. • • • Figure 1: The simplest regular 2D polygons. In three-dimensional space, there are just five regular polyhedra -- the Platonic solids, and they can readily be depicted by using shaded perspective renderings (Fig.2). As we contemplate higher dimensions and the regular polytopes that they may admit, it becomes progressively harder to understand the geometry of these objects. Projections down to two (printable) dimensions discard a fair amount of information, and people often have difficulties comprehending even 4D polytopes, when only shown pictures or 2D graphs. -
REGULAR POLYTOPES in Zn Contents 1. Introduction 1 2. Some
REGULAR POLYTOPES IN Zn ANDREI MARKOV Abstract. In [3], all embeddings of regular polyhedra in the three dimen- sional integer lattice were characterized. Here, we prove some results toward solving this problem for all higher dimensions. Similarly to [3], we consider a few special polytopes in dimension 4 that do not have analogues in higher dimensions. We then begin a classification of hypercubes, and consequently regular cross polytopes in terms of the generalized duality between the two. Fi- nally, we investigate lattice embeddings of regular simplices, specifically when such simplices can be inscribed in hypercubes of the same dimension. Contents 1. Introduction 1 2. Some special cases, and a word on convexity. 2 3. Hypercubes and Cross Polytopes 4 4. Simplices 4 5. Corrections 6 Acknowledgments 7 References 7 1. Introduction Definition 1.1. A regular polytope is a polytope such that its symmetry group action is transitive on its flags. This definition is not all that important to what we wish to accomplish, as the classification of all such polytopes is well known, and can be found, for example, in [2]. The particulars will be introduced as each case comes up. Definition 1.2. The dual of a polytope of dimension n is the polytope formed by taking the centroids of the n − 1 cells to be its vertices. The dual of the dual of a regular polytope is homothetic to the original with respect to their mutual center. We will say that a polytope is \in Zn" if one can choose a set of points in the lattice of integer points in Rn such that the points can form the vertices of said polytope. -
Regular Incidence Complexes, Polytopes, and C-Groups
Regular Incidence Complexes, Polytopes, and C-Groups Egon Schulte∗ Department of Mathematics Northeastern University, Boston, MA 02115, USA March 13, 2018 Abstract Regular incidence complexes are combinatorial incidence structures generalizing regu- lar convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of abstract regular poly- topes has been well-studied. The paper describes the combinatorial structure of a regular incidence complex in terms of a system of distinguished generating subgroups of its automorphism group or a flag-transitive subgroup. Then the groups admitting a flag-transitive action on an incidence complex are characterized as generalized string C-groups. Further, extensions of regular incidence complexes are studied, and certain incidence complexes particularly close to abstract polytopes, called abstract polytope complexes, are investigated. Key words. abstract polytope, regular polytope, C-group, incidence geometries MSC 2010. Primary: 51M20. Secondary: 52B15; 51E24. arXiv:1711.02297v1 [math.CO] 7 Nov 2017 1 Introduction Regular incidence complexes are combinatorial incidence structures with very high combina- torial symmetry. The concept was introduced by Danzer [12, 13] building on Gr¨unbaum’s [17] notion of a polystroma. Regular incidence complexes generalize regular convex polytopes [7], regular complex polytopes [8, 42], various types of incidence geometries [4, 5, 21, 44], and many other highly symmetric objects. The terminology and notation is patterned after con- vex polytopes [16] and was ultimately inspired by Coxeter’s work on regular figures [7, 8]. ∗Email: [email protected] 1 The first systematic study of incidence complexes from the discrete geometry perspective occurred in [33] and the related publications [13, 34, 35, 36]. -
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. -
Realizing Regular Polytopes Over Q
REGULAR POLYTOPES REALIZED OVER Q TREVOR HYDE A regular polytope is a d-dimensional generalization of a regular polygon and a Platonic solid. Roughly, they are convex geometric objects with maximal rotational symmetry. To avoid stating a precise definition we appeal to a known complete classification of all regular polytopes. The codimension 1 parts of a polytope are called facets. For example, the facets of a cube are squares. • Dimension 2: In two dimensions, the regular polytopes are the familiar regular polygons. There is a regular n-gon for every n ≥ 3. • Dimension 3: There are five regular polytopes in three dimensions, these are the Platonic solids: ◦ Tetrahedron, or 3-simplex ◦ Cube, or 3-cube ◦ Octahedron, or 3-orthoplex ◦ Dodecahedron ◦ Icosahedron Each regular polytope P has a dual—also a regular polytope—formed by placing vertices at the center of each facet of P and then taking the convex hull of these vertices. Taking the dual of the resulting polytope gives us a copy of the original polytope, hence the terminology. Each regular n-gon is its own dual. The tetrahedron is its own dual. The cube and octahedron are dual, as are the dodecahedron and icosahedron. • Dimension 4: We have six regular polytopes in four dimensions described below by their three dimensional facets: ◦ 4-simplex (3-simplex facets) ◦ 4-cube (3-cube facets) ◦ 4-orthoplex (3-simplex facets) ◦ 24-cell (octahedral facets) ◦ 120-cell (dodecahedral facets) ◦ 600-cell (3-simplex facets) The 4-simplex and the 24-cell are both self-dual. The 4-cube and 4-orthoplex are dual, as are the 120-cell and 600-cell.