Geometric realizations for Dyck's regular map on a surface of genus 3

Klein's and Dyck's regular maps on Riemann surfaces of genus 3 were one impetus for the theory of regular maps, automorphic functions, and algebraic curves. Recently a polyhedral realization inE ³ of Klein's map was discovered [18], thereby underlining the strong analogy to the icosahedron. In this paper we show that Dyck's map can be realized inE ³ as a polyhedron of Kepler-Poinsot-type, i.e., with maximal symmetry and minimal self-intersections. Furthermore some closely related polyhedra and a Kepler-Poinsot-type realization of Sherk's regular map of genus 5 are discussed.     

The combinatorially regular polyhedra of index 2

Summary We investigate polyhedral realizations of regular maps with self-intersections in E³, whose symmetry group is a subgroup of index 2 in their automorphism group. We show that there are exactly 5 such polyhedra. The polyhedral sets have been more or less known for about 100 years; but the fact that they are realizations of regular maps is new in at least one case, a self-dual icosahedron of genus 11. Our polyhedra are closely related to the 5 regular compounds, which can be interpreted as discontinuous polyhedral realizations of regular maps.The author was born on March 5, 1937; so exactly half a century after Otto Haupt.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

On regular polyhedra with hidden symmetries

We consider polyhedral realizations of oriented regular maps with or without self-intersections in E³ whose symmetry group is a subsgroup of small index in their. automorphism group. The four classical kepler-poinsot polyhedra are the only ones of index 1. There are exactly five of Index 2, all with icosahedral symmetry group [W2] as the Kepler-poinsot polyhedra. In this paper we show that there are no such polyhedral realizations with octahedral (tetrahedral) symmetry group and index 2 or 3 (2,3,4,5), which is best possible in the octahedral case.     

A Polyhedral Model in Euclidean 3-Space of the Six-Pentagon Map of the Projective Plane

In a private communication, Branko Grünbaum asked: “I wonder whether you know anything about the possibility of realizing as a polyhedron in Euclidean 3-space the family of six pentagons, that is a model of the projective plane arising by identifying antipodal points of the regular dodecahedron. Naturally, any realization must have some self-intersections—but is there any realization that is not completely contained in a plane?”We show that it is possible to realize this polyhedron; in our realization five of the six faces are simple polygons. In this model there are sets of three faces, which form a realization of the Möbius strip without self-intersections. There are four variants of the model. We conjecture that in any model of this polyhedron there must be at least one self-intersecting face.     

Orbits of linear maps and regular languages

The equivalence is established of the problem of hitting a polyhedral set by the orbit of a linear map and the intersection of a regular language and a language of permutations of binary words ( P_\mathbbBP_\mathbb{B}-realizability problem). The decidability of the both problems is presently unknown, and the first one is a straightforward generalization of the famous Skolem problem and the nonnegativity problem in the theory of linear recurrent sequences.     

The two Pappus-tori

Two polyhedral embeddings of the regular maps {3, 6}_3,0 and {6, 3}_3,0 in E ³ are given. They are the smallest combinatorially regular tori of type {3, 6} and {6, 3} embedded in E ³, and their automorphism group is that of the famous Pappus-configuration.     

Polyhedral 2-manifolds inE 3 with unusually large genus

An equivelar polyhedral 2-manifold in the class ?_p,q is one embedded inE ³ in which every face is a convexp-gon and every vertex isq-valent. In this paper, examples are constructed, to show that each of the classes ?_3,q (q≧7), ?_4,q (q≧5) and ?_p,4 (p≧5) contains infinitely many distinct combinatorial types. As particular examples, there are polyhedral 2-manifolds with 576 vertices and genus 577, and with 4096 faces and genus 4097. A modification of one construction shows that there is a constantk, such that for eachg≧2, there exists a closed polyhedral 2-manifold inE ³ of genusg with at mostkg/logg vertices.     

Regular combinatorial maps

The classical approach to maps is by cell decomposition of a surface. A combinatorial map is a graph-theoretic generalization of a map on a surface. Besides maps on orientable and non-orientable surfaces, combinatorial maps include tessellations, hypermaps, higher dimensional analogues of maps, and certain toroidal complexes of Coxeter, Shephard, and Grünbaum. In a previous paper the incidence structure, diagram, and underlying topological space of a combinatorial map were investigated. This paper treats highly symmetric combinatorial maps. With regularity defined in terms of the automorphism group, necessary and sufficient conditions for a combinatorial map to be regular are given both graph- and group-theoretically. A classification of regular combinatorial maps on closed simply connected manifolds generalizes the well-known classification of metrically regular polytopes. On any closed manifold with nonzero Euler characteristic there are at most finitely many regular combinatorial maps. However, it is shown that, for nearly any diagram D, there are infinitely many regular combinatorial maps with diagram D. A necessary and sufficient condition for the regularity of rank 3 combinatorial maps is given in terms of Coxeter groups. This condition reveals the difficulty in classifying the regular maps on surfaces. In light of this difficulty an algorithm for generating a large class of regular combinatorial maps that are obtained as cyclic coverings of a given regular combinatorial map is given.     

Hyperbolic polyhedral surfaces with regular faces

We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least 2π. The combinatorial information of these surfaces is shown to be identified with that of Euclidean polyhedral surfaces with negative combinatorial curvature everywhere. We prove that there is a gap between areas of non-smooth hyperbolic polyhedral surfaces and the area of smooth hyperbolic surfaces. The numerical result for the gap is obtained for hyperbolic polyhedral surfaces, homeomorphic to the double torus, whose 1-skeletons are cubic graphs.     

On chirality groups and regular coverings of regular oriented hypermaps

We prove that if the Walsh bipartite map M = W (ℋ) of a regular oriented hypermap ℋ is also orientably regular then both M and ℋ have the same chirality group, the covering core of M (the smallest regular map covering M) is the Walsh bipartite map of the covering core of ℋ and the closure cover of M (the greatest regular map covered by M) is the Walsh bipartite map of the closure cover of ℋ. We apply these results to the family of toroidal chiral hypermaps (3, 3, 3) _b,c = W ⁻¹{6, 3} _b,c induced by the family of toroidal bipartite maps {6, 3} _b,c .     

Analysis of Curve Reconstruction by Meshless Parameterization

This paper proposes and analyzes a method called meshless parameterization for reconstructing curves from unordered point samples. The method solves a linear system of equations based on convex combinations so as to map the sampled points into corresponding parameter values, whose natural ordering provides the ordering of the points. Using the theory of M-matrices, we derive natural conditions on the point sample which guarantee the correct ordering. A sufficient condition is that the underlying curve be tangent-continuous and free of self-intersections and that the sample is dense enough.     

Coxeter–Petrie Complexes of Regular Maps

Coxeter–Petrie complexes naturally arise as thin diagram geometries whose rank 3 residues contain all of the dual forms of a regular algebraic map M. Corresponding to an algebraic map is its classical dual, which is obtained simply by interchanging the vertices and faces, as well as its Petrie dual, which comes about by replacing the faces by the so-called Petrie polygons. Jones and Thornton have shown that these involutory duality operations generate the symmetric groupS₃ , giving in all six dual forms, and whose source is the outer automorphism group of the infinite triangle group generated by involutions s₁, s₂, s₃, subject to the additional relation s₁s₃ =  s₃s₁. In fact, this outer automorphism group is parametrized by the permutations of the three commuting involutions s₁,s₃ , s₁s₃. These involutions together with the involutions₂ can be taken to define the nodes of a Coxeter diagram of shape D₄(with the involution s₂at the central node), and when the original map M is regular, there is a natural extension from M to a thin Coxeter complex of rank 4 all of whose rank 3 residues are isomorphic to the various dual forms of M. These are fully explicated in case the original algebraic map is a Platonic map.     

Infinite series of combinatorially regular polyhedra in three-space

The paper discusses polyhedral realizations in ordinary Euclidean 3-space of Coxeter's regular skew polyhedra {4, p|4 ^p/2]–1} and their duals on an orientable surface of genus 2 ^p–3(p–4)+1. Our considerations are based on work of Coxeter, Ringel and McMullen et al., revealing that certain polyhedral manifolds discovered by the last three authors are in fact the polyhedra in question. We also describe Kepler-Poinsot-type polyhedra in 3-space obtained by projections from Coxeter's regular skew star-polyhedra in 4 dimensions.     

Three results on the regularity of the curves that are invariant by an exact symplectic twist map

A theorem due to G. D. Birkhoff states that every essential curve which is invariant under a symplectic twist map of the annulus is the graph of a Lipschitz map. We prove: if the graph of a Lipschitz map h:T→R is invariant under a symplectic twist map, then h is a little bit more regular than simply Lipschitz (Theorem 1); we deduce that there exists a Lipschitz map h:T→R whose graph is invariant under no symplectic twist map (Corollary 2). Assuming that the dynamic of a twist map restricted to a Lipschitz graph is bi-Lipschitz conjugate to a rotation, we obtain that the graph is even C ¹ (Theorem 3). Then we consider the case of the C ⁰ integrable symplectic twist maps and we prove that for such a map, there exists a dense G _δ subset of the set of its invariant curves such that every curve of this G _δ subset is C ¹ (Theorem 4).     

Some isomorphically polyhedral orlicz sequence spaces

A Banach space is polyhedral if the unit ball of each of its finite dimensional subspaces is a polyhedron. It is known that a polyhedral Banach space has a separable dual and isc ₀-saturated, i.e., each closed infinite dimensional subspace contains an isomorph ofc ₀. In this paper, we show that the Orlicz sequence spaceh _M is isomorphic to a polyhedral Banach space if lim _t→0 M(Kt)/M(t)=∞ for someK<∞. We also construct an Orlicz sequence spaceh _M which isc ₀-saturated, but which is not isomorphic to any polyhedral Banach space. This shows that beingc ₀-saturated and having a separable dual are not sufficient for a Banach space to be isomorphic to a polyhedral Banach space.     

A homotopy continuation method for solving normal equations

In this paper, we present a continuation method for solving normal equations generated byC ² functions and polyhedral convex sets. We embed the normal map into a homotopyH, and study the existence and characteristics of curves inH ¹(0) starting at a specificd point. We prove the convergence of such curves to a solution of the normal equation under some conditions on the polyhedral convex setC and the functionf. We prove that the curve will have finite are length if the normal map, associated with the derivative df(·) and the critical coneK, is coherently oriented at each zero of the normal mapf _c inside a certain ball of ⁿ . © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.This research was performed at the Department of Industrial Engineering, University of Wisconsin-Madison, Madison, WI, USA.     

Type 3 diminimal maps on the torus

A polyhedral map on the torus is diminimal if either shrinking or removing an edge yields a nonpolyhedral map. We show that all such maps on the torus fall into one of two classes, type 2 and type 3, and show that there are exactly two type 3 ones, which are given explicitly.     

The weakly neighborly polyhedral maps on the 2-manifold with euler characteristic −1

A weakly neighborly polyhedral map (w.n.p. map) is a two-dimensional cell-complex which decomposes a closed 2-manifold without boundary, such that for every two vertices there is a 2-cell containing them. We prove that there are just four w.n.p. maps with Euler characteristic –1 and we describe them.     

Light paths with an odd number of vertices in large polyhedral maps

LetP _k be a path onk vertices. In this paper we prove that (1) every polyhedral map on the torus and the Klein bottle contains a pathP _k such that each of its vertices has degree 6k–2 ifk is odd,k3, (2) every large polyhedral map on any compact 2-manifoldM with Euler characteristic (M)<0 contains a pathP _k such that each of its vertices has degree 6k – 2 ifk is odd,k3, (3) moreover, these bounds are attained. Fork=1 ork even,k2, the bound is 6k which has been proved in our previous paper.     

Totally chiral maps and hypermaps of small genus

An orientably regular hypermap is totally chiral if it and its mirror image have no non-trivial common quotients. We classify the totally chiral hypermaps of genus up to 1001, and prove that the least genus of any totally chiral hypermap is 211, attained by twelve orientably regular hypermaps with monodromy group A₇ and type (3,4,4) (up to triality). The least genus of any totally chiral map is 631, attained by a chiral pair of orientably regular maps of type {11,4}, together with their duals; their monodromy group is the Mathieu group M₁₁. This is also the least genus of any totally chiral hypermap with non-simple monodromy group, in this case the perfect triple covering 3.A₇ of A₇. The least genus of any totally chiral map with non-simple monodromy group is 1457, attained by 48 maps with monodromy group isomorphic to the central extension 2.Sz(8).