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Articles 31 - 43 of 43
Full-Text Articles in Geometry and Topology
Quest For Tilings On Riemann Surfaces Of Genus Six And Seven, Robert Dirks, Maria Sloughter
Quest For Tilings On Riemann Surfaces Of Genus Six And Seven, Robert Dirks, Maria Sloughter
Mathematical Sciences Technical Reports (MSTR)
The problem of kaleidoscopically tiling a surface by congruent triangles is equivalent to finding groups generated in certain ways. In order to admit a tiling, a group must have a specific set of generators as well as an involutary automorphism, T, that acts to reverse the orientation of the tiles. The purpose of this paper is to explore group theoretic and computational methods for determining the existence of symmetry groups and tiling groups, as well as to classify the symmetry and tiling groups on hyperbolic Riemann surfaces of genus 6 and 7.
Lengths Of Geodesics On Klein’S Quartic Curve, Ryan Derby-Talbot
Lengths Of Geodesics On Klein’S Quartic Curve, Ryan Derby-Talbot
Mathematical Sciences Technical Reports (MSTR)
A well-known and much studied Riemann surface is Klein’s quartic curve. This surface is interesting since it is the smallest complex curve with maximal symmetry. In addition to this high degree of symmetry, Klein’s quartic curve can be tiled by triangles,giving rise to a tiling group generated by reflections. Using the tiling group and the universal cover of the tiling group we are able to compile a list of the lengths of the short,simple,closed geodesics on this surface. In particular,w e are able to determine whether the geodesic loops generated by the tiling are the systoles,i.e.,the shortest closed geodesics.
Splitting Tiled Surfaces With Abelian Conformal Tiling Group, Sean A. Broughton
Splitting Tiled Surfaces With Abelian Conformal Tiling Group, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
Let p be a reflection on a closed Riemann Surface S, i.e., an anti-conformal involutary isometry of S with a non-empty fixed point subset. Let Sp denote the fixed point subset of p, which is also called the mirror of p. If S −Sp has two components, then p is called separating and we say that S splits at the mirror Sp. Otherwise p is called non-separating. We assume that the system of mirrors, Sq, as q varies over all reflections in the isometry group Aut*(S) defines a tiling of the surface, consisting of triangles. In turn, the tiling determines …
Divisible Tilings In The Hyperbolic Plane, Sean A. Broughton, Dawn M. Haney, Lori T. Mckeough, Brandy M. Smith
Divisible Tilings In The Hyperbolic Plane, Sean A. Broughton, Dawn M. Haney, Lori T. Mckeough, Brandy M. Smith
Mathematical Sciences Technical Reports (MSTR)
We consider triangle-quadrilateral pairs in the hyperbolic plane which "kaleidoscopically" tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisible tiling. All possible such divisible tilings are classified. There are a finite number of 1,2, and 3 parameter families as well as a finite number of exceptional cases.
Tilings Which Split A Mirror, Jim Belk
Tilings Which Split A Mirror, Jim Belk
Mathematical Sciences Technical Reports (MSTR)
We consider the mirror of a reflection which consists of its subset of fixed points. We investigate a number of conditions on the tiling that guarantee that the surface splits at a mirror.
Constructing Kaleidscopic Tiling Polygons In The Hyperbolic Plane, Sean A. Broughton
Constructing Kaleidscopic Tiling Polygons In The Hyperbolic Plane, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
We have all seen many of the beautiful patterns obtained by tiling the hyperbolic plane H by repeated reflection in the sides of a "kaleidoscopic" polygon. Though there are such patterns on the sphere and the euclidean plane, these positively curved and fiat geometries lack the richness we see in the hyperbolic plane. Many of these patterns have been popularized by the beautiful art of M.C. Escher. For a list of references and a more complete discussion on the construction of artistic tilings see [6].
Symmetry And Tiling Groups For Genus 4 And 5, C. Ryan Vinroot
Symmetry And Tiling Groups For Genus 4 And 5, C. Ryan Vinroot
Mathematical Sciences Technical Reports (MSTR)
All symmetry groups for surfaces of genus 2 and 3 are known. In this paper, we classify symmetry groups and tiling groups with three branch points for surfaces of genus 4 and 5. Also, a class of symmetry groups that are not tiling groups is presented, as well as a class of odd order non-abelian tiling groups.
Quadrilaterals Subdivided By Triangles In The Hyperbolic Plane, Dawn M. Haney, Lori T. Mckeough
Quadrilaterals Subdivided By Triangles In The Hyperbolic Plane, Dawn M. Haney, Lori T. Mckeough
Mathematical Sciences Technical Reports (MSTR)
In this paper, we consider triangle-quadrilateral pairs in the hyperbolic plane which “kaleidoscopically” tile the plane simultaneously. These tilings are called divisible tilings or subdivided tilings. We restrict our attention to the simplest case of divisible tilings, satisfying the corner condition, in which a single triangle occurs at each vertexof the quadrilateral. All possible such divisible tilings are catalogued as well as determining the minimal genus surface on which the divisible tiling exists. The tiling groups of these surfaces are also determined.
Oval Intersections In Tilings On Surfaces, Dennis A. Schmidt
Oval Intersections In Tilings On Surfaces, Dennis A. Schmidt
Mathematical Sciences Technical Reports (MSTR)
A tiling is a covering by polygons, without gaps or overlapping, of a compact, orientable surface. We are particularly interested in tilings by triangles that generate a large symmetry group of the surface. An oval of the tiling is a simple, closed curve that is a union of edges of the tiling. We investigate the number of points of intersection of two ovals. We have found that the number of intersections is bounded when the subgroup of orientation preserving symmetries is abelian. However, there is no upper bound on the number of intersections in the non-abelian case.
Counting Ovals On A Symmetric Riemann Surface, Sean A. Broughton
Counting Ovals On A Symmetric Riemann Surface, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
Let S be a compact Riemann surface without boundary. A symmetry of S is an anti-conformal, involutary automorphism. Its fixed point set is a disjoint union of circles, each of which is called an oval. A method is presented for counting the ovals of a symmetry when S admits a large group G of automorphisms. The method involves only calculations in G, based on the geometric description of S/G, and the knowledge of the action of the symmetry on G.
A Stronger Triangle Inequality, Herb Bailey
A Stronger Triangle Inequality, Herb Bailey
Mathematical Sciences Technical Reports (MSTR)
The triangle inequality is basic for many results in real and complex analysis. The geometric form states that the sum of any two sides of a triangle is greater than the third. This was included as Proposition XX in the first book of Euclid's Elements. Many geometric triangle inequalities involving sides, angles, altitudes, inscribed circles and circumscribed circles have been found. Hundreds of these inequalities are summarized in [l] and [2]. A nice geometric proof of the triangle inequality is given in [3].
Symmetries Of Accola-Maclachlan And Kulkarni Surfaces, Sean A. Broughton, E Bujalance, A F. Costa, J M. Gamboa, G Gromadzki
Symmetries Of Accola-Maclachlan And Kulkarni Surfaces, Sean A. Broughton, E Bujalance, A F. Costa, J M. Gamboa, G Gromadzki
Mathematical Sciences Technical Reports (MSTR)
For all g greater than or equal to 2, there is a Riemann surface of genus g whose automorphism group has order 8g+8, establishing a lower bound for the possible orders of automorphism groups of Riemann surfaces. Accola and MacLachlan established the existence of such surfaces; we shall call them Accola-MacLachlan surfaces. In this paper we determine the symmetries of surfaces with genus g = 3(mod 4), computing the number of ovals and the separability of the symmetries. The results are then applied to classify the real forms of these complex algebraic curves.
Integer Triangles With Rational Medians, Bart Goddard, Dale Mesner
Integer Triangles With Rational Medians, Bart Goddard, Dale Mesner
Mathematical Sciences Technical Reports (MSTR)
A characterization of all integer-sided triangles with a rational median is given, similar to the categorization of Pythagorean triangles. An infinite family of integer-sided triangles with two rational medians is given, along with several examples of three rational medians. All examples come from solutions to systems of quadratic Diophantine equations.