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Home CCSU Theses & Dissertations The Euclidean and hyperbolic geometry underlying M.C. Escher's regular division designs / Karen...

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The Euclidean and hyperbolic geometry underlying M.C. Escher's regular division designs / Karen Kleinschmidt Haxhi

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Identifier	Thesis 1491
Author	Haxhi, Karen Kleinschmidt
Title	The Euclidean and hyperbolic geometry underlying M.C. Escher's regular division designs / Karen Kleinschmidt Haxhi
Publisher	Central Connecticut State University
Date	1998
Resource Type	Master's Thesis
Notes	In this paper Euclidean and hyperbolic discontinuous isometries* are discussed and compared. The graphic artist M.C. Escher (1898-1972) produced works which have for the structure points, lines, and angles, which repeat in the plane (Euclidean and hyperbolic) at regular intervals. These regular systems of points, lines and angles can be described with mathematics - - with groups of discontinuous isometries and the tilings they produce. This paper is therefore also a comparison of Escher's Euclidean and hyperbolic regular division (the plane is divided into congruent shapes which cover the plane) designs. The isometries of the Euclidean and the hyperbolic planes are discussed empirically. Specific isometries are described and diagrammed in thirteen of Escher's Euclidean designs (Escher made dozens of regular division Euclidean designs.) and in two of Escher's four "Circle Limit" pictures, which are hyperbolic regular division designs. In the empirical spirit of this paper, I have also provided instructions of how to construct with ruler and compass a hyperbolic tiling. The construction of Euclidean tilings are straightforward when compared to the construction of hyperbolic tilings. Euclidean tilings follow directly from the diagrams in this paper of the Euclidean isometry groups. Indeed since childhood we have all stared at Euclidean tile patterns on walls and floors and can produce many tilings with no understanding of the underlying mathematics. The creation of hyperbolic tilings is not so straightforward. We do not often see hyperbolic patterns, and when we do, as when looking at an Escher "Circle Limit" print, it seems strange that similar hyperbolic shapes in our model (Escher used the Poincare disk model) of the hyperbolic plane. Additionally, our Euclidean notion of straight lines is not visually apparent in Escher's hyperbolic designs. In the Poincare disk model hyperbolic lines not passing through the center of the disk are represented by circular arcs orthogonal to the bounding circle of the disk. This paper is designed for the secondary school mathematics teacher who wants to gain a qualitative understanding of the differences between the Euclidean and hyperbolic planes, and their tilings, without the abstract notation that often accompanies such discussions. Tilings of the Euclidean plane and the hyperbolic plane may then be used to illustrate, visually, the differences between the two planes. These visual differences can then be used to illustrate the differences in the theorems of the Euclidean and the hyperbolic planes. All differences between the two planes stem from their different parallel axioms. The introduction of hyperbolic geometry on the secondary school level may serve to show students that the proofs they do in class may not be mere formalities to show something that is visually obvious. In our models of the hyperbolic plane, congruence between line segments, polygons, circles, etc. may not be visually obvious. Therefore a proof written to prove a theorem for the hyperbolic plane may give a student a clearer understanding of the importance of applying the rules of logic to a set of axioms and theorems to prove, or disprove, a conjecture they may have. Copies of two Escher prints below will visually give a preview of the differences between Euclidean designs/tilings and hyperbolic designs/tilings that will be discussed in this paper. Figure A has for its structure a Euclidean isometry group. Figure B has for its structure a hyperbolic isometry group.
Subject	Geometry, Descriptive Geometrical drawing Hyperbolic spaces Geometry, Non-Euclidean
Department	Department of Mathematical Sciences
Advisor	McGowan, Jeffrey K
Type	Text
Digital Format	application/pdf
Language	eng
OCLC number	40251155
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Home CCSU Theses & Dissertations The Euclidean and hyperbolic geometry underlying M.C. Escher's regular division designs / Karen...

The Euclidean and hyperbolic geometry underlying M.C. Escher's regular division designs / Karen Kleinschmidt Haxhi

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