October 30

Transformational Geometry and Tessellations

Transformational Geometry Brief Introduction Arthur Cayley (1821-1895) and James Sylvester (1814-1897) were the earliest mathematicians to write about transformations. Their importance is because sets of transformations can be used to classify geometries. High school geometry now covers the transformations of rotation, translations, reflections and glide reflections. In 1872, Felix Klein classified geometries by applying the definitions below. Of particular importance is the application of transformations to crystallography which led to lattice theory and the famous 17 wallpaper groups. (see Web Resources). Definitions A transformation is a mapping f of A onto B such that each elements of B is the image of exactly one element of A. A transformation f is called an isometry of A onto B if it preserves distances. NOTE: The four basic Euclidean transformations: rotation, translations, reflection and glide reflections, are all isometries. Types of Euclidean Transformations 1. A translation is a correspondence between points and their image points so that each image is the same distance in the same direction from the original point. 2. A rotation is a correspondence between points and their image points where one point is fixed and the image points are transformed at a new angle position. The example below shows 5 rotations of the original shape around the center point: 3. A reflection is a correspondence between points and their image points so that each image is transformed as a mirror image over a horizontal(vertical or other) line. 4. A glide reflection is a correspondence between points and their image points where the image points are the product of a reflection and a translation parallel to the fixed line of reflection. This is often used in ornamental patterns - seen especially in the Alhambra in Grenada, Spain.

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Web Resources Elementary and Middle School Children's Comprehension of Euclidean Transformations - a brief report on a study by F. Richard Kidder on 9, 11 and 13 year olds and Euclidean Transformations http://www.nctm.org/jrme/abstracts/volume_07/vol07-01-jan1976.html#school Euclidean Isometries - fun animated explanation of the isometries http://www.math.ubc.ca/~robles/hyperbolic/eucl/isom/ Euclidean Transformations - using matrices to work with the transformations http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html Geometry of Transformations - workshop materials from Eastern Illinois University which explores and explains the geometric transformations http://www.ux1.eiu.edu/~cfpga/workshop99/outline.html Java Kali - an applet that lets you draw symmetrical patterns based on any of the 17 wallpaper groups, as well as several frieze and rosette groups. (http://www.geom.umn.edu/java/Kali/) NCTM Standard 8: Geometry from an Algebraic Perspective - geometric transformations using algebra http://timss.enc.org/TIMSS/standrds/standard/enc2280/280161s8.htm Penrose Tilings at the Geometry Junkyard - a page of links to Penrose tilings http://www.ics.uci.edu/~eppstein/junkyard/penrose.html QuasiTiler from the Geometry Center which will generate Penrose tilings interactively - http://www.geom.umn.edu/apps/quasitiler/about.html Tessellation Resources - compiled by Professor Doris Schattschneider from Moravian College (schattdo@moravian.edu) of online resources: http://www.geom.umn.edu/software/tilings/ Tessellation Tutorials from the Math Forum - a complete introduction to tessellations. http://mathforum.org/sum95/suzanne/tess.intro.html Totally Tessellated - an outstanding ThinkQuest site with lots of theory, references, links, and examples! http://library.thinkquest.org/16661/ Wallpaper Group Links - many links to other sites and resources from the University of Saskatchewan Department of Mathematics and Statistics http://math.usask.ca/~dlp537/links.html Wallpaper Groups - probably the best introduction to transformation and the 17 wallpaper groups (http://www.clarku.edu/~djoyce/wallpaper/) World of Escher - basic site for material on Escher, even if it is a commercial site http://www.WorldOfEscher.com World of Sir Roger Penrose - a brief biography of Sir Roger Penrose http://www.WorldOfEscher.com/misc/penrose.html

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