December 4

Elliptic Geometry: Congruence and Quadrilaterals

Terms 1. A sphere is defined to be a set of point in space equidistance from a fixed point. 2. A diameter is defined to be a line passing through the center of the sphere with endpoints on the surface of the sphere. 3. A line is defined to be a great circle on the sphere. ttt Spherical Parallel Postulate Given a line and a point P not on , there are no lines through P parallel to . An alternative statement of this is: Any two lines in the plane meet at an ordinary point. Questions 1) Download this Sketchpad File. elliptic14.gsp Move the three vertices A, B, and C of the triangle around. a) How would you complete this theorem?   The sum of the angles in a triangle in spherical geometry is: _____________   b) What is the largest angle sum that a triangle in spherical geometry may have? ttt 2) Since a line in spherical geometry is a great circle, a) What is the length of a line? b) A line of latitude that is not the equator of a sphere represents what other concept in spherical geometry? c) Do two points always determine a unique line in spherical geometry? d) What is the shortest path between two points? ... 3) Complete the following statements: a) The sum of the measures of the angles in a quadrilateral is _______ 360o. b) In spherical geometry , a Sacchari quadrilateral has congruent and ________ summit angles. c) Similar but non-congruent triangles _______ exist in spherical geometry. d) The Pythagorean Theorem is ______ in spherical geometry. e) In spherical geometry, a Lambert quadrilateral has its fourth angle _________ . ... 4) Answer these questions: a) Do regular polygons exist in spherical geometry? b) Can you state at least 2 theorems which are true in Euclidean geometry and would also hold for spherical geometry? c) Which finite geometries are spherical?

Web Resources The following resources all help in the understanding of spherical (elliptic) geoemtry: The Applications Of Non-Euclidean Geometry - discussion of cosmology and geometry with applications of Spherical Geometry (http://members.tripod.com/~noneuclidean/applications.html) Cinderella: Elliptic Cycloid - an animation showing a cycloid in elliptic space (I only was able to see this when I viewed it in IE, not in Netscape!) (http://www.cinderella.de/en/demo/gallery/EllipticCycloid.html) Escher's tessellation in spherical geometry - illustration of Escher's spherical tessellation of angles and devils (http://plus.maths.org/issue18/xfile/) Geometry in Space - worksheet and material to compare the three geometries as applications in space. (http://universe.sonoma.edu/materials/lesson_plans/geometry.html) Geometry of the Sphere - the best resource on the web for a complete course on spherical (elliptic) geometry (http://math.rice.edu/~pcmi/sphere/) Great Circles - explanation of the different circles with an illustration as well as a revolving sphere showing shortest distance (http://mathworld.wolfram.com/GreatCircle.html) Lénárt Sphere™ Construction Materials: Construction Materials for Another World of Geometry - new classroom materials to explore spherical geometry (http://www.keypress.com/catalog/products/tools/Prod_LnrtSphre.html) Lesson Plan in Spherical Geometry - complete lesson plan with solutions based on Non-Euclidean Adventures on the Lenart Sphere by Istvan Lenart (http://www.mccallie.org/myates/Spherical%20Geometry/Table%20of%20Contents/table%20of%20contents.htm) Model of a Sphere - draggable (and rotating) model with the great circles and regular circles (http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/math/s/s556.htm) Non-Euclidean Geometry with LOGO - an article on LOGO's use in non-Educlidean geometries, which is under development at the Math Department in Cardiff, Wales - file is in PDF format (http://www.bham.ac.uk/ctimath/reviews/aug91/LOGO.PDF) Spherical Triangle - explanation of the spherical triangle with an illustration as well as a revolving sphere showing an example (http://mathworld.wolfram.com/SphericalTriangle.html) Triangles in Elliptic Geometry - properties of triangles, generated with the software called Fnord. (http://www.geom.umn.edu/~banchoff/elliptic/elliptic.html)

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