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Geometry in Colleges at the Undergraduate Level. The DOCUMENT RESUME ED 024 578 SE 004 992 By- Durst, Lincoln K., Ed. Committee of the Undergraduate Program in MathematicsGeometry Conference, Part III: Geometric Transformation Groups and Other Topics. Committee on she Undergraduate Program in Mathematics,Berkeley, Calif. Spons Agency-National Science Foundation, Washington,D.C. Report No-18 Pub Date Oct 67 Note- 174p. EDRS Price MF-$0.75 HC-$8.80 Descriptors-*College Mathematics, *Conference Reports,Conferences, *Curriculum, CurriculumDevelopment, *Geometry, Instruction, *Mathematics, UndergraduateStudy Identifiers-California, National Science Foundation,Santa Barbara This is Part III of the first volumeof the proceedings of theCommittee on the Undergraduate Program in Mathematics(CUPM) Geometry Conference,held at Santa the status of Barbara June, 1967. The purpose of theconference was to consider geometry in colleges at theundergraduate level.The conference, attendedby undergraduate mathematicsteachers,involved lectures on variousaspects of geometry, analyses o f materialpresented, and an examinationof the relevance of the material to the undergraduatecurriculum. In Part III ofthe proceedings are contained the following lectures:(1) "TransformationGroups from the Geometric Viewpoint," by H.S.M Coxeter, (2)"Two Applications of Geometry,"by Herbert Busemann, inFunctional (3)"Some ComputationalIllustrationsofGeometricalProperties Iteration," by Glen Culler, (4)"Generalizations in Geometry," byPreston Hammend, (5) "The Nature and Importance ofElementary Geometry in aModern Education," byPaul J. Kelly, and (6) "Joining andExtending as GeometricOperations (A Coordinate-Free Approach to N-Space)," by WalterPrenowitz. (RP) COMVITTEE ONItit HNOERGRAOHATt PROGRAV IN MAIM/ATIH *.J04/ co ORT f=3 OCTOBER 1967 NUMBER 18 CUPM GEOMETRY CONFERENCE PROCEEDINGS PART III: GEOMETRIC TRANSFORMATION GROUPS AND OTHER TOPICS Lectures by H. S.M. Coxeter and Others U.S. DEPARTMENT OF HEALTIi, EDUCATION & WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIGINATING IT.POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILY REPRESENT OFFICIAL OFFICE OF EDUCATION POSITION OR POLICY. MATHEMATICAL ASSOCIATION OF AMER ICA CUPM GEOMETRY CONFERENCE Santa Barbara, California June 12 - June 30, 1967 PROCEEDINGS OF THE CONFERENCE Edited by Lincoln K. Durst PART III: GEOMETRIC TRANSFORMATION GROUPS AND OTHER TOPICS Lectures by H. S. M. Coxeter and Others COMMITTEE ON THE UNDERGRADUATE PROGRAM IN MATHEMATICS Mathematical Association of America @) 1968, Mathematical Associationof America, Inc. "PERMISSION TO REPRODUCE THIS COPYRIGHTED MATERIAL HAS BEEN GRANTED E. A. Cameron BY Treasurer, Ma-- TO ERIC AND ORGANIZATIONS OPERATING UNDER AGREEMENTS WITH THE U.S. OFFICE OF EDUCATION. FURTHER REPRODUCTION OUTSIDE THE ERIC SYSTEM REQUIRES PERMISSION OF THE COPYRIGHT OWNER." Financial support for the Committee onthe Undergraduate Program in Mathematicshas been provided by theNational Science Foundation.A limited number of copiesof these Proceedings is available fromCUM, Post Office Box1024 Berkeley, California 94701. FOREWORD This is the final volumeof the Proceedings ofthe CUPM GeometryConference, held at SantaBarbara in June, contains an Introduction 1967. Part I of the Proceedings by Walter Prenowitz andthe lectures of BrankoGrUnbaum and Victor Klee on Convexity. Part II contains thelectures of Andrew Gleason and NormanSteenrod on Geometry inOther Subjects. The texts are based onrecordings made of the lectures and discussions, and wereprepared for publication bythe assistants, Melvin Hausner,John Reay and Paul Yale.The lecturers were able to makeminor changes andcorrections on the finalsheets, hut an earlydeadline prevented major revision or extensivepolishing of thc texts.The typing for offset was done by Mrs.K. Black and thefigures were prepared by W. David M.Youngdahl. Lincoln K. Durst Claremont Men's College MEMBERS OF THE CONFERENCE Murray S. Klamkin Russell V. Benson California State College,Fullerton Ford Scientific Laboratory Victor L. Klee, Jr. (Lecturer) Gavin Bjork University of Washington Portland State College Rev. John E. Koehler John W. Blattner Seattle University San Fernando ValleyState College Sister M. Justin Markham Herbert Busemann (Lecturer) University of SouthernCalifornia St. Joseph College Michael H. Millar Jack G. Ceder (Visitor) University of California, Stanford University Santa Barbara H. Stewart Moredock Sacramento State College G. D. Chakerian University of California,Davis Richard B. Paine Colorado College H. S. M. Coxeter (Lecturer) University of Toronto Walter Prenawitz (Chairman) Glen J. Culler (Lecturer) Brooklyn College University of California, John R. Reay (Assistant) Santa Barbara Western Washington StateCollege Andrew/4. Gleason (Lecturer) Paul T. Rygg Harvard University Western Washington StateCollege Neil R. Gray Geo e T, Sallee Western Washington StateCollege University of California,Davis Helmut Groemar James M. Sloss (Visitor) University of Arizona University of California, Santa Barbara Branko GrUnbaum (Lecturer) University of Washington Norman E. Steenrod (Lecturer) Preston C. Hammer (Lecturer) Princeton University Pennsylvania State University George Stratopoulos Weber State College Melvin Hausner (Assistant) New York University Robert M. Vogt Norman W. Johnson San Jose State College Michigan State University William B. Woolf University of Washington Mervin L. Keedy Purdue University Paul B. Yale (Assistant) Paul J. Kelly(Lecturer) Pomona College University of California, Santa Barbara Raymond B. Killgrove California State College, Los Angeles CONTENTS TRANSFORMATION GROUPSFROM THE GEOMETRIC 1 VIEWPOINT H. S. M. Coxeter 73 TWO APPLICATIONS OFGEOMETRY Herbert Busemann SOME COMPUTATIONALILLUSTRATIONS OF GEOMETRICAL PROPERTIES IN FUNCTIONALITERATION 107 Glen Culler GENERALIZATIONS IN GEOMETRY 115 Preston Hammer THE NATURE ANDIMPORTANCE OF ELEMENTARY GEOMETRY IN A MODERNEDUCATION 133 Paul J. Kelly JOINING AND EXTENDINGAS GEOMETRICOPERATIONS 149 A coordinate-freeapproach to n-space Walter Prenowitz REPORT ON CURRICULUMDISCUSSION 167 Walter Prenawitz TRANSFORMATION GROUPS FROM THE GEOMETRIC VIEWPOINT Lectures by H. S. M. Coxeter (Lecture notes by Paul Yale) Lecture I. Euclidean GeomIrx: The group of similarities. When a geometry is characterized inaccordance with Klein's Erlangen program, two groups arise: a groupGunder which all the propositions remain valid, and a normal subgroup H under whichthe concepts and their properties are maintained. For instance, in the case of Euclidean geometry,G is the group of similarities(because every true theorem remains true when anysimi- larity is applied to the figure involved)and His the group cf displacements (or direct isometries) because displacements preservedistance, area, and so on. It is accordingly desirable toclassify similarities and to discusstranslations, rotations, twists, and the way they can bebuilt up from reflections. believe that some version of this materialshould be in the curriculum of our schools. I have a feeling that littlebits of it should come in from the first grade on, because everything is sosimple, not requiring coordinates, still less any calculus.We live in a space, so,not only formathematicians, but also for physicists, engineers, andarchitects, it is desirable to have some feeling for space. Of course the first kind of space weshould think about is Euclidean space. Some educators have suggested that spaceshould bc studied in kindergarten, and the plane infirst grade, since solid objects are more natural tothink of than flat figures. Euclidean geometry is essentially a study of congruenceand similarity, so zlt an early age oneshould see the relation between a figure and a congruent orsimilar figure. Although one would present itdifferently to young people, perhapsthe of the easiest way to define a similarity(or "similarity transformation") Euclidean plane is in terms of barycentric coordinates. Suppose we are given two similar scalene triangles. The transformation carrying a point with barycentric coordinates (x,y,z) for the first triangle to the point with barycentric coordinates (x,y,z) for the second is called a similarity. It is a one to one transformation of the plane onto itself. It has a certain ratio of magnification, p, which means that segments are increased by the ratio 2 p to 1 and areas by the ratio p to 1. The similarity is direct if the two triangles have the same sense and is opposite if they have opposite senses. If p = 1, the transformation is called a congruent transformation or isometry. It is desirable first to classify isometries and then to classify similarities. The first theorem is that every isometry of the plane can be expressed as the product of three or fewer reflections, a reflection being a transformation that leaves fixed every point on a line, the mirror, and takes points on one side of this line to points at the same distance on the other side.Reflection is an involutory transformation; i.e., its period is two: the image of the image is the object itself. To see that three reflections suffice, consider two congruent triangles that are related by the given isometry. You may need one reflection for the first pair of corresponding vertices, another for the second pair, and a third for the third. Since each reflection reverses sense, a product of an even number of reflections preserves sense; hence for displace- ments (direct isometries) the number three may be reduced to two; i.e., each displacement is the product
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