DOCUMENT RESUME ED 116 939 SE 020 142 AUTHOR Wenninger, Magnus J. TITLE Polyhedron Models for the Classroom. Second INSTITUTION National Council of Teachers of Mathematics, Inc., Reston, Va. PUB DATE 75 NOTE 64p.; For an earlier edition, see ED 038 271 AVAILABLE FROM National Council of Teachers of Mathematics, 1906 Association. Drive, Reston, Virginia 22091 ($1.40, discounts on quantity orders) EDRS PRICE Mr-$0.76 Plus Postage. HC Not Available from EDRS. DESCRIPTORS *Activity Learn'ng; College Mathematics; Construction (Process); Geom tric Concepts; *Geometry; *Instructional aterials; Mathematical Enrichment; *Mathematical Mo els; Mathematics Materials; Secondary Educat on; *Secondary School Mathematics IDENTIFIERS National. Council f Teachers of Mathematics; NCTM; *Polyhedrons ABSTRACT .This second edition explains the historical bac1ground and techniques for constructing various types of polyhedra. Seven center-fold sheets are included, containing full-scale drawings from which nets or templates may be made to construct the models shown and described in the text. Details are provided for construction of the five Platonic solids, the thirteen Archimedean solids, stellations or compounds, and other miscellaneous polyhedra. The models may be used to illustrate the ideas of symmetry, reflection, rotation, and translation. Included is.a bibliography of related sources.(Author/JBW) *********************************************************************** * Documents acquired by ERIC include many informal unpublished *materials not available from other sources. ERIC makes every effort* *to obtain the best copy available. Nevertheless, items of marginal * *reproducibility are often encountered and this affects the quality * *of the microfiche and hardcopy reproductions ERIC makes available * *via,the ERIC Document Reproduction Service (EDRS). EDRS is not *responsible for the quality of the original document. Reproductions* *supplied by EDRS are the best that can be made from the original. * *********************************************************************** r. e U S DEPARTMENT OF HEALTH, EDUCATION & WELFARE s44- NATIO"AL INSTITUTE OF EuUCATION 4.41,4,;.alatL_ THIS DOCUMENT HAS BEEN REPRO- DUCED EXACTLY AS RECEIVED FROM N' THE PERSON OR ORGANIZATION ORIGIN. -7 ATING IT POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILY REPRE- SENT OFFICIAL NATIONAL INSTITUTE OF EDUCATION POSITION OR POLICY ',L rt, CY' PERMISSION TO REPRODUCE THIS rat COPYRIGHTED MATERIAL sY MICRO FICHE ONLY HAS BEEN GRANTED BY Charles R.HTcka N.C.T.M. TO ERIL AND ORGANIZATIONS DPERAT ING UNDER AGREEMENTS WITH THE NA Ci TIONAL INSTITUTE OF EDUCATION 1°.FURTHER REPRODUCTION OUTSIDE THE ERIC SYSTEM REQUIRES PERM'S SION OF THE COPYRIGHT OWNER O POLYHEDRON MODELS for the Classroom Second Edition MAGNUS J. WENNINGER St. Augustine's College Nassau, Bahamas A NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS 1906 Association Drive, Reston, Virginia 22091 3 Copyright @ 1966, 1975 by THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, INC. All rights reserved Printed in the United States of America 4 Contents Introduction vii Introduction to the First Edition 1 The Five Platonic Solids 3 The Thirteen Archimedean Solids 6 Prisms, Antiprisms, and Other Polyhedra 9 The Four Kepler-Poinsot Solids 11 Other Stellations or Compounds 21 Some Other Uniform Polyhedra 28 Conclusion 41 Notes 42 Bibliography 43 Photographs: Stanley Toogood's International Film Productions, Nassau, Bahamas 4 v Introduction This booklet, first published in 1966,went through its sixth print- ing in 1973, makingup a total of 35 000 printed copies. That fact alone attests to the continuing interest inthis work on the part of teachers and students alike. The second edition now being presentedadds a new feature seven center-fold sheets containing full-scale drawingsfrom which nets or templates, may be derived to make the modelsshown in the photographs and described in the text. The following procedure is recommended for makingthe tem- plates. Carefully lopsen the staples at thecenter of the booklet and remove the center-fold pages. Having done this, turn the staplesback down so that the booklet remains intact forreading and reference. The center-fold pages may now be cut with scissors.Leave a quarter- inch margin all around the edge of eachpattern. This will provide the proper net or template for making the models. If you do not wish to cut the center-foldpages, or if you would like to make a more durable netor template, you may copy the pattern onto heavier paper. First, place the patternover a sheet of heavier paper (perhaps index card stock) andthen prick through the vertices of the pattern witha divider point or a needle. Use a straightedge to draw the outline of thepattern onto the heavy paper and trim as described before, leavinga quarter-inch margin all around the edge. This net or template isto be used to multiply parts, which are treated as describedon page 5. Do net trim the net or template. Trim only the parts to be used for makingthe model. Spe- cific directions are given for trimming the parts shcwnon pages 4, 13, 15, and 26. It is assumed thatyour own experienc.: and ingenuity will dictate what is best in all othercases. The templates may then be saved and used for makingmore models of the same kind or, in some instances, for making different n?odels where a particular tem- plate is used in combination with others. vii viii Polyhedron Models for the Classroom If you should need or want more explicitinstructions for making the models described on pages 28-40, youwill find them in a larger work of mine, a book entitledPolyhedron Models, published by Cambridge University Press (New York, 1974)in paperback form. This is a source book for all 75 knownuniform polyhedra and for some stellated forms. It might be of interest for readers to knowthat a definitive enu- meration of uniform polyhedra has been made.John Skil ling, of the Department of Applied Mathematicsand Theoretical Physics at Cambridge University, has just published theresults of his computer analysis of the problem. He shows that "thelist of Coxeter et al. is indeed complete as regards uniform polyhedra inwhich only two faces meet at any edge. The naturalgeneralization that any even number of faces may meet at an edge allowsjust one extra poly- hedron to be included in the set." (SeeJohn Skil ling, "The Complete Set of Uniform Polyhedra," PhilosophicalTransactions of the Royal Society of London, ser. A, vol. 278, no. 1278[March 1975].) M. J. W. Nassau, Bahamas April 1975 Introductien to the First Edition The study of polyhedra is an ancient one, going back to the dawn of history.It is especially those polyhedra that are called uniform that have evoked the greatest interest and provided the most fascina- tion.It should therefore be of special usefulness for mathematics stu- dents and teachers in their classrooms today to see and handle these geometrical solids in aesthetically pleasing mo5iels and to be delighted with their beauty and form. Most students show immediate interest in this kind of work, and teachers are often surprised to see the quality of the results a student obtains in making the models. It is a genuine outlet for the creative instinct; in addition, it calls for care and accuracy, as well as persever- ance and pertinacity in models that have many parts and an intricate color arrangement. It is also surprising the models can stimulate interest in some of the basic theorems of solid geometry. And, when a project is finished, the models will enhance the appearance of the classroom, where they can be put on permanent display. 41110 1 a alLmf arINI HEXAHEDRONk OR CUBE OCTAHEDRON THE FIVE PLATONIC SOLIDS TETRAHEDRON DODECAHEDRON ht ICOSAHEDRON The Five Platonic Solids The most ancient polyhedra are the set of five known as the Platonic solids.They derive their name from the great Greek philosopher Plato, who discovered them independently about 400 B.C., though they were probably known before Plato. The ancient Egyptians knew four of them: the tetrahedron, octahedron, and cube are.found in their archi- tectural design, and Egyptian icosahedral dice are to be found in an exhibit in the British Museum. According to Heath, the Etruscans were acquainted with the dodecahedron before 500 n.c.i All five were studied by the early Pythagoreans before the time of Plato and Euclid. It is in the Elements of Euclid, however, that we find the most exten- sive treatment of the geometry of these five solids. Today models of these solids, usually in plastic, are featured in the catalogues of scientific and educational supply houses. But models in heavy paper are so easily made and so useful as a project for stu- dents that it is well worth the effort to make a set. The nets, or pat- terns, for making these models are given in many geometry textbooks. It will be found that the models are even more attractive when they are made with facial polygons of various colors.(Suggested color patterns are set out in Figures 1, 2, 3, 4, and 5.) FIG.1 FIG. 2 3 10 R B FIG. 3 FIG. 4 FIG. 6 FIG. 7 FIG. S FIG. 8 11 Polyhedron Models for the Classroom 5 For easiest construction and sharpest edges, the material to use is heavy paper with a somewhat hard finishthe type used for file cards. It can be bought in larger sheets, in colors, under the name "colored tag."Pastels are very suitable, and they are a good alternative to deeper colors. In the method of construction suggested here you need only one triangle, square, or pentagon as a net. Three, four, or five sheets of colored paper may be stapled together and the net placed on top of them. Then, using a sharp needle and pricking through all the sheets at one time, make a hole in the paper at each vertex of the net, which is held as a guide or template. In this way exact copies of each part are quickly obtained.