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Cons=Ucticn (Process) DOCUMENT RESUME ED 038 271 SE 007 847 AUTHOR Wenninger, Magnus J. TITLE Polyhedron Models for the Classroom. INSTITUTION National Council of Teachers of Mathematics, Inc., Washington, D.C. PUB DATE 68 NOTE 47p. AVAILABLE FROM National Council of Teachers of Mathematics,1201 16th St., N.V., Washington, D.C. 20036 ED RS PRICE EDRS Pr:ce NF -$0.25 HC Not Available from EDRS. DESCRIPTORS *Cons=ucticn (Process), *Geometric Concepts, *Geometry, *Instructional MateriAls,Mathematical Enrichment, Mathematical Models, Mathematics Materials IDENTIFIERS National Council of Teachers of Mathematics ABSTRACT This booklet explains the historical backgroundand construction techniques for various sets of uniformpolyhedra. The author indicates that the practical sianificanceof the constructions arises in illustrations for the ideas of symmetry,reflection, rotation, translation, group theory and topology.Details for constructing hollow paper models are provided for thefive Platonic solids, miscellaneous irregular polyhedra and somecompounds arising from the stellation process. (RS) PR WON WITHMICROFICHE AND PUBLISHER'SPRICES. MICROFICHEREPRODUCTION f ONLY. '..0.`rag let-7j... ow/A U.S. MOM Of NUM. INCIII01 a WWII WIC Of MAW us num us us ammo taco as mums NON at Ot Widel/A11011 01116111111 IT.P01115 OF VOW 01 OPENS SIAS SO 101 IIKISAMIT IMRE Offlaat WC Of MANN POMO OS POW. OD PROCESS WITH MICROFICHE AND PUBLISHER'S PRICES. reit WeROFICHE REPRODUCTION Pvim ONLY. (%1 00 O O POLYHEDRON MODELS for the Classroom MAGNUS J. WENNINGER St. Augustine's College Nassau, Bahama. rn ErNATIONAL COUNCIL. OF Ka TEACHERS OF MATHEMATICS 1201 Sixteenth Street, N.W., Washington, D. C. 20036 ivetmIssromrrIPRODUCE TmscortnIGMED Al"..Mt IAL BY MICROFICHE ONLY HAS IEEE rano By Mat __Comic _ TeachMar 10 ERIC MID ORGANIZATIONS OPERATING UNSER AGREEMENTS WHIM U. S. OFFICE OF EDUCATION. FURRIER 71PRODUCTION OUTSIDE RE ERIC SYSTEM REQUIRES PERMISSION OF RE COPYRIGHT OMER." Copyright 0 1966 by THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, INC. All rights reserved Second printiag 1968 Permissionto reproduce this copyrighted workhas been granted to the EducationalResources information (ERIC) and Center to the organizationoperating under with the Officea Education to centred reproducedocuments in- cluded in theERIC systemby means of but this rightis not cinferred microfiche only fiche received to any users ofthe micro- from the ERICDocument Reproduction 1 Service. Further reprodudion ofany part requires mission of thecopyright owner. per- Printed in the United States of America Contents 1 Introduction 3 The Five PlatonicSolids 6 The ThirteenArchimedean Solids 9 Prisms, Antiprisms, andOther Polyhedra 11 The Four Kepler-PoinsotSolids 2-1 Other Stellations orCompounds 28 Some Other UniformPolyhedra 41 Conclusion 42 Notes 43 Bibliography Photographs: Stanley Toogood'sInternational Film Productions, Nassau,Bahamas iii Introduction 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 awl handle these geometrical solids in aesthetically pleasing models 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 how 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. 1 HEXAHEDRON, OR CUBE OCTAHEDRON THE FIVE PLATONIC SOLIDS _.....111.-__ TETRAHEDRON i DODECAHEDRON ICOSAHEDRON The FivePlatonic Solids The most ancient polyhedra arethe set of five known asthe Platonic solids. They derive their namefrom the great Greekphilosopher though they Plato, who discovered themindependently about 400 a.c., four of were probablyknown before Plato. Theancient Egyptians knew them: the tetrahedron,octahedron, and cube arefound in their archi- tectural design, andEgyptian icosahedraldice are to be found in an exhibit in the BritishMuseum. According toHeath, the Etruscans five were were acquaintedwith the dodecahedronbefore 500 B.c.1 All studied by the earlyPythagoreans before the timeof Plato and Euclid. It is in the Elements ofEuclid, however, that wefind the most exten- sive treatment of the geometryof these five solids. Today models of thesesolids, usually in plastic, arefeatured in the catalogues of scientific andeducational supply houses.But models in heavy paper are soeasily made and so useful as aproject for stu- dents that it is wellworth the effort to make a set.The nets, or pat- terns, for makingthese models are given in manygeometry textbooks. It will be found thatthe models are even moreattractive when they are madewith facial polygons ofvarious colors.(Suggested color patterns are set out inFigures 1, 2, 3, 4, and 5.) Pio. 1 Fm. 2 3 Y Y us FE. 3 FIG. 4 Flo. 6 Fig. 7 Flo. 5 no. 6 5 Polyhedron Models for theClassroom the material to useis For easiest constructionand sharpest edges, used for file cards. heavy paper with asomewhat hard finishthe type "colored It can be bought inlarger sheets, in colors,under tue name tag." Pastels are verysuitable, and they are agood alternative to deeper colors. In themethod of constructionsuggested here you need Three, four, or five only one triangle, square, orpentagon as a net. and the net placed on sheets oi colored paper maybe stapled together through all the top of them.Then, using a sharpneedle and pricking each vertex of the net, sheets at one time, make ahole in the paper at copies of each which is held as a guide ortemplate. In this way exact initial trimming with part are quicklyobtained. Next give the paper an scissors, with all thesheets still held firmlytogether by the staples. all You must be careful toprovide about aquarter-inch margin After around to be used forflaps or tabs to cementthe parts together. Experience will soon this it will be best to treateach part individually. teach you that the accuracyof your completedmodel is directly pro- portional to the care youhave lavished on eachindividual part. With compass, you must nowscore a sharppoint, such as that of a geometry guide to connect the paper, using astraightedge or set square as a needed, since the the needle holeswith lines.(Pencil lines are not part.) More process ofscoring sufficiently outlinesthe shape of each how to do accurate trimmingis next to be done.(For suggestions on folding of this, see Figures 6, 7,and 8.) The scoredlines then make the tabs a simple and accurateoperation. adhesive, since it is A good household cementprovides the best clothespins of the coiled- strong andquick-drying. A few wooden spring variety that havebeen turned inside outmake excellent clamps. the When these are used,the cementing canproceed rapidly and of clamps can be movedfrom one part to the nextin almost a matter minutes as each part issuccessively cut andtrimmed. The last part and will not give you toomuch trouble if youfirst cement one edge the other edges and let it set firmly, thenproceed to put cement on A close down the lastpolygon as you wouldclose the lid on a box. maneuver the needle or compass pointmakes a good instrument to last edges into accurateposition. Deft fingersand a little practice will do the rest. The Thirteen Archimedean Solids Once a set of the five Platonic solids has been made, the next proj- ect will certainly be to make a set of the thirteenArchimedean solids. These too have an ancient history. Plato is said to have known at least one of them, the cuboctahedron.Archimedes wrote about the entire set, though his book on them is lost. Kepler is thefirst of the moderns to have treated these solids in a systematic way.He was also the first to observe that two infinite sets of polyhedra, the setof prisms and the set of antiprims, have something in common with the thirteen Archimedeans, namely, membership in the set known as the semi- regular polyhedra.= (A semiregular polyhedron is one that admits a variety of polygons as faces, provided that they are all regular and that all the vertices are the same.) As in the case of the Platonic solids, so too in that of the Archime- deans the beauty of the set is greatly enhanced by suitable color ar- rangements for the faces. Since it is evident that many different color arrangements are possible, you may find it interesting to work out a suitable arrangement for yourself. The general principle is to work for some kind of symmetry and to avoid having adjacent faces with the same color. This may remind you of the map-coloring problem. The fact is that a polyhedron surface is a map, and as such is studied in the branch of mathematics known as topology. In making these models, however, you need not enter into any deep mathematical analysis to get what you want. Your own good sense will suggest suitable procei -res. (See page 8.) The actual technique of construction is the same here as that given above: namely, only one polygona triangle, square, pentagon, hexagon, octagon, or decagonwill serve as a net. However, it is important to note that in any one model all the edges must be of the same length. If you want to make a set having all edgesequa!, you will find the volumes growing rather large with some models in the set. Of course a large model takes up more display space, so you must gauge your models with that fact in mind.
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