& 4Ԇj:.Ň<~!@]'a~@M1W{+^ ~4{(HjO>BQFD8{SOpƓ/Y +Nؠjœ(E6{/yċ81Kî}6G,_HħX __X.7hyA+?=?9K(}_pN0|\v&O?6NDd 0 #AB8~d~$x T8~d~$x"T2Q<xYoTYڤMn`S[^ BlcM#͚ƙBU4 1 4'7/ xq`c줛Mbw;gװ3ǫx'{aٓi3WšdzFő܌.iMûFsV2Iݞ8w7xt)-Yɥmҧrxtms*v"u!ѢQ&-[ƫOY}/ܛe2p*4䢘M]̕ It8;D\g&+HX(bJ"m b/QЗ d&#\VLe$Is>iͦ]+^\ :ESew>$z]"1TַTƜn6eoŒMve^qg($oF˥uQ-A1:ÞFVӕ4ɡeŒbA, Q̋E+&]nr@%KD4ȅ%Gm3Y9%ƄUJ (!e||M*c,=+ːAT1u']ys* 2yWJ6yq ]l{R$yI x=$N\VP!}ג%2RRi :|-rX?l=Sqfؚb ĵ߬!!6$VGssO\$dwe`C,b,Ba+K+zyLb{Y/ahZׯԠc}!G!jBk'~;[P{bJ2]@34ʼ=w":=>4E?>rzY{.GreUw3ϙDr(xق)hI[$k{$haPFϊ8F))fǂj]U ccTuzsQJW=ي%@ B=_H>"IIm;i_??JABC=DDEFGKmLNPRkTV4\]:^cWe9km)o2vwy{]|?~/В)?.ƫ/R0VrTa$x}t\nOmf1R'r~ #PRCE+,.{}"#$&35BTV*,0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000JJM(OVNzf(+[U8 I.wlJ!k}8 iUnknownBranko Grnbaum8GM!aa˷")*24C\eAINXZaszʹ˹07gkpźƺʺ !"%&ABJM_`cdnsûͻλѻֻ3:OYZ^ckpzʼ̼ռּSV4;msؿFPJP:?GO (*/SZjwx| !#'(-7=>IN[^fhimqT[[_msz~FPQY}BH^cdhiv{ %*23@ETU]`ijmnz{~&'34<?PQTUabefqruv#(56>AKT\~IRV\]fgoswx(+.046;=@BFTVnqsx#.4JN/6KUVZ_glv%.>~^fRRVwW^^``oa/bbb-f\f$i,iu'v߷EKZq˺̺Mͻ|l~ƽҿOZPR^whkjr?AluzFRLNsxcQqH}1>Nc8j:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Branko GrnbaumdMacintosh HD:Branko: Active:Noble polyhedra:Your polyhedra, my polyhedra:Your polyhedra... paper.3Branko Grnbaum<Macintosh HD:Temporary Items:AutoRecovery save of Your polyhBranko Grnbaum<Macintosh HD:Temporary Items:AutoRecovery save of Your polyhBranko GrnbaumMacintosh HD:Users:grunbaum:Desktop:Desktop Folder:Branko: Active:Noble polyhedra:Your polyhedra, my polyhedra:Your polyhedra... paper.3Branko Grnbaum<Macintosh HD:Temporary Items:AutoRecovery save of Your polyhBranko GrnbaumMacintosh HD:Users:grunbaum:Desktop:Desktop Folder:Branko: Active:Noble polyhedra:Your polyhedra, my polyhedra:Your polyhedra... paper.3Branko GrnbaumMacintosh HD:Users:grunbaum:Desktop:Desktop Folder:Branko: Active:Noble polyhedra:Your polyhedra, my polyhedra:Your polyhedra... paper.3Branko GrnbaummMacintosh HD:Users:grunbaum:Documents:Branko: Active:Your polyhedra, my polyhedra:Your polyhedra... paper.3@aaqcaa)nH*CJ GCJOJQJmH FGCJOJQJmH C$Eƀb'mHnHx*,.137393A3B3CJJ(Q(RXXaammmnmpmrmss't(t{ttttuuuuu2v2x ||=|>|}}()BC;; ƞ֨ߨ 357;BEno`h9\x}ĸƽǽɽ lmqrLOmnghz{LM ! 0@1"0H@020l@180|@1D0@0P0R0@120ض@1b0@180@180n@101@11@1@121@121l@121$@1@121@120@00$@121+@120,@120.@1205@121:@121=@120>@00`@141k@141s@0s@100v@120<@1̊@1Ҋ@140܌@1,41z@1p41@1v41@1414141@1414141@1.1@141$@141b@140ڛ@141@1>50@160~@160:@1@1@0@1@1 @161б@0ر@1ڱ@160@160>@160@160@160@160@160@160@160@160@1|@160@160@160@160@160@160<@1600@060@00@0.@1l.@180.@181.@181.@1.@180.@0.@0.@GTimes New Roman5Symbol3Arial9New York3Times"h$esF i$!j!+xxd0+Branko GrnbaumSummaryInformation(DocumentSummaryInformation8CompObjX0Tabletvxx2d+Branko Grnbaum i4@4NormalCJOJQJkH'mH <A@<Default Paragraph Font,@,Header !FC`FBody Text Indent$`a$6 FMicrosoft Word DocumentNB6WWord.Document.8 ՜.+,0`hpx 'j Title Oh+'0T (4<DL'ososososNormalfBranko Grnbaum9anMicrosoft Word 10.1@@/@!i$on9March182003 i4@4NormalCJOJQJkH'mH <A@<Default Paragraph Font,@,Header !6+6!zzzzzzzz z zzz zzzzzzzzzzzzzzzz3 jbjb^^^^^^^^:h<Kl0111114RQRQRQ8Q\QL@r$|R|R(RRRRRRHjJjJjJjJjJjJj,t ,vHvjr1RRRRRvjb11RRFR6bbbRT 1R1RHjb2NV3N1111RHjb\b4j114j2RP@RQ_4j4jrr4jtvbtv4jbBranko Grnbaum: Are your polyhedra the same as my polyhedra ? Dedicated to J. E. Goodman and R. Pollack, in admiration for their many-sided contributions to discrete geometry 1. Introduction. "Polyhedron" means different things to different people. There is very little in common between the meaning of the word in topology and in geometry. But even if we confine attention to geometry of the 3-dimensional Euclidean space as we shall do from now on "polyhedron" can mean either a solid (as in "Platonic solids", convex polyhedron, and other contexts), or a surface (such as the polyhedral models constructed from cardboard using "nets", which were introduced by Albrecht Drer [17] in 1525, or, in a more modern version, by Aleksandrov [1]), or the 1-dimensional complex consisting of points ("vertices") and line-segments ("edges") organized in a suitable way into polygons ("faces") subject to certain restrictions ("skeletal polyhedra", diagrams of which have been presented first by Luca Pacioli [44] in 1498 and attributed to Leonardo da Vinci). The last alternative is the least usual one but it is close to what seems to be the most useful approach to the theory of general polyhedra. Indeed, it does not restrict faces to be planar, and it makes possible to retrieve the other characterizations in circumstances in which they reasonably apply: If the faces of a "surface" polyhedron are simple polygons, in most cases the polyhedron is unambiguously determined by the boundary circuits of the faces. And if the polyhedron itself is without selfintersections, then the "solid" can be found from the faces. These reasons, as well as some others, seem to warrant the choice of our approach. Before deciding on the particular choice of definition, the following facts which I often mention at the start of courses or lectures on polyhedra should be considered. The regular polyhedra were enumerated by the mathematicians of ancient Greece; an account of these five "Platonic solids" is the final topic of Euclid's "Elements" [18]. Although this list was considered to be complete, two millennia later Kepler [38] found two additional regular polyhedra, and in the early 1800's Poinsot [45] found these two as well as two more; Cauchy [7] soon proved that there are no others. But in the 1920's Petrie and Coxeter found (see [8]) three new regular polyhedra, and proved the completeness of that enumeration. 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Dedicated to J. E. Goodman and R. Pollack, in admiration for their many-sided contributions to discrete geometry 1. Introduction. "Polyhedron" means different things to different people. There is very little in common between the meaning of the word in topology and in geometry. But even if we confine attention to geometry of the 3-dimensional Euclidean space as we shall do from now on "polyhedron" can mean either a solid (as in "Platonic solids", convex polyhedron, and other contexts), or a surface (such as the polyhedral models constructed from cardboard using "nets", which were introduced by Albrecht Drer [17] in 1525, or, in a more modern version, by Aleksandrov [1]), or the 1-dimensional complex consisting of points ("vertices") and line-segments ("edges") organized in a suitable way into polygons ("faces") subject to certain restrictions ("skeletal polyhedra", diagrams of which have been presented first by Luca Pacioli [44] in 1498 and attributed to Leonardo da Vinci). The last alternative is the least usual one but it is close to what seems to be the most useful approach to the theory of general polyhedra. Indeed, it does not restrict faces to be planar, and it makes possible to retrieve the other characterizations in circumstances in which they reasonably apply: If the faces of a "surface" polyhedron are simple polygons, in most cases the polyhedron is unambiguously determined by the boundary circuits of the faces. 0@1"0H@020l@180|@1D0@0P0R0@120ض@1b0@101@11@1@121@121l@121$@1@121@120@00$@121+@120,@120.@1205@121:@121=@120>@00`@141k@141s@0s@100v@120<@1̊@1Ҋ@140܌@1,41z@1p41@1v41@1414141@1414141@1.1@141$@141b@140ڛ@141@1>50@0|@00@0.@1l.@1H50.@121.@121.@1.@130.@0.@0.@GTimes New Roman5Symbol3Arial9New York3Times"Ah$eƍf&ki$!j!+xx2d+Branko Grnbaumkln 6JK} $]a$ $dh]a$ $dh]a$$a$$dha$ $dha$ $dha$UVWXYZ Z[\]^_`abcde=>EFHIUVop{|,,...0002 33H5L5N5666888mH jU jU jUCJF}~,.03N5666666668 $`a$$a$) 0 00P/ =!"#$%|HH@ Rc(hh@d-19May52002 ) 0 00P/ =!"#$%|HH@ Rc(hh@d-517May82002 16 ) 0 00P/ =!"#$%|HH@ Rc(hh@d-s ], ] and andis --,.-,avoidance9June172002 manyseveral isohedral constructions of such polyhedra the constructions in this case as welland results inthatin [49] available that is both general andalso 18 ) 0 00P/ =!"#$%|HH@ Rc(hh@d- 21