; TeX output 1998.04.09:14123ڍ&ffG Ս"V cmbx10POL Y.MP:TDEFINITIONOFREGULARCONVEXPOLYHEDRAINMETAPOST>ɉffGK`y cmr10versionUU1.0,8April1998DenisTRoQegel(mightbGeusefultoremindtheEulerformulalinkingthenumbGeroffaces b> cmmi10f,verticesvandedgeseinpGolyhedraUUwithnoholeindimension3:qDZfLo+8v !", cmsy10eUP=2.ThisUUcanbGeveri edinthefollowingtable:K㎍typGe'facesvertices&:edgesKEtetrahedron4u42v6KEcubGe6u8-v12KEoGctahedron8u6-v12KEdoGdecahedron12u20-v30KEicosahedron20u12-v30F*orUUmoredetails,seethedoGcumentationenclosedinthedistribution.message+32d23|=fhFcosO(2[=5)33Lʉfe0` 2sin6*2([=5)`5%3ڍ&defsetىffdo}'decahedronadffrvpoints|2(exprcinst*Ϋ)UP= newffp}'ointsDA4(fc;])(10);UU%facecenterssetىffp}'oint;Oq(fc;]qƮ1 б;0;0;0);msetىffobj^ff pp}'oint?Sc(1;1;0;0);setىffobj^ff pp}'ointM6(2;cosd(72);sindp(72);0);UU%72=360/5r}'otate]\ffnninadffrvplaneV}(3;fc uqƮ1~;1;2);r}'otate]\ffnninadffrvplaneV}(4;fc uqƮ1~;2;3);r}'otate]\ffnninadffrvplaneV}(5;fc uqƮ1~;3;4);c}'osan0[=UPcosd(72)=(28sind8(36)sind(36));msinan!=UPsqrt\n(1c}'osan;ɷ)t2);an"m=UP1808angle((c}'osan ;sinanK));newffabso"ff4fac}'e]\ffnnpointgi(fc;]qƮ2 б;fc uqƮ1~;1;2;anG3);newffabso"ff4fac}'e]\ffnnpointgi(fc;]qƮ3 б;fc uqƮ1~;2;3;anG3);newffabso"ff4fac}'e]\ffnnpointgi(fc;]qƮ4 б;fc uqƮ1~;3;4;anG3);newffabso"ff4fac}'e]\ffnnpointgi(fc;]qƮ5 б;fc uqƮ1~;4;5;anG3);newffabso"ff4fac}'e]\ffnnpointgi(fc;]qƮ6 б;fc uqƮ1~;5;1;anG3);r}'otate]\ffnninadffrvplaneV}(6;fc uqƮ2~;2;1);r}'otate]\ffnninadffrvplaneV}(7;fc uqƮ2~;1;6);r}'otate]\ffnninadffrvplaneV}(8;fc uqƮ2~;6;7);r}'otate]\ffnninadffrvplaneV}(9;fc uqƮ3~;2;8);r}'otate]\ffnninadffrvplaneV}(10;fc uqƮ3~;8;9);r}'otate]\ffnninadffrvplaneV}(11;fc uqƮ4~;3;10);r}'otate]\ffnninadffrvplaneV}(12;fc uqƮ4~;10;11);r}'otate]\ffnninadffrvplaneV}(13;fc uqƮ5~;4;12);r}'otate]\ffnninadffrvplaneV}(14;fc uqƮ5~;12;13);r}'otate]\ffnninadffrvplaneV}(15;fc uqƮ6~;5;14);newffabso"ff4fac}'e]\ffnnpointgi(fc;]qƮ7 б;fc uqƮ2~;6;7;anG3);newffabso"ff4fac}'e]\ffnnpointgi(fc;]qƮ8 б;fc uqƮ3~;8;9;anG3);newffabso"ff4fac}'e]\ffnnpointgi(fc;]qƮ9 б;fc uqƮ4~;10;11;anG3);newffabso"ff4fac}'e]\ffnnpointgi(fc;]qƮ10C;fc uqƮ5~;12;13;anG3);r}'otate]\ffnninadffrvplaneV}(16;fc uqƮ7~;6;15);r}'otate]\ffnninadffrvplaneV}(17;fc uqƮ7~;15;16);r}'otate]\ffnninadffrvplaneV}(18;fc uqƮ8~;7;17);r}'otate]\ffnninadffrvplaneV}(19;fc uqƮ9~;9;18);r}'otate]\ffnninadffrvplaneV}(20;fc uqƮ10{[;11;19);normalize]\ffnnobjQ4(inst k)(2;20);UU%oppGositeverticessetىffobj^ff pp}'ointM6(21;0;0;0);UU%centerofdoGdecahedronfr}'ee]\ffnnpointsB,(fc;])(10);enddef"O;`6+3ڍ&v\rardef#_defff̧do}'decahedrono (exprcinst*Ϋ)UP= sa9ve,./c}'osanD8;sinanK;anG3;newffobj^ff pp}'ointsV((inst k;21);UU%21pGointsinthede nitionnewffobj^ff pfac}'esQM(inst k;12);UU%12facesinthede nitionsetىffdo}'decahedronadffrvpoints|E(inst k);setىffobj^ff pfac}'eH)v(1;"5,4,3,2,1"<;"ff0fa1",);setىffobj^ff pfac}'eH)v(2;"8,7,6,1,2"<;"b40000",);setىffobj^ff pfac}'eH)v(3;"10,9,8,2,3"A;"b49b49",);setىffobj^ff pfac}'eH)v(4;"12,11,10,3,4"LF;"b49bc0",);setىffobj^ff pfac}'eH)v(5;"14,13,12,4,5"LF;"b4c8fe",);setىffobj^ff pfac}'eH)v(6;"6,15,14,5,1"G;"b4fefe",);setىffobj^ff pfac}'eH)v(7;"6,7,17,16,15"LF;"b4fe40",);setىffobj^ff pfac}'eH)v(8;"8,9,18,17,7"G;"45d040",);setىffobj^ff pfac}'eH)v(9;"10,11,19,18,9"Q;"45a114",);setىffobj^ff pfac}'eH)v(10;"11,12,13,20,19"V;"45a1d4",);setىffobj^ff pfac}'eH)v(11;"14,15,16,20,13"V;"4569d4",);setىffobj^ff pfac}'eH)v(12;"16,17,18,19,20"V;"112da1",);enddef"O;`7=3ڍ&ffGV'ҍICOSAHEDRONUU(20faces)s4PSfile=icosa.ps llx=-45 lly=-9 urx=45 ury=74 rwi=900ɉffG+TwoUUfacesofanicosahedronarelinkedinthefollowingway:+퍍%5PSfile=vect-fig.4 llx=-1 lly=-3 urx=85 ury=60 rwi=860YwhereUUhUP=cPpUWPfeE3 UX=2]ȍdUUisalsoadiagonalinapGentagon:4q8PSfile=vect-fig.5 llx=-47 lly=-55 urx=57 ury=55 rwi=1040C4cUP=2r5sinUS([=5),UUd=2r5sin(2[=5)=2ccos*(=5)(̍TheUUanglebGetweenUUtwofacesiscomputedasfollows:̍q8PSfile=vect-fig.6 llx=-1 lly=-17 urx=100 ury=39 rwi=1010+ cmmi10ٓRcmr7K`y cmr10\