dim 150 90 animation_frames 70 8 point O 0 0 point Oa 0 0 360 0 distance d O Oa %S is the center of the pentagon point S 77 115 getx Sx S gety Sy S %x is the trace of the plane in which the pentago is point X1 0 45 point X2 1 45 line x X1 X2 perp s S x %S' is the normal projection point S' Sx 22 perp s' S' s intersec Sx s x circle ks Sx S intersec2 S"' S" ks s' line i Sx S" %pentagon A1A2A3A4A5 point A1 50 150 rotate A1 S d A1 perp a1 A1 x intersec A1x a1 x circle ka1 A1x A1 parallel iA1 A1x i intersec2 A1"' A1" ka1 iA1 perp a1' A1" s intersec A1' a1' a1 %cmark A1' rotate A2 S 72 A1 perp a2 A2 x intersec A2x a2 x circle ka2 A2x A2 parallel iA2 A2x i intersec2 A2"' A2" ka2 iA2 perp a2' A2" s intersec A2' a2' a2 %cmark A2' rotate A3 S 72 A2 perp a3 A3 x intersec A3x a3 x circle ka3 A3x A3 parallel iA3 A3x i intersec2 A3"' A3" ka3 iA3 perp a3' A3" s intersec A3' a3' a3 %cmark A3' rotate A4 S 72 A3 perp a4 A4 x intersec A4x a4 x circle ka4 A4x A4 parallel iA4 A4x i intersec2 A4"' A4" ka4 iA4 perp a4' A4" s intersec A4' a4' a4 %cmark A4' rotate A5 S 72 A4 perp a5 A5 x intersec A5x a5 x circle ka5 A5x A5 parallel iA5 A5x i intersec2 A5"' A5" ka5 iA5 perp a5' A5" s intersec A5' a5' a5 %cmark A5' %center S1' of the counterbase midpoint A12 A1 A2 distance R S A1 distance r S A12 distance a A1 A2 expression h {sqrt((a*a)-((a*a)/4))} expression H {sqrt((h*h)-(R-r)*(R-r))} perp i" S" i turtle S1" Sx S" 90 H perp i1" S1" s intersec S1' i1" s %translated base translate B1' S' S1' A1' translate B2' S' S1' A2' translate B3' S' S1' A3' translate B4' S' S1' A4' translate B5' S' S1' A5' %rotated base towards C1 A1 S 2 towards C2 A2 S 2 towards C3 A3 S 2 towards C4 A4 S 2 towards C5 A5 S 2 perp c1 C1 x intersec C1x c1 x circle kc1 C1x C1 parallel iC1 C1x i intersec2 C1"' C1" kc1 iC1 perp c1' C1" s intersec C1' c1' c1 %cmark C1' perp c2 C2 x intersec C2x c2 x circle kc2 C2x C2 parallel iC2 C2x i intersec2 C2"' C2" kc2 iC2 perp c2' C2" s intersec C2' c2' c2 %cmark C2' perp c3 C3 x intersec C3x c3 x circle kc3 C3x C3 parallel iC3 C3x i intersec2 C3"' C3" kc3 iC3 perp c3' C3" s intersec C3' c3' c3 %cmark C3' perp c4 C4 x intersec C4x c4 x circle kc4 C4x C4 parallel iC4 C4x i intersec2 C4"' C4" kc4 iC4 perp c4' C4" s intersec C4' c4' c4 %cmark C4' perp c5 C5 x intersec C5x c5 x circle kc5 C5x C5 parallel iC5 C5x i intersec2 C5"' C5" kc5 iC5 perp c5' C5" s intersec C5' c5' c5 %cmark C5' %upper pyramid expression H1 {sqrt((a*a)-(R*R))} expression H1+ {H1+H} turtle V1" Sx S" 90 H1+ perp ii1" V1" s intersec V1' ii1" s %lower pyramid turtle V2" Sx S" -90 H1 perp iii1" V2" s intersec V2' iii1" s %***************************** %***************************** %dual number thickness 1.2 midpoint C12' C1' C2' midpoint C23' C2' C3' midpoint C34' C3' C4' midpoint C45' C4' C5' midpoint C51' C5' C1' towards D1 V2' C12' 0.666666 towards D2 V2' C23' 0.666666 towards D3 V2' C34' 0.666666 towards D4 V2' C45' 0.666666 towards D5 V2' C51' 0.666666 linethickness thickness drawsegment D1 D2 drawsegment D2 D3 drawsegment D3 D4 normal linethickness thickness drawsegment D4 D5 drawsegment D5 D1 cmark D1 cmark D2 cmark D3 cmark D4 cmark D5 %************************** midpoint B12' B1' B2' midpoint B23' B2' B3' midpoint B34' B3' B4' midpoint B45' B4' B5' midpoint B51' B5' B1' towards D6 V1' B12' 0.666666 towards D7 V1' B23' 0.666666 towards D8 V1' B34' 0.666666 towards D9 V1' B45' 0.666666 towards D10 V1' B51' 0.666666 linethickness thickness drawsegment D6 D7 drawsegment D7 D8 drawsegment D8 D9 drawsegment D9 D10 drawsegment D10 D6 normal cmark D6 cmark D7 cmark D8 cmark D9 cmark D10 %************************ towards D11 B4' C12' 0.666666 towards D12 B5' C23' 0.666666 towards D13 B1' C34' 0.666666 towards D14 B2' C45' 0.666666 towards D15 B3' C51' 0.666666 cmark D11 cmark D12 cmark D13 cmark D14 cmark D15 %************************ towards D16 C4' B12' 0.666666 towards D17 C5' B23' 0.666666 towards D18 C1' B34' 0.666666 towards D19 C2' B45' 0.666666 towards D20 C3' B51' 0.666666 cmark D16 cmark D17 cmark D18 cmark D19 cmark D20 %************************** linethickness thickness drawsegment D1 D11 drawsegment D2 D12 drawsegment D3 D13 drawsegment D4 D14 normal linethickness thickness drawsegment D5 D15 linethickness thickness drawsegment D6 D16 normal linethickness thickness drawsegment D7 D17 drawsegment D8 D18 linethickness thickness drawsegment D9 D19 drawsegment D10 D20 drawsegment D11 D19 drawsegment D12 D19 drawsegment D12 D20 drawsegment D13 D20 drawsegment D13 D16 drawsegment D14 D16 normal linethickness thickness drawsegment D14 D17 drawsegment D15 D17 drawsegment D15 D18 drawsegment D11 D18 %the cube in the dodecahedron linethickness 0.5 drawdashsegment D6 D8 drawdashsegment D19 D20 drawdashsegment D6 D20 drawdashsegment D8 D19 drawdashsegment D1 D3 drawdashsegment D3 D14 normal drawdashsegment D14 D15 drawdashsegment D1 D15 linethickness 0.5 drawdashsegment D1 D19 drawdashsegment D3 D20 drawdashsegment D14 D6 normal drawdashsegment D15 D8 normal %Euklid's construction XIII.17 midpoint P D6 D19 midpoint Q D1 D20 midpoint H D19 D20 midpoint N D6 D20 midpoint O D8 D19 towards R P N 0.618033 towards S P O 0.618033 towards T Q H 0.618033 printat_l D20 {B} printat_r D19 {C} printat_lt D6 {E} printat_rt D8 {F} cmark_rt P cmark_lb Q cmark_t H cmark_lb N cmark_rt O cmark_lt R cmark_lt S cmark_l T printat_t D10 {U} printat_rt D9 {V} printat_rb D12 {W} drawsegment N O drawsegment H Q drawsegment R D10 drawsegment S D9 drawsegment T D12