正多面体を角材や板材で作る Polyhedron made of …

1 1 Polyhedron made of Timber or Plate --- Collaboration of Architecture and Mathematics --- 2016/11/06 Kimihiro Morishita Math. KanazawaMunicipalTechnicalHighschool 2018/06/28 ( ) ( Marking ) NHK Made of Timber 4 3 Regular Tetrahedron & Scalene Regular Triangular Pyramid 6 Regular Hexahedron 8 4 Regular Octahedron & Equilateral Regular Quadrangular Pyramid 12 Regular Dodecahedron 20 Regular Icosahedron Equilateral Fullerene Conclusion Made of Plate 4 3 Regular Tetrahedron & Scalene Regular Triangular Pyramid 6 Regular Hexahedron 8 4 Regular Octahedron & Equilateral Regular Quadrangular Pyramid 12 Regular Dodecahedron 20 Regular Icosahedron Equilateral Fullerene Conclusion 2 Made of Timber 4 Marking of Regular Tetrahedron L 1 M

2 4 ABCD CD E ABE A BE F AF 3 BCD F 3 BCD 3 BCD 3 F 3 BCD = BAF ( Vertical Joint Angle ) LAB= LBEAE23== LBEBF3132= = 31sin= 32cos= 21tan= A 3 A = 12031 AGG 31 GGAB = 6021 AGG G1AG2 ( Horizontal Joint Angle ) MGG231= MGG2221= 360tan221= =AGGG MGGAG613212== Fig 1a A B C D Fig 1b A B C D E Fig 2a A B E Fig 2b A B E F 3 ( Joint Depth ) 4 12 3 1 tan2Mt= 2b 3 A 4 A1 3 G1 G2 G3 4 G 3 A 4 A4 A5 A6 A1A3 MAA231= 2b BA1A6 36161 AAAABA == 633121tanAAAA== MAAAA231263==.

3 MAA=42 3 AG2 4 A5G MGA615= Fig 4 A1 B A2 A3 G A4 A5 A6 Fi g 3 B A G1 G2 G3 4 GAAA561 45421 AGAAAA MAA661= 263161214154 ===AAAAAAAAGAGA MGAGA21534== +=+=.. (Under Length) (Side Length) A1G GA6 G A1 A6 6 L=77cm M= 5 26 3 1 L 3 kL kL L 2 3 5a kLADACAB=== LDBCDBC=== CD E A AB,AC,AD BC,CD,DB ( )2144122222 = = =kLLkLkLAE yLAF= xLBF= =+ =+ 41423222222kyxkyx 414232222 = kkxx 413432222+ = + kkxxx 13=x 31=x F 3 ABC 31sinkkLxL== 2231sink= 222313coskk = 131tan22 =k 131tan2 =k A t 4 =60 6tan2 MMt== A3A6 632131tan2 AAMk= = MkAA132632 = A2G MkkMkktAAGA21363213sin263222+ = + =+= Fig 5a A B C D E Fig 6a A B E F 6 A B B 1 L 3 9 LCB=11 = 30321 BBB MBB=31 MBB332= MLCB3222+= 10a Fig 7 A1 B A2 A3 G A4 A5 A6 Fig 9 B1 B2 B3 C1 C2 7 6b kLAB= LBE23= LkAE2142 =

4 Kkkk3123241443cos22= += 2231cosk= 222313sinkk = 13tan22 =k 13tan2 =k B B 10b MHB26= MBB=41 = = = 4871451 BBBBBBHAB MkkMBB313sin152 == MkMBB31cos45== 61213tan2 BBMk= = MkBB132612 = MkBBMBB = =31254274 877413tan2 BBBBk= = Fig 10a B A G Fig 6b A B E 8 MkkkkBBBB1331613748722 = = I B1GH 11 HBM12sin= MkkMHB136sin212 == Fig 11 B G G H I O 30 J B8 Fig 10b B1 A G B4 I H B5 B6 B7 B8 B9 B10 N 9 10b B1H 11 BH B 10b B1 B4 MkkBH1362 = MkkBHOH132322 == 22 MGGOG== kkOHOG313tan2 == 22222cossin313tan= =kk 1613sin222 =kk 163cos222 =kk 163cos2 =kk 3130tan= =BJJI kkHJJI313tan2 == HJkkBJJI31332 == HJkkBJ132 = MkkBHHJkkkHJkkHJBJ13613113222 ==+ = + =+ ()MkkkkHJ+ =13136222 163cos2 ==kkHIHJ ()MkkkkkHJkkHI+ = =13131623162222 B4B8 B9B10 749487109 BBBBBBBB= ()()MkkkkkMkkkkkMkkkBBBBBBBB133126133161 6126133163123122748794109222 = = = = B 10b I IN B1B4 B4B8 11 I N A4A8 B4B8 30 B1B4 B1B4 B4B8 B4B8 30 N IN 10 2=k = +=+ = B = = = = = = ()()

5 +=+ = k=2 L=46cm M=40mm 11 12 B1B4 B1B4 B4B8 B4B8 30 N IN IN B1B4 k=2 L= M=105mm 13 14 2016 15 6 ( ) Marking of Regular Hexahedron 6 4 3 ABCD 4 BAC 60 BAC CAD BAD 90 3 BCD LADACAB=== LDBCDBC2=== CD E 2 3 3 ABE A BE F 4 F 3 BCD = BAF LAB= LAE22= LBE26= LBEBF3632= = 3236sin== 31cos= 2tan= 4 A 3 4 MGGAG613212== 2b 4 3 A 4 A1 3 G1 G2 G3 4 G 3 A 4 A4 A5 A6 A1A3 MAA231= MAA2221= Fig 1b A B C D E Fig 2a A B E Fig 2b A B E F Fig 1a A B C D 16 2b BA1A6 36161 AAAABA == 63312tanAAAA== MAAAA== 242 MAA= 3 AG2 4 A5G MGA615= GAAA561 45421 AGAAAA MAA361= 233161214154 ===AAAAAAAAGAGA

6 MGAGA215234== MMGA= += 12 Fig 4 A1 B A2 A3 G A4 A5 A6 Fig3 B A G1 G2 G3 17 18 8 Marking of Regular Octahedron L 3 8 4 3 ABC LACAB== LBC2= 3 ABC 2 3 A BC F CAF 45 1tan= 4 A 3 M MGG231= 22232 MMGG== 3 AG2G3 2 3 22 MAG= 4 == = 4536161 AAAABA MAAAA26331==.. 3 A1A2A4 A4A5G 2 3 A5G 3 AG2 Fig 1b A B C Fig 2a A B C Fig 2b A B C F Fig 1a 19 MAA2242= MAAGA22455== MGA=4 +=.. A1G A6G 12 8 Fig 4 A1 B A2 A3 G A4 A5 A6 Fig 3 A G1 G2 G3 20 12 6 4 8 4 4 8 0 MAAAA26331==.

7 MGA222=.. G A1 A6 Fig 4 A1 A2 A3 G A6 21 4 22 12 Marking of Regular Dodecahedron 12 5 12 1 5 3 3 4 6 5 1 1 5 1 2 3 2 3 3 5 x 3 72 2 2 2 3 )1(:11: =xx 1)1(= xx 012= xx 251 =x 0>x 251+=x 4151512152cos72cos =+= == x 8531652652cos2 = = 85552sin2+= 5254)53()55(535552tan2+=++= += 52552tan+= 2 3 415125cos36cos+= == x 853165265cos2+=+= 8555sin2 = 5254)53()55(53555tan2 = =+ = 5255tan = 12 72 36 36 36 108 23 4 6 F 3 BCD LADACAB=== LDBCDBC215+=== LDECE415+== LBCBE431523+== LBEBF631532+= = 32156315sin+=+= 65312526sin2+=+= 653cos2 = ()4535353tan22+= += 253tan+= A 4 6 3 M MAG612= 2b 4 3 A A4 A5 2b BA1A6 36161 AAAABA == 6331253tanAAAA=+= () = =+=+=.

8 Fig 1b A C D B E Fig 2a A B E Fig 2b A B E F Fig 1a A B C D 24 MMAAAA410232253632142 = == 3 AG2 4 A5G MGA615= 45421 AGAAAA MMAA539253761 =+ = 25393161214154 ===AAAAAAAAGAGA MMMMMGAGA421022152252625312539525394 = = = = = = == + =.. 30 Fig3 B A G1 G2 G3 Fig 4 A1 B A2 A3 G A4 A5 A6 25 20 Marking of Regular Icosahedron 3 20 3 5 5 CD E 3 ABE LCDADACAB==== LDBBC215+== LDECE21== LAE23= LLLBE2525415262121522+= += += A BE F 3 BCD 2 3 Fig 1a A B C D Fig 1b A B C D E Fig 2a A B E Fig 2b A B E F 26 F xLAF= yLBF= 122=+yx 222232525 = ++yx 4352545251=+ ++y 2534526452541525+=+=++=+y 525253++=y ()()()()10551030515514352025252553752545 6142+= + = +=++=y 1055+=y 1055122 = =yx BAF ()

9 41515155555tan2222+= += +==xy 215tan+= A A 5 527231 = = AGG 31 GGAB 521 = AGG MGG231= MGG2221= 5255tan221 == AGGG MMMGGAG105252025252552522525212+= += = = Fig 3 A G1 G2 G3 27 2b 3 A A4 A5 2b BA1A6 36161 AAAABA == 6331215tanAAAA=+= () = = =+=+=.. MAAAA4210632142 == 3 AG2 4 A5G MGA105255+= 45421 AGAAAA MMAA552526261 = += 2553161214154 ===AAAAAAAAGAGA MMMMMMMGAGA42102215225262535255152010555 10251025255552554+=+=+=+=+= +=+ = = = ++ =+=.. 30 C60 Fig 4 A1 B A2 A3 G A4 A5 A6 28 Marking of Fullerene 20 5 12 6 20 60 5 1 6 2 2 2 6 P 30 6 5 Q 60 90 3 ABC 3 ABD 6 3 ACD 5 AB P AD AC Q LADACAB=== LDBBC3== LCD215+= LDECE415+== LLCEACAE4521016526122 =+ = = Fig 1a A C D B E Fig 2a A B E F B A D C 29 LLLCEBCBE225214524216526322 = =+ = = AF BE A 3 BCD 5 5 6 6 3 BE 2a P( AB) F 3 BCD 20 xLAF= yLBF= 122=+yx 2224521045242 = +yx 165210252421652421 = +y 38558521125242= += y 52426 =y ()

10 2185918943652118521185242362+=+= = =y 2185929122 = =yx BAF ()()()()()()2222215345269109452186549109 4455295921609943659295219592959189tan +=+= += ++ +=++= +==xy ()2153tan+= P () + Q 1a 1b 2b Q DAF F 3 BCD BF=DF 3 ADF 3 ABF = P Q 3 B C D Fig 1b A C D B E F 30 LADACAB52123 === LDBBC3== LCD215+= P G2AG3(= 1 ) BAD 2 1 Q 6 P 5 CAD 2 1 CAI1(= 2 ) 3 ABD ACD 2 3 3 3 A BD J Fig 2b A D F Fig3 B A G1=H1 G2 G3 C D H2 I1 I2 6 6 5 P Q Q 31 ()()109456173109451851109410934365211843 521182352118222 += += += = = =BJABAJ ()()561710956171095617109561731094231tan = =+=+ ==AJBJ P Q 6 t1 ()MMMt10925617561721tan21 += == A CD K ()()109857342910981095343652118165265211 8415524236222 = + +=+ = + = =CKACAK 57342910921557342910984152tan += +==AKCK Q 5 t2 ()109573429152tan22 +==MMt P Q 6 36161 AAAABA == Fig 4a A1 B A2 A3 G A4 A5 A6 32 ()63312153tanAAAA=+= ()()() = = =+=+=.