56 - izumi-math.jp

1 ()562006128()36I23(A)414!17191030022()23 136I 1 2 3 (1)(2)(3)2332() EXERCISES IN PAPER FOLDINGS undara RowDOVER SCIENCE & ARTK oryoMiura(Ed.) 19947. Hull(Ed.) Geretschl ager() Geometric OrigamiRona GurkewitzDOVER OrigamiDavid MitchellBurlington Press Origami PolyhedraLewis SimonDOVER Plethora of Polyhedra in OrigamiJohn MontrollDOVER 3(A) CAB30 ABCD ABC40 CDA(1)(180 BCA = 180 (30 +ABCDE||||40 ) = 110 , DCA = 30 + 40 = 70 AB=CDABCDCEABCABCDAC=DE, ACE = 70 + 40 = 110 = DECACEDAD // CE CDA = DCE= ABC = 40 (2)BDBA=BEE ABDABCDE||40 30 qqABDE4 ABE BAE = BEA = (180 40 ))

2 2 = 70 AB=CDBE=CDBC=ED 1 ACE4 ACB ACE = 30 + 40 =70 4 ACEAC=AE 2 ACB = AED 3 = 110 1 ,2 ,3 4 ACB4 AED CDA = EDA = CBA = 40 (3)4 ABCACACEABCDEF||||||40 40 30 30 4 ABCAC ACE = ACB = 180 (30 +40 ) = 110 , DCA =30 + 40 = 70 FCE = 110 70 = 40 = FEC4 FCEFC=FE 1 AE=AB=CD 2 1 ,2 FA=FDFAD AFD = EFC CDA = FAD = (40 + 40 ) 2 = 40 12a2aa215a225a2a10a2110a2310a2aaa2 15a2 4 =15a2a2 110a2 6 =25a253.(1) ABE = 15 , CBF =ABCDEF15 30 x ABCDEFR15 30 x 15 30 BEF(2)ABCDPQABCDPQR PADDQ=AP BP(3)4 ABCAB=AC, A =ABCD100 ABCDE100 60 40 100 ACAD=BC CBD6(4)ABCABCD CAB = ABC = 20 CDAABCDABCDE||||(5)ABCABCD CAB = 80 , BCA = 20 DACABCDABCDE||||||||||||80 74.

3 (1)|||||| ABCDE(1) BAC = DAC, BCA = DCA(2)AC BD, BE=DE(3)AB=AD, CB=CD(2) z= x+ yABCDzxyCDzxyyxyAB(3)ABCDDABC15 30 DABC15 30 |||| ABD= BAC= ACBABDC(4) ADBDABC36 48 12 24 DABC36 48 12 24 ||||||||||AB4 ABD8(5)ABCBDDC ACD = 15 , DAC = 30 CBAAD4 ADCABCDABCDE|||||||||||15 30 (6) ADB4 BDCABEDDABC30 20 40 70 DABC30 20 40 70 E4 ABC4 BECEB=ECDABC30 20 40 70 O|||DABC30 20 40 70 EFG9(7) ADB4 BDCAECDDABC20 20 50 80 DABC20 20 50 80 E(8) CAD4 ABCABFDDABC60 80 70 DABC60 80 70 EF(9) ADB4 ABCB NCD,B AODABCD10 10 40 110 x ABCD10 10 40 110 x B ()1(12)(12)2 OOO||(1)()(2)3O||||||O||||||11124pattern 7pattern13pattern23pattern() (1)4 PBC,4 OADABCD(2) ABE = EBP = AOB = BOG = APE = 15 (3) CBE = PEB = BEA = BAP = 75 (4)AB=PB=OP=OA2 CDABP PBC = PCB = 15 PAPDP APD(xy150 ||||||()15 30 xABCD||||()ABCD|||||||xABCD|||||||xABCD ACB =?)

4 |||30 ABCDE||||15 x(AD // BCCD=CE||||ABCDE(45 30 x 131922 Edward 724300300 ABCD 4 3 5 6 8 1 2 7 1+ 2+ 3+ 8= 180 2+ 3+ 4+ 5= 180 6+ 7= 2+ 3sin 1sin 3sin 5sin 7= sin 2sin 4sin 6sin 8180 nn= 8n= ,20 ,50 ,80 =(ECAB= 50 50 20 80 )ABCD10 20 50 80 60 40 60 30 GHEFEDC BAE = 80 FDC BAF = 60 BD AE, BC AFGBCAFHBDAE4 AGC4 FCGAG=FG4 ABG 4 FBG FBG = 30 ABF + AEF = 180 ABFE FBE = FAE = 20 EBC = 10 EBH = 30 EDH = 30 4 EBH 4 EDHBH=HD4 ABH 4 ADH ADB = 10 (LangleyWEBABCDP ABCDP(6=A, B, C, D) PAB = , PBC = , PCD = , PDA = , PBA = , PCB = , PDC = , PAD = ,sin sin sin sin = sin sin sin sin ABCD bacdxABCD ABD =a, CBD =b, ACB =c, ACD =d, ADB =xsinxsin(a+b+c) sinbsind+sin(b+c+d) sin(x b c) sinasinc= 02sin 3 = sin (2 cos 2 + 1)cos 3 = cos (2 cos 2 1)sin 5 = sin {2(cos 4 + cos 2 ) + 1}cos 5 = cos {2(cos 4 cos 2 ) + 1}= 10 ,30 ,30 ,20 ABCD30 10 30 20 x sinx sin 70 sin 30 sin 20 +sin 80 sin(x 60 ) sin 10 sin 30 = 0sinx cos 20 sin 30 sin 20 + cos 10 sin(x 60 ))))

5 Sin 10 sin 30 = 0 sin 30 sinx cos 20 sin 20 + cos 10 sin(x 60 ) sin 10 = 0 2,2 sinx cos 20 sin 20 + sin(x 60 ) sin 20 = 0 sin 20 2 sinx cos 20 + sin(x 60 ) = 0sin(x+ 20 ) + sin(x 20 ) + sin(x 60 ) = 015{sin(x+ 20 ) + sin(x 60 )}+ sin(x 20 ) = 02 sin(2x 40 )/2 cos(80 /2) + sin(x 20 ) = 0(2 cos 40 + 1) sin(x 20 ) = 0 (2 cos 40 + 1)sin(x 20 ) = 00 < x <180 x= 20 XAY = , YAB = , YBX = , XBA = = YXBtan =cos( + 2 ) sin cos( + 2 ) sin sin( )sin( + 2 ) sin + sin( + 2 ) sin = 10 ,30 ,30 ,20 = 20 , = 30 , = 30 , = 10 = cos(10 + 2 30 ) sin 20 cos(20 + 2 30 ) sin 10 sin 10 = cos 70 sin 20 sin 80 sin 10 sin 10 = sin220 sin210 sin 10 =1 cos 40 2 1 cos 20 2 sin 10 = 12(cos 40 cos 20 ) sin 10 = 12( 2 sin 30 sin 10 ) sin 10 = sin 30 sin 10 sin 10 = 12sin 10 = sin(10 + 2 30 ) sin 20 sin(20 + 2 30 ) sin 10 = sin 70 sin 20 + sin 80 sin 10 = cos 20 sin 20 + cos 10 sin 10 =12sin 40 +12sin 20 =12(sin 40 + sin 20 )

6 =12 2 sin 30 cos 10 =12cos 10 tan = sin 10 cos 10 = tan( 10 ) = 10 ,x= 10 + 30 = 20 16!1.(1)|||||| ABCB E(2)4 OABOB |||||| ABA O(3) AOB |||||| ABB O(4)OB ||||||ABO2.()10300(1)(2)12345()63624? ? ??? ? ??? ? ??? ? ??? ? ??? ? ?? (a)ABCFDE (b)ABCFDE(c)ABC CAF()2 CFCF[(a)](b),(c)CDFEFE // BC, FD // AC4 AFE4 FBD4 ABCFE=FDAF:FB=FE:BD=FD:BD=AC:BC(c)AC6=BC 18(1)AB=AC, BD=BC, AD // BCCD=CE()()4 ABDAB4BD CB4 CDD ||||ABCDE||||ABCDED ||()4 BDCBC4 DBD ||||ABCDED |||||||||||ABCDE()||||ABCDE||||ABCDEF19( 2) ADB4 BDC4 BCC C 4 ABCABCD30 40 80 20 x ABCD30 40 80 20 x C ||||||||||||||||(3) ADB4 ABDA 4 EBC4 DBA 4 CBA ABCD20 10 40 40 x ABCD20 10 40 40 x A E(4) ADB4 ABCACBNDC.

7 N4AB DABCD60 40 30 70 x ABCD60 40 30 70 x B NM20(5) ADBBCE4 BCDEFCD4 CDE4 ACE 4 ACDABCD30 10 70 30 x ABCD30 10 70 30 x EF (6) ADB4 BCDBDEBCA10 ,30 ,70 ,30 AECDABCD20 10 100 20 x ABCD30 10 70 30 30 x E(7) ACD = 2 ABDBC4 ACDB4 ACD36(300)ABCD 2 2110300) 1 2 3 4 1 2 3 4 1 2 3 4110102080101012010130301002013050606010 2101030402010220202080102023050704040310 1030110101032020305020203305080403041010 4030301042020301001020430702050105101040 7020105202040403020530703040206101040110 1010620204070202063070505010710105050301 0720204010010207307060402081010508020108 2020606030208308030501091010604040109202 0805040209308040402010101060603011020201 0030802103010020401011101060802011120201 0040502113010030401012101010030701122020 1005020212401030110201310101004050113202 0110308021340105080401410101005030114202 0110403021440105010030151010110307011520 2012030602154010501102016101011040401162 0302060102164010806070171010120201001172 0303040202174010807060181010120306011820 3030801021840101006070191010120402011920 3050304021940202011010201010130201001202 0305060202204020407030211010130303012120 3050801022140204011010221010140207012220 3070404022240205080302310202040101232030 8030602234020607040241020307010124203080 4040224402060803025102040203012520308060 1022540208050702610204040201262030100306 0226402080703027102040801012720301004020 2274020100507028102050303012820301103040 2284030307020291020505020129204020501022 9403030100103010205080101302040403030230 4030406030311020703040131204040701023140 3040802032102070403013220406040302324030 5080203310208020601332040605020233403070 6040341020804030134204070403023440307070 2035102080502013520408030502354030805060 3610201002070136204080501023640308060303 7102010030401372040100303023740402080103 8102010040201382050406010238404040702039 1020110207013920507030402394040606030401 0201103030140205070402024040408050404110 2011040101412060204010241406030701042102 0120206014220603050102424060505030431020 1302030143206050303024340702060104410302 0301014420605050102444070305020451030302 0201452060703030245407040601046103030501 0146206070401024640705050204710304030201 4720608030202475010401003048103040601014 8208030401024850106080504910306020401492 0804030202495010601003050103060402015020 8040401025050204080305110307030301512080 5030202515020401002052103070402015220100 2030102525020708030531030802050153201004 0301025350302010010541030803030154301020 130102545030308020551030805010155301030l 0020255503040703056103010205015630103013 0102565030401001057103010030201573010408 0302575030608020581030110204015830104011 0202585030707030591030110301015930106060 5025950308060506010301202020160301060804 0260505020801061104030401016130106010020 2615050606030621040502030162301070705026 2601030120206310405030201633010708040263 6010401202064104080204016430108050702646 0107080606510408030201653010807050265602 0201201066104080401016630101005080266602 0301002067104010020301673010100605026760 2030201068105060203016830101104010026860 205080406910506040101693010110507026960