Mircea V. Diudea
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1 1 Platonics & Archimedeans Platonics & Archimedeans Mircea Mircea V. Diudea V. Diudea Faculty of Chemistry and Chemical Engineering Faculty of Chemistry and Chemical Engineering Babes Babes - - Bolyai Bolyai University University 400084 400084 Cluj Cluj , ROMANIA , ROMANIA [email protected] [email protected]
Transcript of Mircea V. Diudea
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Platonics & ArchimedeansPlatonics & Archimedeans
MirceaMircea V. DiudeaV. Diudea
Faculty of Chemistry and Chemical EngineeringFaculty of Chemistry and Chemical EngineeringBabesBabes--BolyaiBolyai UniversityUniversity400084400084 ClujCluj, ROMANIA, ROMANIA
[email protected]@chem.ubbcluj.ro
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ContentsContents
1.1. Platonic OjbectsPlatonic Ojbects
2.2. ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects1,21,2
1. Catalan objects (i.e., duals of the Archimedean solids).2. B. de La Vaissière, P. W. Fowler, and M. Deza, J. Chem. Inf. Comput. Sci.,
2001, 41, 376-386.
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EulerEuler Theorem on PolyhedraTheorem on Polyhedra
v v –– e + fe + f = = χχ = = 22((11 –– gg)) (1)
χχ = Euler= Euler’’s s characteristiccharacteristicv = v = number of vertices, number of vertices, e = e = number of edges,number of edges,f = f = number of faces,number of faces,gg = genus ; = genus ; ((gg = 0 for a sphere; 1 for a = 0 for a sphere; 1 for a torustorus))..
A A consequenceconsequence::A sphereA sphere can not be tessellated only by hexagons.can not be tessellated only by hexagons.
Fullerenes need Fullerenes need 12 pentagons12 pentagons (for(for closingclosing the the cage) and cage) and (N/2(N/2--10) hexagons10) hexagons..In the opposite, a tube In the opposite, a tube andand a a torustorus allow pure hexagonal nets.allow pure hexagonal nets.
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By substituting By substituting vv, , ee and and ff (as below) in the Euler relation one obtains:(as below) in the Euler relation one obtains:
(2) (2)
(3) (3)
(4) (4)
••For For a givena given genusgenus of the surface, (of the surface, (44) gives the number of ) gives the number of ss--polygons. polygons. This condition is This condition is independent of the number of hexagonsindependent of the number of hexagons, which is , which is therefore therefore arbitraryarbitrary..
••The The Platonic tilingsPlatonic tilings consist of consist of a single kind of polygons.a single kind of polygons.••In In Platonic polyhedraPlatonic polyhedra ((gg = 0): from (4), = 0): from (4), ff55 =12, =12, or or ff44 == 6 or 6 or ff33 == 4.4.
••TheThe Archimedean tilingsArchimedean tilings show show two different kinds of polygonstwo different kinds of polygons. .
∑= s sff
)1(12)6( gfs ss −=−∑
Fullerene countingFullerene counting
efsv s s 23 =⋅= ∑
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••Platonic SolidsPlatonic Solids
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DuDu((I I ) = ) = DuDu((SnSn((T T ))))DodecahedronDodecahedronDD55SnSn((T T ))IcosahedronIcosahedronII44DuDu((OO))=Du(Me=Du(Me((T T ))))Cube (hexahedron)Cube (hexahedron)CC33MeMe((T T ))OctahedronOctahedronOO22--TetrahedronTetrahedronTT11
FormulaFormulaPolyhedronPolyhedronSymbolSymbol
Platonic PolyhedraPlatonic Polyhedra(derived from Tetrahedron)(derived from Tetrahedron)
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Schlegel ProjectionSchlegel ProjectionT etrahedronT etrahedron
Platonic SolidsPlatonic Solids
1. Catalan objects (i.e., duals of the Archimedean solids).2. B. de La Vaissière, P. W. Fowler, and M. Deza, J. Chem. Inf. Comput. Sci.,
2001, 41, 376-386.
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Schlegel ProjectionSchlegel ProjectionCubeCube
Platonic SolidsPlatonic Solids
1. Catalan objects (i.e., duals of the Archimedean solids).2. B. de La Vaissière, P. W. Fowler, and M. Deza, J. Chem. Inf. Comput. Sci.,
2001, 41, 376-386.
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Schlegel ProjectionSchlegel ProjectionOctahedronOctahedron
Platonic SolidsPlatonic Solids
1. Catalan objects (i.e., duals of the Archimedean solids).2. B. de La Vaissière, P. W. Fowler, and M. Deza, J. Chem. Inf. Comput. Sci.,
2001, 41, 376-386.
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Schlegel ProjectionSchlegel ProjectionDodecahedronDodecahedron
Platonic SolidsPlatonic Solids
1. Catalan objects (i.e., duals of the Archimedean solids).2. B. de La Vaissière, P. W. Fowler, and M. Deza, J. Chem. Inf. Comput. Sci.,
2001, 41, 376-386.
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Schlegel ProjectionSchlegel ProjectionIcosahedronIcosahedron
Platonic SolidsPlatonic Solids
1. Catalan objects (i.e., duals of the Archimedean solids).2. B. de La Vaissière, P. W. Fowler, and M. Deza, J. Chem. Inf. Comput. Sci.,
2001, 41, 376-386.
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••Archimedean SolidsArchimedean Solids
•• CatalanCatalan objects (objects (i.ei.e., ., duals duals of the of the Archimedean Archimedean solids).solids).
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SnSn((DD)) = Du= Du((PP55((DD)) = )) = DuDu((OpOp((CaCa((DD))))))Snub dodecahedronSnub dodecahedronSDSD1313SnSn((CC)) = Du= Du((PP55((CC)) = )) = DuDu((OpOp((CaCa((C C ))))))Snub cubeSnub cubeSCSC1212TrTr((IDID) = ) = TrTr((MeMe((SnSn((T T ))))))Truncated icosidodecahedronTruncated icosidodecahedronTIDTID1111TrTr((COCO) = ) = TrTr((MeMe((MeMe((T T ))))))Truncated cuboctahedronTruncated cuboctahedronTCOTCO1010MeMe((IDID) = ) = MeMe((MeMe((II)) = )) = DuDu((PP44((II))))RhombicosidodecahedronRhombicosidodecahedronRIDRID99MeMe((COCO) ) = = MeMe((MeMe((CC)) = )) = DuDu((PP44((C C ))))RhombicuboctahedronRhombicuboctahedronRCORCO88MeMe((II) = ) = MeMe((DD) = ) = MeMe((SnSn((T T ))))IcosidodecahedronIcosidodecahedronIDID77MeMe((CC)) = Me= Me((OO))= Me= Me((MeMe((T T ))))CuboctahedronCuboctahedronCOCO66TrTr((DD) ) = Tr= Tr((DuDu((SnSn((T T ))))))Truncated dodecahedronTruncated dodecahedronTDTD55TrTr((II) ) = Tr= Tr((SnSn((TT))))Truncated icosahedronTruncated icosahedronTITI44TrTr((CC) ) = Tr= Tr((DuDu((MeMe((T T ))))))Truncated cubeTruncated cubeTCTC33TrTr((OO) ) = Tr= Tr((MeMe((T T ))))Truncated octahedronTruncated octahedronTOTO22TrTr((T T ))Truncated tetrahedronTruncated tetrahedronTTTT11FormulaFormulaPolyhedronPolyhedronSymbolSymbol
Archimedean Polyhedra (derived from Tetrahedron)Archimedean Polyhedra (derived from Tetrahedron)
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Du Du ((Tr Tr ((T T )) = C 1)) = C 1Tr Tr ((T T ) = A 1) = A 1
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects1,21,2
1.1. CatalanCatalan objects (objects (i.ei.e., ., duals duals of the of the Archimedean Archimedean solids).solids).2. 2. B. de La Vaissière, P. W. Fowler, and M. Deza, B. de La Vaissière, P. W. Fowler, and M. Deza, JJ. . ChemChem. . InfInf. . ComputComput. . SciSci., .,
20012001, , 4141, 376, 376--386.386.
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Du Du ((Tr Tr ((O O )) = C 2)) = C 2Tr Tr ((O O ) = A 2) = A 2
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects11
1.1. For other operation names see For other operation names see www.georgehart.comwww.georgehart.com\\virtualvirtual--polyhedrapolyhedra\\conway_notation.html conway_notation.html
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Du Du ((Tr Tr ((C C )) = C 3)) = C 3Tr Tr ((C C ) = A 3) = A 3
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects1,21,2
1. 1. CatalanCatalan objects (objects (i.ei.e., ., duals duals of the of the Archimedean Archimedean solids).solids).2. 2. B. de La Vaissière, P. W. Fowler, and M. Deza, B. de La Vaissière, P. W. Fowler, and M. Deza, JJ. . ChemChem. . InfInf. . ComputComput. . SciSci., .,
20012001, , 4141, 376, 376--386. 386.
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Du Du ((Tr Tr ((I I )) = C 4)) = C 4Tr Tr ((I I ) = A 4) = A 4
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects11
1.1. For other operation names see For other operation names see www.georgehart.comwww.georgehart.com\\virtualvirtual--polyhedrapolyhedra\\conway_notation.html conway_notation.html
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Du Du ((Tr Tr ((DD)) = C 5)) = C 5Tr Tr ((D D ) = A 5) = A 5
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects1,21,2
1. CatalanCatalan objects (objects (i.ei.e., ., duals duals of the of the Archimedean Archimedean solids).solids).2. 2. B. de La Vaissière, P. W. Fowler, and M. Deza, B. de La Vaissière, P. W. Fowler, and M. Deza, JJ. . ChemChem. . InfInf. . ComputComput. . SciSci., .,
20012001, , 4141, 376, 376--386. 386.
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Du Du ((Me Me ((C C )) = C 6)) = C 6Me Me ((C C ) = A 6) = A 6
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects11
1.1. For other operation names see For other operation names see www.georgehart.comwww.georgehart.com\\virtualvirtual--polyhedrapolyhedra\\conway_notation.htmlconway_notation.html
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Du Du ((Me Me ((I I )) = C 7)) = C 7Me Me ((I I ) = A 7) = A 7
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects1,21,2
1. CatalanCatalan objects (objects (i.ei.e., ., duals duals of the of the Archimedean Archimedean solids).solids).2. 2. B. de La Vaissière, P. W. Fowler, and M. Deza, B. de La Vaissière, P. W. Fowler, and M. Deza, JJ. . ChemChem. . InfInf. . ComputComput. . SciSci., .,
20012001, , 4141, 376, 376--386. 386.
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Du Du ((Me Me ((CO CO )) = P)) = P44(C) = C8(C) = C8Me Me ((CO CO ) = A 8) = A 8
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects11
1.1. For other operation names see For other operation names see www.georgehart.comwww.georgehart.com\\virtualvirtual--polyhedrapolyhedra\\conway_notation.html conway_notation.html
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Du Du ((Me Me ((I D I D )) = )) = PP44((I I )= C 9)= C 9Me Me ((I D I D ) = A 9) = A 9
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects1,21,2
1. CatalanCatalan objects (objects (i.ei.e., ., duals duals of the of the Archimedean Archimedean solids).solids).2. 2. B. de La Vaissière, P. W. Fowler, and M. Deza, B. de La Vaissière, P. W. Fowler, and M. Deza, JJ. . ChemChem. . InfInf. . ComputComput. . SciSci., .,
20012001, , 4141, 376, 376--386. 386.
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Du Du ((Tr Tr ((CO CO )) = C 10)) = C 10Tr Tr ((CO CO ) = A 10) = A 10
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects11
1.1. For other operation names see For other operation names see www.georgehart.comwww.georgehart.com\\virtualvirtual--polyhedrapolyhedra\\conway_notation.htmlconway_notation.html
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Du Du ((Tr Tr ((I D I D )) = C 11)) = C 11Tr Tr ((I D I D ) = A 11) = A 11
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects1,21,2
1.1. CatalanCatalan objects (objects (i.ei.e., ., duals duals of the of the Archimedean Archimedean solids).solids).2. 2. B. de La Vaissière, P. W. Fowler, and M. Deza, B. de La Vaissière, P. W. Fowler, and M. Deza, JJ. . ChemChem. . InfInf. . ComputComput. . SciSci., .,
20012001, , 4141, 376, 376--386.386.
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PP55((D D ) = ) = DuDu ((Sn Sn ((C C )) = C12)) = C12Sn Sn ((C C ) = A12) = A12
1.1. For other operation names see For other operation names see www.georgehart.comwww.georgehart.com\\virtualvirtual--polyhedrapolyhedra\\conway_notation.htmlconway_notation.html
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects11
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PP55((DD) = ) = Du Du ((Sn Sn ((D D )) = C13)) = C13Sn Sn ((D D ) = A13) = A13
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects1,21,2
1.1. CatalanCatalan objects (objects (i.ei.e., ., duals duals of the of the Archimedean Archimedean solids).solids).2. 2. B. de La Vaissière, P. W. Fowler, and M. Deza, B. de La Vaissière, P. W. Fowler, and M. Deza, JJ. . ChemChem. . InfInf. . ComputComput. . SciSci., .,
20012001, , 4141, 376, 376--386. 386.
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PP44(C(C6060) ; ) ; NN = 182= 182Du Du ((PP44(C(C6060)) ; )) ; NN = 180= 180
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects1,21,2
1.1. CatalanCatalan objects (objects (i.ei.e., ., duals duals of the of the Archimedean Archimedean solids).solids).2. 2. B. de La Vaissière, P. W. Fowler, and M. Deza, B. de La Vaissière, P. W. Fowler, and M. Deza, JJ. . ChemChem. . InfInf. . ComputComput. . SciSci., .,
20012001, , 4141, 376, 376--386. 386.
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PP55(C(C6060) ; ) ; NN = 272= 272Sn Sn (C(C6060) ; ) ; NN = 180= 180
ArchimedeanArchimedean--Catalan,Catalan, Dual Dual ObjectsObjects1,21,2
1.1. CatalanCatalan objects (objects (i.ei.e., ., duals duals of the of the Archimedean Archimedean solids).solids).2. 2. B. de La Vaissière, P. W. Fowler, and M. Deza, B. de La Vaissière, P. W. Fowler, and M. Deza, JJ. . ChemChem. . InfInf. . ComputComput. . SciSci., .,
20012001, , 4141, 376, 376--386. 386.
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