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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.
33
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.
44
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 =⋅= ∑
55
••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)
77
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.
88
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.
99
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.
1010
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.
1111
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.
1212
••Archimedean SolidsArchimedean Solids
•• CatalanCatalan objects (objects (i.ei.e., ., duals duals of the of the Archimedean Archimedean solids).solids).
1313
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)
1414
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.
1515
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
1616
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.
1717
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
1818
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.
1919
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
2020
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.
2121
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
2222
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.
2323
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
2424
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.
2525
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
2626
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.
2727
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.
2828
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.