11

Platonics & ArchimedeansPlatonics & Archimedeans

MirceaMircea V. DiudeaV. Diudea

Faculty of Chemistry and Chemical EngineeringFaculty of Chemistry and Chemical EngineeringBabesBabes--BolyaiBolyai UniversityUniversity400084400084 ClujCluj, ROMANIA, ROMANIA

diudea@chem.ubbcluj.rodiudea@chem.ubbcluj.ro

22

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

66

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.