11

Operations on MapsOperations on Maps

MirceaMircea V. DiudeaV. Diudea

Faculty of Chemistry and Chemical EngineeringFaculty of Chemistry and Chemical EngineeringBabesBabes--BolyaiBolyai UniversityUniversity400028400028 ClujCluj, ROMANIA, ROMANIA

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

22

ContentsContents

Cage Building by Map OperationsCage Building by Map Operations1.1. Capra, Ca/SCapra, Ca/S112.2. Septupling SSeptupling S22

33

Operations on MapsOperations on Maps

•• A A map map M M is a combinatorial representation of a closed is a combinatorial representation of a closed surface.surface.11

•• Several operations on a map allow its transformation Several operations on a map allow its transformation into new maps (into new maps (convex polyhedraconvex polyhedra).).

•• PlatonicPlatonic polyhedra:polyhedra: Tetrahedron, Cube, Octahedron,Tetrahedron, Cube, Octahedron,Dodecahedron and IcosahedronDodecahedron and Icosahedron

1. Pisanski, T.; Randić, M. Bridges between Geometry and Graph Theory. In: Geometry at Work, M. A. A. Notes, 2000, 53, 174-194.

44

EulerEuler Theorem on PolyhedraTheorem on Polyhedra

v v –– e + fe + f = = χχ = = 22((11 –– gg))

χχ = = EulerEuler’’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))..

Any map Any map MM and its transforms by map operations will obey the and its transforms by map operations will obey the EulerEuler theoremtheorem11

1. L. Euler, Elementa doctrinae solidorum, Novi Comment. Acad. Sci. I. PetropolitanaeComment. Acad. Sci. I. Petropolitanae1758, 4, 109-140.

55

Platonic SolidsPlatonic Solids

OctahedronOctahedronCubeCubeTetrahedronTetrahedron

66

Platonic SolidsPlatonic Solids

IcosahedronIcosahedronDodecahedronDodecahedron

Dual of a triangulation is always a cubic net.Dual of a triangulation is always a cubic net.

77

Schlegel Schlegel projectionprojection

•• A projection of a sphereA projection of a sphere--like polyhedron like polyhedron on a plane is called aon a plane is called a SchlegelSchlegel diagram.diagram.

•• In a polyhedron, theIn a polyhedron, the centercenter of diagramof diagram is is taken either a taken either a vertexvertex, the , the center of ancenter of anedgeedge or the or the center of a facecenter of a face

88

Cube Cube and its dualand its dual,, OctahedronOctahedron

CubeCube

OctahedronOctahedron

Du

Du

SchlegelSchlegel diagramsdiagrams

99

PolygonalPolygonal PPss CappingCapping

•• CappingCapping PPss (s = 3, 4, 5) of a face is achieved as follows:(s = 3, 4, 5) of a face is achieved as follows:11•• Add a Add a new vertex in the centernew vertex in the center of the of the faceface. Put . Put s s --33 points on the points on the

boundary edgesboundary edges. . •• Connect the central point with one vertex (the end points includConnect the central point with one vertex (the end points included) ed)

on each edge. on each edge. •• The parent face is thus covered by: The parent face is thus covered by: trigonstrigons ((ss = 3), = 3),

tetragonstetragons ((ss = 4) = 4) pentagonspentagons ((ss = 5). = 5).

•• The The PP33 operation is also called operation is also called stellationstellation oror(centered) (centered) triangulationtriangulation. .

•• When all the faces of a map are thus operated, it is referred toWhen all the faces of a map are thus operated, it is referred to as as an omnicapping an omnicapping PPss operation. operation.

1. M. V. Diudea, Covering forms in nanostructures, Forma 2004 (submitted)

1010

The resulting map shows the relations:The resulting map shows the relations:

PPss ((M M ):):;;

Maps transformed as above form Maps transformed as above form dual pairs dual pairs ::

Vertex Vertex multiplication multiplication ratio by this ratio by this dualizationdualization is always: is always:

Since:Since:

PolygonalPolygonal PPss CappingCapping

000 )3( fesvv +−+=0ese =

00 fsf =

)())(( 3 MLeMPDu =

))(())(( 4 MMeMeMPDu =

)())(( 5 MSnMPDu =

00/)( dvDuv =

0000))(()( vdfsMPfDuv s ===

0000 )2( seesfse =−+=

1111

PP33((DodecahedronDodecahedron))PP33((CubeCube))

PP3 3 Capping = TriangulationCapping = Triangulation

1212

PP44((DodecahedronDodecahedron))11PP44((CubeCube))11

PP4 4 Capping = TetrangulationCapping = Tetrangulation

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.

1313

PP55((DodecahedronDodecahedron))PP55((CubeCube))

PP5 5 Capping = PentangulationCapping = Pentangulation11

1. For other operation names see www.georgehart.com\virtual-polyhedra\conway_notation.html

1414

7.7. Capra Capra 11 Ca Ca ((M M )) / Septupling/ Septupling

•• GoldbergGoldberg22 relation:relation:mm = (= (a a 22 + + abab + + b b 22); ); a a ≥≥ b b ; a + ; a + b >b > 00

Le Le : (1, 1); : (1, 1); mm = 3= 3QQ : (2, 0); : (2, 0); mm = 4= 4CaCa : (2, 1); : (2, 1); mm = 7= 7

1.1. M. V. Diudea, M. V. Diudea, Studia Univ. BabesStudia Univ. Babes--BolyaiBolyai, 2003, 48, 3, 2003, 48, 3--21212.2. M. Goldberg, M. Goldberg, Tohoku Math. JTohoku Math. J., 1937, ., 1937, 4343, 104, 104--108.108.

Ca Ca ((M M ) = ) = TrTrP P 55((PP55((M M ))))

1515

CapraCapra

77Ca: Ca: (2, 1)(2, 1)Septupling (Capra) Septupling (Capra) 44Q:Q: (2, 0)(2, 0)Quadrupling (Chamfering) Quadrupling (Chamfering) 33Le: Le: (1, 1)(1, 1)Tripling (Leapfrog )Tripling (Leapfrog )11II: (1,0): (1,0)Identity Identity

mmSymbolSymbolOperationOperation

GoldbergGoldberg factors factors mm by operations on a 3by operations on a 3--valent mapvalent map

The The mm factor is also related to the formula giving factor is also related to the formula giving the volume of truncated pyramid, of height the volume of truncated pyramid, of height h h : :

VV = = mhmh/3/3coming from the ancient Egypt.coming from the ancient Egypt.

1616

Ca Ca ((MM) =) =TrTrP P 55((PP55((M M )), )), examplesexamples

Square faceSquare face

.E2(M) P5(M) TrP5(P5(M))

.... . . . .

....

.... ....

1717

Theorem onTheorem on Ca Ca ((M M ))

•• The vertex multiplication ratio inThe vertex multiplication ratio in Ca Ca ((M M )) iiss22dd0 0 ++ 11 irrespective of the tiling type ofirrespective of the tiling type of MM..

•• DemonstrationDemonstration: Observe that, for each vertex of : Observe that, for each vertex of MM, , 22dd00 new vertices result in new vertices result in Ca Ca ((M M )) and and the old vertexthe old vertex isis

preservedpreserved, thus demonstrating the theorem:, thus demonstrating the theorem:11

vv = 2= 2dd00vv00 + + vv00v v //vv00 = (2= (2dd00+1)+1)vv0 0 //vv00 = 2= 2dd00+1+1

1. M. V. Diudea, 1. M. V. Diudea, FormaForma, 2004 (submitted)., 2004 (submitted).

1818

Capra,Capra, continuedcontinued

TheThe transformed parameters are:transformed parameters are:11

Ca Ca ((M M ):): vv 11 = (2= (2dd00+1)+1)vv0 0 ==vv0 0 + 2+ 2ee0 0 + + ss00ff00ee = 7= 7ee0 0 = 3= 3ee0 0 + 2+ 2ss00ff00ff = (s= (s00+1)+1)ff00

It involves rotation byIt involves rotation by ππ/2/2ss00 of the parent facesof the parent faces

1.1. M. V. Diudea, M. V. Diudea, Studia Univ. BabesStudia Univ. Babes--BolyaiBolyai, 2003, 48, 3, 2003, 48, 3--2121

1919

CaCann ((M M );); Iterative operationIterative operation

dd00 > 3> 3vvnn = 8= 8vvn n --11 –– 77vvn n --22; ; n n ≥≥ 22

dd00 = 3= 3vvnn = 7= 7nn ⋅⋅ vv0 0 eenn = 7= 7n n ⋅⋅ ee00 = 7= 7nn ⋅⋅ 33vv0 0 /2/2ffnn = = ff00 + (7+ (7nn --1)1)⋅⋅vv0 0 /2/2

Capra,Capra, continuedcontinued

2020

Capra, Capra, continuedcontinued

•• CaCa insulates any face of insulates any face of MM by its own by its own hexagons, which hexagons, which are not sharedare not shared with any with any old face (in contrast to old face (in contrast to LeLe or or Q Q ).).

•• In case of a In case of a 33--valentvalent regular map the regular map the vertex multiplication ratio by Capra is vertex multiplication ratio by Capra is 77. .

•• Maps of Maps of d d 00 >> 33 are not regular anymore. are not regular anymore.

2121

Capra Capra -- ChiralChiral, , continuedcontinued

Enantiomeric pair of the Enantiomeric pair of the Ca Ca ((C C )), in Schlegel projection, in Schlegel projection

Since Since PP55 can be done either clockwise or countercan be done either clockwise or counter--clockwise, it resultsclockwise, it resultsin an in an enantiomericenantiomeric pair: pair: CaS CaS ((M M ) and ) and CaR CaR ((M M ), in terms of ), in terms of the the sinister/rectussinister/rectus stereochemical isomers stereochemical isomers

2222

CaR CaR ((CaS CaS ((T T )))); ; NN = 196= 196Ca Ca ((T T )); ; NN = 28= 28

CapraCapra, , examplesexamples

The sequence The sequence CaS CaS ((CaS CaS ((M M )))) results in a results in a twistedtwisted transform, while transform, while CaR CaR ((CaS CaS ((M M )))) will will straighten straighten the structure.the structure.

2323

CaR CaR ((CaS CaS ((C C )))); ; NN = 392= 392Ca Ca ((C C )); ; NN = 56= 56

CapraCapra, , examplesexamplesThe sequence The sequence Du Du ((Op Op ((Ca Ca ((M M )))))) is just the snub is just the snub Sn Sn ((MM)), , being this transform a being this transform a chiralchiral one. one.

2424

CapraCapra, , examplesexamples

Enantiomeric pair of a Enantiomeric pair of a Ca Ca (SWNT)(SWNT)

2525

Capra Capra -- OpenOpenThe The OpenOpen operation operation OpOpnn can be written as:can be written as:

OpOpn n ((Ω Ω ((M M )) = )) = EEn n Ω Ω ((M M ))EEn n ΩΩ is the is the n n --homeomorphichomeomorphic transformation of the transformation of the edges edges bounding the parentbounding the parent--like faces, like faces, namely those resulted by namely those resulted by

the the ΩΩ operation.operation.

In case In case Ω Ω ((M M ) = ) = Ca Ca ((M M ) ) the resulting open map has all the resulting open map has all polygons of the polygons of the same (6+same (6+nn) size) size. It is the . It is the only knownonly knownoperationoperation leading to a leading to a Platonic tessellationPlatonic tessellation in in

open/infinite objects.open/infinite objects.

2626

Capra Capra –– Open,Open, continuedcontinued

TheThe transformed parameters are:transformed parameters are:11

Op Op ((Ca Ca ((M M )):)): vv OpOp = = v v + s+ s00ff0 0 ==vv0 0 + 6+ 6e e 0 0 ee OpOp = 9= 9e e 0 0 f f OpOp = = ff --ff00 = s= s00ff00

1.1. M. V. Diudea, M. V. Diudea, Studia Univ. BabesStudia Univ. Babes--BolyaiBolyai, 2003, 48, 3, 2003, 48, 3--2121

The open (transformed) maps The open (transformed) maps cannot longer be embeddedcannot longer be embedded in in a surface of a surface of genus zerogenus zero ((e.ge.g., in the plane or the sphere).., in the plane or the sphere).

2727

Capra Capra -- OpenOpen, , continuedcontinued

Open polyhedraOpen polyhedra are embedded in surfaces are embedded in surfaces of of negativenegative Euler Euler χχ values and values and gg > 1> 1

The The genusgenus gg of an of an open Capraopen Capra object is calculable as:object is calculable as:

The spherical character of the parent polyhedron involves the EuThe spherical character of the parent polyhedron involves the Euler ler relation written as:relation written as:

The number of The number of parent facesparent faces, which , which equals that of open facesequals that of open faces, is:, is:

thus demonstrating the thus demonstrating the gg ––formulaformula, which can be , which can be generalisedgeneralised for for any any open objectopen object..

2/2/)e2( 000 fvg =+−=

gevfevMCaOp OpOpOp 2-2)))((( 00 =−=+−=χ

2000 =+− fev

000 2 evf +−=

2828

Capra Capra -- OpenOpen, , continuedcontinued

LLattices with attices with gg > 1> 1 will have will have negativenegativeand consequently and consequently negative curvaturenegative curvature. .

The The genus genus of the five of the five PlatonicPlatonic solids transformed by solids transformed by

is: is: 22 (Tetrahedron); (Tetrahedron);

33 (Cube); (Cube);

44 (Octahedron); (Octahedron);

66 (Dodecahedron) and (Dodecahedron) and

1010 (Icosahedron)(Icosahedron)

))(( MOpχ

))(( MCaOp

2929

Op Op ((Ca Ca ((C C )))); ; SS = 2.171 = 2.171 NN = 80; = 80; EE = 108; = 108; FF77 = 24; = 24; gg = 3 = 3

Op Op ((Ca Ca ((T T )))); ; SS = 10.432= 10.432NN = 40; = 40; EE = 54; = 54; FF77 = 12; = 12; gg = 2= 2

Capra Capra -- OpenOpen, , examplesexamples

3030

CaR CaR ((Op Op ((CaS CaS ((C C )))))); ; S = 0.428S = 0.428N = 464; E = 660; F = 192; g = 3; N = 464; E = 660; F = 192; g = 3;

CaR CaR ((Op Op ((CaS CaS ((T T )))))); ; S = 1.114S = 1.114N N = 232;= 232; E E = 330;= 330; F F = 96;= 96; g g = 2= 2

Capra Capra -- OpenOpen, , examplesexamples

3131

Op Op ((Ca Ca ((D D )))); ; S S = 1.574= 1.574N N = 200; = 200; E E = 270; = 270; FF7 7 = 60;= 60; g g = 6 = 6

Ca Ca ((D D )); ; N = 140N = 140

Capra Capra -- OpenOpen, , examplesexamples

Op Op ((Ca Ca ((M M )))) leads to leads to all heptagonall heptagon coveringcovering of negatively curved unitsof negatively curved units

3232

CaRCaR(C(C260260); ); NN = 1820= 1820CC260260 ; (; (IIhh))

Capra Capra -- OpenOpen, , examplesexamples

The The FowlerFowler’s phantasmagoric fullerene ’s phantasmagoric fullerene CC260260, covered only by , covered only by pentagons and heptagons and its pentagons and heptagons and its CaCa transformtransform

3333

Capra Capra -- OpenOpen, , examplesexamples

CaR CaR ((Op Op ((CaS CaS ((DD))))))

NN = 1160; = 1160; EE = 1650; = 1650; FF = 480; g = 6; = 480; g = 6; SS = 0.335= 0.335

3434

Capra Capra -- OpenOpen

Ca Ca ((SupraSupra--D D ((Op Op ((Ca Ca ((T T )))))))); ; SS = 1.075= 1.075NN = 4100; = 4100; gg = 21 = 21

SupraSupra--D D ((Op Op ((Ca Ca ((T T )))))); ; S = S = 7.876 7.876 N N = 620;= 620; E E = 900;= 900; FF7 7 = 240;= 240; g g = 21= 21

The building block The building block Op Op ((Ca Ca ((T T )),)), enables construction of a enables construction of a suprasupra--molecularmolecular(open) (open) dodecahedrondodecahedron,,11 a multi torus of a multi torus of genus 21genus 21, entirely covered by , entirely covered by heptagons heptagons FF77 ((Platonic Platonic tessellation)tessellation)

1. M. V. Diudea, 1. M. V. Diudea, FormaForma, 2004 (submitted), 2004 (submitted)

3535

POAV POAV –– Strain EnergyStrain Energy

θθp p = = θθσπσπ -- 9090oo pyramidalizationpyramidalization angle angle

SESE = 200(= 200(θθp p ))2 2 strain energystrain energy120 120 -- (1/3) (1/3) ΣΣθθij ij deviation to planarity deviation to planarity

In theIn the POAV1POAV1 theorytheory1,21,2 the the ππ--orbital axis vector makes orbital axis vector makes equal angles to the three equal angles to the three σσ--bonds of the spbonds of the sp22 carbon:carbon:

1. R.C. Haddon, 1. R.C. Haddon, J. Am. Chem. SocJ. Am. Chem. Soc., 112, 3385 (1990).., 112, 3385 (1990).2. R.C. Haddon, 2. R.C. Haddon, J. Phys. Chem. AJ. Phys. Chem. A, 105, 4164 (2001). , 105, 4164 (2001).

3636

Capra Capra -- Open Open -- LeapfrogLeapfrog

Le Le ((Op Op ((Ca Ca ((C C ))))))NN = 240; = 240; EE = 348; = 348; FF = 104; = 104; gg = 3 = 3

Le Le ((Op Op ((Ca Ca ((T T ))))))NN = 120; = 120; EE = 174; = 174; FF = 52; = 52; gg = 2 = 2

The sequence The sequence Le Le ((Op Op ((Ca Ca ((MM)))))) enables a covering by disjoint azulenic (5,7) unitsenables a covering by disjoint azulenic (5,7) units11

1. M. V. Diudea, 1. M. V. Diudea, FormaForma, 2004 (submitted), 2004 (submitted)

3737

Capra Capra -- Open Open -- LeapfrogLeapfrog

SupraSupra--((azulenicazulenic))--D D ((Le Le ((Op Op ((CaCa((TT))))))))N = 2400; g = 21N = 2400; g = 21

Le Le ((Op Op ((Ca Ca ((D D ))))))NN = 600; = 600; EE = 870; = 870; FF = 260; = 260; gg = 6= 6

The sequence The sequence Le Le ((Op Op ((Ca Ca ((MM)))))) enables a covering by disjoint azulenic (5,7) unitsenables a covering by disjoint azulenic (5,7) units11

1. M. V. Diudea, 1. M. V. Diudea, FormaForma, 2004 (submitted), 2004 (submitted)

3838

RetroRetro--CapraCapra

PP--5a5a((DD);); NN = 80= 80Ca Ca ((DD); ); NN = 140= 140

MM0 0 = P= P--55(TrTr--(P5(P5))(M(M11)))) = = EE--22((PP--5a5a(M(M11)) ))

Delete the smallest faces and continue with Delete the smallest faces and continue with EE--22

3939

Septupling OperationsSeptupling Operations

•• FormulasFormulas

)())(()( 51 5 MCaMPTrMS P ==

))(( 4)3,1(2 MEJS +−=

4040

SS11 & & SS22 OperationsOperations

OpOp22aa((SS22((C C )) )) SS22((C C ) ) EE44((C C ))

OpOp((SS11((C C ))))SS11((C C ))PP55((C C ))

4141

Lattice counting by the Lattice counting by the SeptuplingSeptupling--OpenOpensequence of map operationssequence of map operations

44

33

22

11

OperationOperationParameter Parameter

000000 )12(2 vdfsevv +=++=07ee =

000 ffsf +=)(1 MS

))(( 1 MSOp0000 )13( vdfsvvOp +=+=

09eeOp =000 fsfffOp =−=

00 )12( vdv +=07ee =

000 ffsf +=)(2 MS

))(( 22 MSOp a00 )14( vdvOp +=

011eeOp =00 fsfOp =

4242

Septupling Septupling SS22

SS2 2 ––OpOp22aa ((T T ); ); NN = 52= 52SS22((T T ); ); NN = 28= 28

4343

Septupling Septupling SS2 2 -- IterativeIterative

SSSSSS2 2 ((T T ); ); NN = 1372 = 4 x 7= 1372 = 4 x 733SSSS2 2 ((T T ); ); NN = 196 = 4 x 7= 196 = 4 x 722

4444

Septupling Septupling SS22

SS2 2 ((O O ); ); NN = 54 = 6 x 9= 54 = 6 x 9SS2 2 ((C C ); ); NN = 56 = 8 x 7= 56 = 8 x 7

4545

Septupling Septupling SS2 2 -- RacemicRacemic

SS2 2 --OpOpR R ((C C ); ); NN = 104= 104SS2 2 --OpOpSS ((C C ); ); NN = 104= 104

4646

Iterative lattice counting by Iterative lattice counting by SeptuplingSeptupling

)(, MS niCase:Case: 30 =d

07 vvn

n ⋅=

07 een

n ⋅=

000 6/)()17( ffsfn

n +⋅−=

4747

Septupling Septupling SS2 2 –– IterativeIterativeFractalizationFractalization

SS22SS22SS2 2 ((C C ); ); NN = 2744 = 8 x 7= 2744 = 8 x 733SS22SS2 2 ((C C ); ); NN = 392 = 8 x 7= 392 = 8 x 722

4848

Septupling Septupling SS2 2 –– IterativeIterativeFractalizationFractalization

SS22SS22((DD); ); NN = = 20 x 7= = 20 x 72 2 = 980= 980SS2 2 ((DD); ); NN = 140 = 20 x 7= 140 = 20 x 7

4949

Septupling Septupling SS2 2 –– IterativeIterativeFractalizationFractalization

SS22SS22SS22((DD); ); NN = 6860 = 20 x 7= 6860 = 20 x 733FiveFive--fold symmetryfold symmetry

SS22SS22SS22((DD); ); NN = 6860 = 20 x 7= 6860 = 20 x 722TwoTwo--fold symmetryfold symmetry

5050

Septupling Septupling SS2 2 –– IterativeIterativeFractalizationFractalization

SS22SS2 2 ((I I ); ); NN = 972 = 12 x 9= 972 = 12 x 922SS22((I I ); ); NN = 132 = 12 x 11= 132 = 12 x 11

5151

Septupling Septupling SS2 2 –– IterativeIterativeFractalizationFractalization

SS22SS22SS2 2 ((I I ); ); NN = 6852= 6852FiveFive--fold symmetryfold symmetry

SS22SS22SS22((I I ); ); NN = 6852= 6852TwoTwo--fold symmetryfold symmetry

5252Sn(D) = Du(P5(D)) = Du(Op(Ca(D)))Snub dodecahedronSD13

Sn(C) = Du(P5(C)) = Du(Op(Ca(C )))Snub cubeSC12

Tr(ID) = Tr(Me(Sn(T )))Truncated icosidodecahedronTID11

Tr(CO) = Tr(Me(Me(T )))Truncated cuboctahedronTCO10

Me(ID) = Me(Me(I)) = Du(P4(I))RhombicosidodecahedronRID9

Me(CO) = Me(Me(C)) = Du(P4(C ))RhombicuboctahedronRCO8

Me(I) = Me(D) = Me(Sn(T ))IcosidodecahedronID7

Me(C) = Me(O)= Me(Me(T ))CuboctahedronCO6

Tr(D) = Tr(Du(Sn(T )))Truncated dodecahedronTD5

Tr(I) = Tr(Sn(T))Truncated icosahedronTI4

Tr(C) = Tr(Du(Me(T )))Truncated cubeTC3

Tr(O) = Tr(Me(T ))Truncated octahedronTO2

Tr(T )Truncated tetrahedronTT1

Du(I ) = Du(Sn(T ))DodecahedronD5

Sn(T )IcosahedronI4

Du(O)=Du(Me(T ))Cube (hexahedron)C3

Me(T )OctahedronO2

-TetrahedronT1

FormulaPolyhedronSymbol

Platonic and Archimedean polyhedra (derived from Tetrahedron)Platonic and Archimedean polyhedra (derived from Tetrahedron)