Operations on Maps. Mircea V. Diudea. Faculty of Chemistry and Chemical Engineering Babes-Bolyai

Transcription

1 Operations on Maps Mircea V. Diudea Faculty of Chemistry and Chemical Engineering Babes-Bolyai Bolyai University Cluj,, ROMANIA 1

2 Contents Cage Building by Map Operations 1. Capra, Ca/S 1 2. Septupling S Septupling S 2 2

3 Operations on Maps A map map M is a combinatorial representation of a closed surface. 1 Several operations on a map allow its transformation into new maps (convex( polyhedra). Platonic polyhedra: Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron 1. Pisanski, T.; Randić, M. Bridges between Geometry and Graph Theory. In: Geometry at Work, M. A. A. Notes, 2000, 53,

4 Euler Theorem on Polyhedra Any map M and its transforms by map operations will obey the Euler theorem 1 v e + f = χ = 2(1 g) χ = Euler s characteristic v = number of vertices, e = number of edges, f = number of faces, g = genus ; (g = 0 for a sphere; 1 for a torus). 1. L. Euler, Elementa doctrinae solidorum, Novi Comment. Acad. Sci. I. Petropolitanae Comment. Acad. Sci. I. Petropolitanae 1758, 4,

5 Platonic Solids Tetrahedron Cube Octahedron 5

6 Platonic Solids Dodecahedron Icosahedron Dual of a triangulation is always a cubic net. 6

7 Schlegel projection A projection of a sphere-like polyhedron on a plane is called a Schlegel diagram. In a polyhedron, the center of diagram is taken either a vertex,, the center of an edge or the center of a face 7

8 Cube and its dual, Octahedron Schlegel diagrams Cube Du Du Octahedron 8

9 Polygonal P s Capping Capping P s (s = 3, 4, 5) of a face is achieved as follows:1 Add a new vertex in the center of the face.. Put s -3 points on the boundary edges. Connect the central point with one vertex (the end points included) on each edge. The parent face is thus covered by: trigons (s = 3), tetragons (s = 4) pentagons (s = 5). The P 3 operation is also called stellation or (centered) triangulation. When all the faces of a map are thus operated, it is referred to as an omnicapping P s operation. 1. M. V. Diudea, Covering forms in nanostructures, Forma 2004 (submitted) 9

10 The resulting map shows the relations: P s (M ): Maps transformed as above form dual pairs : ; Vertex multiplication ratio by this dualization is always: Since: Polygonal P s Capping v = v + e = se 0 e ( e = se f = s 0 f ( s 3) e0 f 0 = s0 f0 + s 2) Du ( P 3 ( M )) = Le( M ) Du ( P 4 ( M )) = Me( Me( M )) Du ( P 5 ( M )) = Sn( M ) v ( Du) / v = d 0 0 v( Du) = f ( Ps ( M )) = s f = d v

11 P 3 Capping = Triangulation P 3 (Cube) P 3 (Dodecahedron) 11

12 P 4 Capping = Tetrangulation P 4 (Cube) 1 P 4 (Dodecahedron) 1 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,

13 P 5 Capping = Pentangulation 1 P 5 (Cube) P 5 (Dodecahedron) 1. For other operation names see \conway_notation.html 13

14 7. Capra 1 Ca (M ) / Septupling Ca (M ) = Tr P 5 (P 5 (M )) Goldberg 2 relation: m = ( (a 2 + ab + b 2 ); a b ; a + b > 0 Le : (1, 1); m = 3 Q : (2, 0); m = 4 Ca : (2, 1); m = 7 1. M. V. Diudea, Studia Univ. Babes-Bolyai Bolyai,, 2003, 48, M. Goldberg, Tohoku Math. J., J 1937, 43,,

15 Capra Goldberg factors m by operations on a 3-valent 3 map Operation Identity Tripling (Leapfrog ) Quadrupling (Chamfering) Septupling (Capra) Symbol I: : (1,0) Le: (1, 1) Q: (2, 0) Ca: (2, 1) m The m factor is also related to the formula giving the volume of truncated pyramid, of height h : V = mh/3 coming from the ancient Egypt. 15

16 Ca (M)) = =Tr P 5 (P 5 (M )), )), examples Square face E 2 (M) P 5 (M) Tr P5 (P 5 (M)) 16

17 Theorem on Ca (M ) The vertex multiplication ratio in Ca (M ) is 2d irrespective of the tiling type of M. Demonstration: : Observe that, for each vertex of M, 2d 0 new vertices result in Ca (M ) and the old vertex is preserved,, thus demonstrating the theorem: 1 v = 2 2d 0 v 0 + v 0 v /v 0 = (2d 0 +1)v 0 /v 0 = 2 2d M. V. Diudea, Forma,, 2004 (submitted). 17

18 Capra, continued The transformed parameters are: 1 Ca (M ): v 1 = (2 = (2d 0 +1)v 0 =v e 0 + s 0 0 f e = 7 7e 0 = 3 3e s 0 0 f f = (s 0 +1)f 0 It involves rotation by π/2s 0 of the parent faces 1. M. V. Diudea, Studia Univ. Babes-Bolyai Bolyai,, 2003, 48,

19 Capra, continued Ca n (M ); Iterative operation d 0 > 3 d 0 = 3 v n = 8 8v n -1 7v n -2 ; n 2 v n = 7 n v 0 e n = 7 n e 0 = 7 n 3v 0 /2 f n = 0 f + (7 n -1) 1) v 0 /2 19

20 Capra, continued Ca insulates any face of M by its own hexagons, which are not shared with any old face (in contrast to Le or Q ). In case of a 3-valent regular map the vertex multiplication ratio by Capra is 7. Maps of d 0 > 3 are not regular anymore. 20

21 Capra - Chiral, continued Since P 5 can be done either clockwise or counter-clockwise, clockwise, it results in an enantiomeric pair: CaS (M ) and CaR (M ), in terms of the sinister/rectus stereochemical isomers Enantiomeric pair of the Ca (C ),, in Schlegel projection 21

22 Capra, examples The sequence CaS (CaS (M )) results in a twisted transform, while CaR (CaS (M )) will straighten the structure. Ca (T ); N = 28 CaR (CaS (T )); N =

23 Capra, examples The sequence Du (Op (Ca (M ))) is just the snub Sn (M), being this transform a chiral one. Ca (C ); N = 56 CaR (CaS (C )); N =

24 Capra, examples Enantiomeric pair of a Ca (SWNT) 24

25 Capra - Open The Open operation Op n can be written as: Op n (Ω (M )) = E n Ω (M ) E n Ω is the n -homeomorphic transformation of the edges bounding the parent-like faces, namely those resulted by the Ω operation. In case Ω (M ) = Ca (M ) the resulting open map has all polygons of the same (6+n) ) size.. It is the only known operation leading to a Platonic tessellation in open/infinite objects. 25

26 Capra Open, Open, continued The transformed parameters are: 1 Op (Ca (M )): )): v Op Op = v + s 0 0 f =v e 0 e Op = 9 9e 0 f Op = f -f 0 = s 0 0 f The open (transformed) maps cannot longer be embedded in a surface of genus zero (e.g.,., in the plane or the sphere). 1. M. V. Diudea, Studia Univ. Babes-Bolyai Bolyai,, 2003, 48,

27 Capra - Open, continued Open polyhedra are embedded in surfaces of negative Euler χ values and g > 1 The genus g of an open Capra object is calculable as: χ( Op( Ca( M ))) = vop eop + f Op = v0 e0 = 2-2g g = ( 2 v 0 + e 0 ) / 2 = f 0 The spherical character of the parent polyhedron involves the Euler relation written as: v e + f = The number of parent faces,, which equals that of open faces,, is: f 0 = 2 v0 + e0 thus demonstrating the g formula,, which can be generalised for any open object. / 2 27

28 Capra - Open, continued Lattices with g > 1 will have negative and consequently negative curvature. The genus of the five Platonic solids transformed by is: 2 (Tetrahedron); 3 (Cube); 4 (Octahedron); 6 (Dodecahedron) and 10 (Icosahedron) Op( Ca( M )) χ( Op( M )) 28

29 Capra - Open, examples Op (Ca (T )); S = N = 40; E = 54; F 7 = 12; g = 2 Op (Ca (C )); S = N = 80; E = 108; F 7 = 24; g = 3 29

30 Capra - Open, examples CaR (Op (CaS (T ))); S = N = 232; E = 330; F = 96; g = 2 CaR (Op (CaS (C ))); S = N = 464; E = 660; F = 192; g = 3; 30

31 Capra - Open, examples Op (Ca (M )) leads to all heptagon covering of negatively curved units Ca (D ); N = 140 Op (Ca (D )); S = N = 200; E = 270; F 7 = 60; g = 6 31

32 Capra - Open, examples The Fowler s phantasmagoric fullerene C 260, covered only by pentagons and heptagons and its Ca transform C 260 ; ( (I h ) CaR(C (C 260 ); ); N =

33 Capra - Open, examples CaR (Op (CaS (D))) N = 1160; E = 1650; F = 480; g = 6; S =

34 Capra - Open The building block Op (Ca (T )), enables construction of a supra-molecular (open) dodecahedron, 1 a multi torus of genus 21,, entirely covered by heptagons F 7 (Platonic tessellation) Supra-D (Op (Ca (T ))); S = N = 620; E = 900; F 7 = 240; g = 21 Ca (Supra-D (Op (Ca (T )))); S = N = 4100; g = M. V. Diudea, Forma,, 2004 (submitted) 34

35 POAV Strain Energy In the POAV1 theory 1,2 the π-orbital axis vector makes equal angles to the three σ-bonds of the sp 2 carbon: θ p = θ σπ - 90 o pyramidalization angle SE = 200(θ p ) 2 strain energy (1/3) Σθij deviation to planarity 1. R.C. Haddon, J. Am. Chem. Soc., 112, 3385 (1990). 2. R.C. Haddon, J. Phys. Chem. A, 105, 4164 (2001). 35

36 Capra - Open - Leapfrog The sequence Le (Op (Ca (M))) enables a covering by disjoint azulenic (5,7) units 1 Le (Op (Ca (T ))) N = 120; E = 174; F = 52; g = 2 Le (Op (Ca (C ))) N = 240; E = 348; F = 104; g = 3 1. M. V. Diudea, Forma,, 2004 (submitted) 36

37 Capra - Open - Leapfrog The sequence Le (Op (Ca (M))) enables a covering by disjoint azulenic (5,7) units 1 Le (Op (Ca (D ))) N = 600; E = 870; F = 260; g = 6 Supra-(azulenic azulenic)-d (Le (Op (Ca(T)))) N = 2400; g = M. V. Diudea, Forma,, 2004 (submitted) 37

38 Retro-Capra M 0 = P -5 (Tr Tr -(P5 (P5)(M (M 1 )) = E -2 (P -5a 5a(M 1 ) ) Delete the smallest faces and continue with E -2 Ca (D); N = 140 P -5a 5a(D); N = 80 38

39 Septupling Operations Formulas S M ) = TrP ( P ( M )) = ( Ca( M ) S + 2 = J ( 1, 3) ( E4 ( M )) 39

40 S 1 & S 2 Operations P 5 (C ) S 1 (C ) Op(S 1 (C )) E 4 (C ) S 2 (C ) Op 2a (S 2 (C )) 40

41 Lattice counting by the Septupling-Open sequence of map operations Parameter Operation 1 v = v0 + 2e0 + s0 f 0 = (2d 0 + 1) v e = 7e 0 f = s f f0 0 S 1 ( M ) 2 v Op = v + s0 f 0 = ( 3d 0 + 1) v e Op = 9e 0 f Op = f f = 0 s0 f0 0 Op( S 1 ( M )) 3 v = ( 2d 0 + 1) v e = 7e 0 f = s f f0 0 S 2 ( M ) 4 v Op = ( 4d + v e Op = 11e 0 f Op = s 0 f 0 0 1) 0 Op 2a ( S 2 ( M )) 41

42 Septupling S 2 S 2 (T ); N = 28 S 2 Op 2a (T ); N = 52 42

43 Septupling S 2 - Iterative SS 2 (T ); N = 196 = 4 x 7 2 SSS 2 (T ); N = 1372 = 4 x

44 Septupling S 2 S 2 (C ); N = 56 = 8 x 7 S 2 (O ); N = 54 = 6 x 9 44

45 Septupling S 2 - Racemic S 2 -Op S (C ); N = 104 S 2 -Op R (C ); N =

46 Iterative lattice counting by Septupling S ( M i, n ) Case: d 0 = 3 v e n n = = n 7 v 0 n 7 e 0 f n n = 7 1) ( s0 f 0 ) / 6 + ( f 0 46

47 Septupling S 2 Iterative Fractalization S 2 S 2 (C ); N = 392 = 8 x 7 2 S 2 S 2 S 2 (C ); N = 2744 = 8 x

48 Septupling S 2 Iterative Fractalization S 2 (D); N = 140 = 20 x 7 S 2 S 2 (D); N = = 20 x 7 2 =

49 Septupling S 2 Iterative Fractalization S 2 S 2 S 2 (D); N = 6860 = 20 x 7 = 6860 = 20 x 7 2 Two-fold symmetry S 2 S 2 S 2 (D); N = 6860 = 20 x 7 = 6860 = 20 x 7 3 Five-fold symmetry 49

50 Septupling S 2 Iterative Fractalization S 2 (I ); N = 132 = 12 x 11 S 2 S 2 (I ); N = 972 = 12 x

51 Septupling S 2 Iterative Fractalization S 2 S 2 S 2 (I ); N = 6852 Two-fold symmetry S 2 S 2 S 2 (I ); N = 6852 Five-fold symmetry 51

52 Platonic and Archimedean polyhedra (derived from Tetrahedron) Symbol Polyhedron Formula 1 T Tetrahedron - 2 O Octahedron Me(T ) 3 C Cube (hexahedron) Du(O)=Du(Me(T )) 4 I Icosahedron Sn(T ) 5 D Dodecahedron Du(I ) = Du(Sn(T )) 1 TT Truncated tetrahedron Tr(T ) 2 TO Truncated octahedron Tr(O) = Tr(Me(T )) 3 TC Truncated cube Tr(C) = Tr(Du(Me(T ))) 4 TI Truncated icosahedron Tr(I) = Tr(Sn(T)) 5 TD Truncated dodecahedron Tr(D) = Tr(Du(Sn(T ))) 6 CO Cuboctahedron Me(C) = Me(O)= Me(Me(T )) 7 ID Icosidodecahedron Me(I) = Me(D) = Me(Sn(T )) 8 RCO Rhombicuboctahedron Me(CO) = Me(Me(C)) = Du(P 4 (C )) 9 RID Rhombicosidodecahedron Me(ID) = Me(Me(I)) = Du(P 4 (I)) 10 TCO Truncated cuboctahedron Tr(CO) = Tr(Me(Me(T ))) 11 TID Truncated icosidodecahedron Tr(ID) = Tr(Me(Sn(T ))) 12 SC Snub cube Sn(C) = Du(P 5 (C)) = Du(Op(Ca(C ))) 13 SD Snub dodecahedron Sn(D) = Du(P 5 (D)) = Du(Op(Ca(D))) 52