Research: Science and Education
Generating Closed Shapes from Regular Tilings William O. J. Boo and Daniell L. Mattern* Department of Chemistry and Biochemistry, University of Mississippi, University, MS 38677;
[email protected] Today’s structural scientists use a large assortment of geometrical shapes to describe the topologies of molecules, ions, clusters, and crystals. The fullerenes have shown us that there are a great number of closed three-dimensional shapes that can be constructed from pentagons and hexagons (1), but the number of closed shapes that can arise from other combinations of the various regular polygons is truly staggering (2). Chemists and other scientists will need to be able to predict and analyze these shapes to effectively describe chemical topologies (3, 4 ). A fullerene can be described by a simple defining rule: its surface has exactly 12 pentagonal faces, along with any number (except one) of hexagonal faces. It should come as no surprise that other closed shapes are also governed by simple rules. We present here rules for generating closed shapes from the three regular tilings. A regular tiling uses identical regular polygons to cover a plane without gaps or overlaps. Such tilings can only be made from equilateral triangles (T’s), squares (S’s), or hexagons (H’s).
P P
P
P
P
P
A P
P
P
P
P
P
B
Rules for Generating Closed Shapes from a Hexagonal Tiling The defining rule for fullerenes can be expressed graphically by replacing exactly 12 H’s in a hexagonal tiling with pentagons (P’s) (5–7 ). Figure 1 shows how this idea can be used to generate the fullerene C60, which has the topology of a truncated icosahedron. Figure 1A shows the generative portion of a hexagonal tiling. Figure 1B shows the tiling with 12 of the H’s replaced by P’s, and also shows which edges must be fused to close the shape. Figure 1C shows the completed [12P:20H]. Any fullerene can, in principle, be generated in this way, reflecting the geometrical composition of the fullerenes, 12P:nH (n = 0, 2, 3, 4, …), and their general chemical formula C20+2n. The fullerene rules involving 12 P’s are only the tip of the iceberg. One can also obtain closure from a hexagonal tiling by replacing six H’s with S’s. The cube [6S:0H], the hexagonal prism [6S:2H], and the truncated octahedron [6S:8H] (demonstrated in Fig. 2A) are examples. The geometrical compositions of this series of structures are 6S:nH (n = 0, 2, 3, 4, …). A hexagonal tiling may likewise be converted into a closed shape by replacing four H’s with T’s. The tetrahedron [4T:0H] and the truncated tetrahedron [4T:4H] (demonstrated in Fig. 2B) are familiar examples. The compositions of this series of structures are 4T:nH (n = 0, 2, 3, 4, …). Closure from a hexagonal tiling is also possible using an assortment of P’s, S’s, and T’s. The possible combinations can be ascertained by using the following hexagonal-tiling point system: each T is worth 3 points; each S, 2 points; each P, 1 point; and each H, 0 points. Closure can be obtained when the sum of the points is exactly 12. Examples of structures
C
[12P:20H] C60
Figure 1. A: Portion of a hexagonal tiling used to generate the [12P:20H] shape. Lettered H’s will be changed to P’s. B: Pattern of polygons needed to generate the [12P:20H] shape. Shaded P’s were changed from H’s. Arcs show which edges will be joined. C: The closed [12P:20H] shape, shown as the structure of C60.
generated from a hexagonal tiling using mixed polygons are [2T:2S:2P:5H], shown in Figure 2C, and [1T:3S:3P:3H], shown in Figure 2D. Table 1 lists some compositions derived using the hexagonal-tiling point system. It contains the first four compositions of eight series of structures, with point groups given for each structure. Series 1 to 3 represent replacement of H’s with only one type of smaller polygon (P, S, or T), as discussed above. Note that in the fullerenes (series 1) the number of H’s can vary greatly (since they contribute zero points), reflecting the huge variety of fullerenes. In series 4 to 8, H’s are replaced by mixtures of the smaller polygons. In Table 1, all of the closed structures have three polygons meeting at each vertex (X3); in fact, only X3 vertices can be generated from a hexagonal tiling. Even so, it is possible to expand the hexagonal-tiling point system to include vertices at which more than three polygons meet, by inspection of familiar closed shapes. A listing of points for the hexagonal tiling, including points for four-polygon vertices (X4), fivepolygon vertices (X5), and six-polygon vertices (X6), is given
JChemEd.chem.wisc.edu • Vol. 79 No. 8 August 2002 • Journal of Chemical Education
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Research: Science and Education Table 1. Combinations of Triangles (T’s), Squares (S’s), and Pentagons (P’s) That Generate Closed Shapes from a Tiling of Hexagons (H’s), When Three Polygons Meet at Each Vertex (X3) No. T
S
P
H
X3
Point Group
1a
–
–
12
–
20
1b
–
–
12
2
1c
–
–
12
1d
–
–
12
2a
–
6
2b
–
2c
–
Table 2. Point System Recipes for Obtaining Closed Shapes from the Three Regular Tilings Tiling
Source of Points
Hexagonal
Tetragonal
Trigonal
3
1
0
No. T
S
P
H
X3
Point Group
Ih
5a
–
3
6
–
14
D3h
Square (S)
2
0
᎑2
24
D6d
5b
–
3
6
2
18
C2
Pentagon (P)
1
᎑1
᎑4
3
26
D3h
5c
–
3
6
3
20
D3h
4
28
Td
5d
–
3
6
4
22
C2
Hexagon (H)
0
᎑2
᎑6
–
–
8
Oh
6a
–
4
4
–
12
D2d
X3 vertex
0
1
3
6
–
2
12
D6h
6b
–
4
4
1
14
C2v
6
–
3
14
D3h
6c
–
4
4
2
16
C2
Triangle (T)
4
X vertex
᎑2
0
2
X5 vertex
᎑4
᎑1
1
᎑6
᎑2
0
12
8
12
2d
–
6
–
4
16
D2d
6d
–
4
4
3
18
C2v
X6 vertex
3a
4
–
–
–
4
Td
7a
1
3
3
–
10
C3v
Total points needed
3b
4
–
–
2
8
D2h
7b
1
3
3
1
12
Cs
3c
4
–
–
4
12
Td
7c
1
3
3
2
14
C1
3d
4
–
–
6
16
Td
7d
1
3
3
3
16
C3v
4a
–
2
8
–
16
D4d
8a
2
2
2
–
8
C2v
4b
–
2
8
1
18
C2v
8b
2
2
2
1
10
C2
4c
–
2
8
2
20
C2v
8c
2
2
2
2
12
C2v
4d
–
2
8
3
22
Cs
8d
2
2
2
3
14
C2
NOTE: Polygon point values sum to 12 in each case.
in the first column of Table 2. The simplest example of a closed structure with mixed vertices is the square pyramid, point group C4v. The composition is [4T:1S:4X3:1X4] with points +12, +2, 0, ᎑2, for a total of 12 points.
A
S
S
C
D
T
T
S
S T
S
There are similar rules for making closed shapes from a tetragonal (square) tiling. Three familiar polyhedra can be generated from a tetragonal tiling by replacing eight S’s with T ’s: the octahedron [8T:6X 4 ], the square antiprism [8T:2S:8X4], and the cuboctahedron [8T:6S:12X4] (illustrated in Fig. 3A). The polygon constituting the generative tiling (S in this case) will contribute 0 points, as does the tiling’s natural vertex (X4). Assigning each T one point, the point total for each of the three polyhedra above is eight.
B T
S
Rules for Generating Closed Shapes from a Tetragonal Tiling
S
T
P
P
P S
S
S
T
S
P
P
T
.
[6S:8H ] C24H24
[4T:4H] C12H12
.
[2T:2S:2P:5H] C18H18
[1T:3S:3P:3H] C16H16
Figure 2. Four closed shapes generated from hexagonal tilings. On the top are the portions of hexagonal tiling used to generate the shapes. Lettered H’s will be changed to the indicated polygons. In the middle are the patterns of polygons needed to generate the shapes. Shaded polygons were changed from H’s. Arcs show which edges will be joined. On the bottom are the closed shapes, represented by the carbon skeletons of the corresponding hydrocarbons, energy-minimized by MM2 using the CS Chem-3D program. A: [6S:8H]; B: [4T:4H]; C: [2T:2S:2P:5H]; D: [1T:3S:3P:3H].
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Research: Science and Education A
B
T T
T T
Table 3. Combinations of Mixed Polygons and Mixed Vertices That Generate Closed Shapes from a Tetragonal Tiling Original
T T
T
No.
X
X
–
–
6
D3h
–
1
6
C2v
–
–
1
4
2
–
–
2
4
3
–
–
6
4
4
–
7
4
4
8
4
9
[20T]
Figure 3. Closed shapes generated from (A) tetragonal and (B) trigonal tilings. On the top of A are the portions of tetragonal tiling used to generate [8T:6S]. The lettered S’s will be changed to T’s. In the middle is the pattern of polygons needed to generate the shape. Shaded T’s were changed from S’s. Arcs show which edges will be joined. On the bottom is the closed shape, a cuboctahedron. In B, no polygon changes are needed to close to the icosahedron, [20T].
It is also possible to reduce vertices from X4 to X3. From the cube [6S:8X3], we induce that each X3 vertex is worth one point. Other point values, similarly obtained by inspection of familiar shapes, are given in the second column of Table 2. Table 3 lists, in the column labeled “original”, some compositions derived from the tetragonal-tiling point system. Included are structures with mixtures of polygons (T’s and S’s) and vertices. The column labeled “pseudo-dual” will be discussed below.
5
H
X4
X3
–
–
3
2
–
–
4
2
1
–
–
1
2
4
2
–
–
2
4
C3v
4
3
–
–
3
4
4
D2h
4
4
–
–
4
4
4
4
D2d
4
4
–
–
4
4
–
4
4
Cs
4
4
–
–
4
4
–
1
2
3
Cs
3
2
1
–
1
6
2
–
–
5
2
C2v
2
5
–
–
2
6
6
2
–
2
1
4
C2
4
1
2
–
2
6
12
6
2
1
–
2
4
C2v
4
2
–
1
2
6
13
6
3
–
–
6
2
D3h
2
6
–
–
3
6
14
6
3
–
–
6
2
C2
2
6
–
–
3
6
15
6
3
–
1
4
3
Cs
3
4
1
–
3
6
16
6
3
–
2
2
4
Cs
4
2
2
–
3
6
17
6
3
–
2
2
4
C2
4
2
2
–
3
6
18
6
3
–
3
–
5
C3v
5
–
3
–
3
6
19
6
3
1
–
3
4
C3v
4
3
–
1
3
6
20
6
4
1
–
4
4
C2v
4
4
–
1
4
6
T
S
X
X
1
2
3
–
2
2
4
–
3
4
1
4
4
5
4
3
S
P
6
–
6
1
C4v
4
4
D2h
3
4
–
4
–
–
4
–
6
1
10
6
11
T
[8T:6S]
Pseudo-Dual Point Group
6
T
NOTE: Polygon plus vertex point values sum to 8 in each case.
Table 4. Combinations of Triangles and Mixed Vertices That Generate Closed Shapes from a Trigonal Tiling No.
T
X7
X6
X5
1
4
–
–
2
6
–
–
3
8
–
Rules for Generating Closed Shapes from a Trigonal Tiling
4
8
–
5
10
6
10
Finally, one can also generate closed shapes from a trigonal tiling. Since there are no smaller polygons than T’s, we obtain only vertex rules from this tiling. From inspection of the icosahedron [20T:12X5] (illustrated in Fig. 3B), the octahedron [8T:6X4], the tetrahedron [4T:4X3], and other familiar shapes, and with the generative polygon (T) and vertex (X6) each contributing zero points, we can induce the point values given in the third column of Table 2. The total number of points needed for closure is again 12. A list of some compositions consisting of T’s and mixed vertices is given in Table 4. These structures are listed in order of increasing numbers of T’s (only even numbers are allowed). Structure 1 is the tetrahedron; structure 2 is the trigonal bipyramid. There are two ways to assemble eight T’s, and the number of possible closed shapes grows quickly with the number of T’s.
7
Point Group
X4
X3
–
–
4
Td
–
3
2
D3h
–
–
6
–
Oh
–
2
2
2
C2v
–
–
2
5
–
D5h
–
–
3
3
1
C3v
10
–
1
3
–
3
C3v
8
10
–
2
–
3
2
C2v C2
9
10
–
1
2
2
2
10
12
–
4
–
–
4
Td
11
12
–
–
6
–
2
D3d
12
12
–
–
4
4
–
D2d
13
12
–
2
2
2
2
C2v
14
12
–
1
4
1
2
C2
15
12
–
2
2
2
2
C2
16
12
–
1
3
3
1
Cs
17
12
–
2
1
4
1
Cs
18
12
–
3
1
1
3
Cs
19
12
1
–
3
2
2
Cs
20
12
1
1
1
3
2
C1
21
12
1
1
2
1
3
C1
NOTE: Vertex point values sum to 12 in each case.
Small Clusters of Identical Spherical Atoms The topologies of small clusters of identical spherical atoms provide many examples of shapes whose faces are all equilateral T’s, of the sort listed in Table 4. For simple clusters
of up to 12 atoms, the closed-shape vertices represent the locations and numbers of the atoms. For clusters of 13 or more atoms, there exists the possibility of internal atoms that do not correspond to any vertices. The structure’s edges
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Research: Science and Education A
B
C
Figure 4. Dual relationships. A: Dual hexagonal (solid) and trigonal (dashed) tilings. B: Cube generated as the dual of an octahedron. C: Octahedron generated as the dual of a cube.
A pseudo-dual [pd] [14T:3X4:6X5]
B original [o]
[3S:6P:14X3]
C original - pseudo-dual
[o-pd] [14T:3S:6P:21X4]
D snub-[o-pd]
[14T:42T':3S:6P:42X5]
E truncated-[pd]
[3S:6P:14H:42X3]
F truncated-[o]
[14T:3O:6D:42X3]
G small rhombi-[o-pd]
[14T:3S:21S':6P:42X4]
H great rhombi-[o-pd]
[21S:14H:3O:6D:84X3]
Figure 5. The evolution of closed structures from [3S:6P:14X3]. A: The [pd] structure [14T:3X4:6X5]. B: The [o] structure [3S:6P:14X3]. C: The [o-pd] structure [14T:3S:6P:21X4]. D: The snub structure [14T:42T′: 3S:6P:42X5]. E: The truncated [pd] structure [3S:6P:14H:42X3]. F: The truncated [o] structure [14T:3O:6D:42X3]. G: The small rhombistructure [14T:3S:21S′:6P:42X4]. H: The great rhombi-structure [21S: 14H:3O:6D:84X3]. The same structures are shown in different orientations in Figure 6.
1020
correspond to places where the atoms touch—that is, the connectivities (or bonds, if the connectivities are localized) between atoms. Up to clusters of six atoms, the number of edges equals the number of bonds. For clusters of seven atoms or more, there exists the possibility of internal bonds. The simplest example is structure 5 in Table 4. This seven-atom cluster has 15 bonds corresponding to structure edges plus one internal bond slightly longer than the others. At low temperatures, the structure with lowest energy—that is, with the most bonds—is the most probable; at higher temperatures, the principle of maximum symmetry (8) suggests that the structure with the highest order of symmetry is the most probable. Dual Tilings and Pseudo-Dual Structures The hexagonal tiling and the trigonal tiling are dual tilings, which means that each of them may be formed from the other by connecting its adjacent polygon centers with line segments, as shown in Figure 4A. The count of polygons in one tiling will therefore equal the count of vertices in its dual. The tetragonal tiling is its own dual. The dualities of the tilings are reflected in the pointvalue recipes of Table 2. The hexagonal-tiling point values for polygons (3, 2, 1, 0 for T, S, P, H, respectively) are the same as the trigonal-tiling point values for vertices (X3, X4, X5, X6). Conversely, the trigonal-tiling point values for polygons (0, ᎑2, ᎑4, ᎑6 for T, S, P, H) match the hexagonal-tiling point values for vertices (X3, X4, X5, X6). This duality relationship can be extended to structures generated from the recipes of these two tilings. Although the concept of duality for polyhedra in general seems to be evasive (2, 9), it is possible to define a pseudo-dual structure in the following way: a pseudo-dual structure has each Xm vertex of the original structure changed to an m-sided face, and each n-sided face of the original structure changed to an Xn vertex; that is, T↔X3, S↔X4, P↔X5, and H↔X6. If Tables 1 and 3 were complete (both would be infinitely large), the pseudo-dual structure for every structure in Table 1 could be found in Table 3, and vice versa. The beauty of this relationship is that pseudo-dual pairs have the same point group symmetry. As an example, consider structure 5a in Table 1, [3S:6P:14X3], point group D3h. Its pseudo-dual [14T:3X4:6X5] (which would appear in an extended Table 3) is also point group D3h. These two structures are shown as Figure 5B and 5A, respectively. The pseudoduality relationship is like receiving a gift. If you know the composition and point group of one closed structure, then you also know the composition and point group of another closed shape, its pseudodual. Table 3 lists the “original” shapes generated from a tetragonal tiling, along with their pseudo-duals. Because the tetragonal-tiling point values for polygons (T, S, P, H) and vertices (X3, X4, X5, X6) are the same (Table 2), those original structures in Table 3 that have the same number of S’s as X4’s and the same number of T’s as X3’s will be their own pseudo-duals (structures 3–8). Euler’s Theorem The numbers of faces (F ), vertices (V ), and edges (E ) of a closed shape are restrained by Euler’s theorem (2):
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F+V=E+2
(1)
This means that two pseudo-duals A and B must have the same number of edges, since FA = VB and VA = FB. Furthermore, E and V can both be calculated for any closed shape if the numbers of component T’s, S’s, P’s, and H’s are known. By definition, F=T+S+P+H (2) (where T is the number of T’s, S is the number of S’s, etc.). Each edge of each polygon comprises 1/2 of an edge in the closed shape; hence E = (3/2)T + 2S + (5/2)P + 3H
(3)
Using Euler’s theorem (by substituting eq 2 and eq 3 into eq 1) gives (T + S + P + H ) + V = (3/2)T + 2S + (5/2)P + 3H + 2 so that V = (1/2)T + S + (3/2)P + 2H + 2
(4)
Having access to V for a set of faces can help to determine the possible closed shapes that can be formed. For example, consider a structure consisting of 3S and 6P. From eq 4, V = (1/2)(0) + 3 + (3/2)(6) + 2(0) + 2 = 14 But V is also restricted by the tiling recipes in Table 2. The hexagonal tiling recipe is 3T + 2S + 1P + 0H + 0X 3 – 2X 4 – 4X 5 – 6X 6 = 12 (where X 3 is the number of X3 vertices, X 4 is the number of X4 vertices, etc.). For [3S:6P], 3(0) + 2(3) + 1(6) + 0(0) + 0X 3 – 2X 4 – 4X 5 – 6X 6 = 12 so that ᎑2X 4 – 4X 5 – 6X 6 = 0, which can only be true if X 4 = X 5 = X 6 = 0. By default, all 14 vertices of [3S:6P] must therefore be X3; in this example, only one set of vertices is possible. The resulting structure [3S:6P:14X3] is entry 5a in Table 1. The Evolution of the Five Groups of Archimedean Solids from the Platonic Solids and the Parallel Behavior of Closed Structures in General Consider the five regular polyhedra (the Platonic solids). The octahedron and the cube are dual polyhedra (Fig. 4B,C), as are the icosahedron and dodecahedron. (The tetrahedron is its own dual.) Combining the faces of such a dual pair, with no polygons of the same type sharing edges, forms the quasi-regular polyhedra (2), with all vertices X4: octahedron [8T:6X4] + cube [6S:8X3] → cuboctahedron [8T:6S:12X4] (Fig. 3A) icosahedron [20T:12X5] + dodecahedron [12P:20X3] → icosidodecahedron [20T:12P:30X4] Furthermore, each of these trios of polyhedra shares the same point group: those in the first group are all Oh, and those in the second group are all Ih. The behavior of pseudo-dual pairs for closed shapes in general is analogous to this dual-pair behavior: original [o] + pseudo-dual [pd] → original-pseudo-dual combination [o-pd]
All vertices of the [o-pd] structure will be X4, and all three structures will belong to the same point group. When combining an [o] structure with its [pd], there is almost always more than one way to assemble the shape. However, in obeying the rule that [o] edge may connect only to [pd] edge, the number of ways is greatly reduced. At least one of these ways produces an [o-pd] that has the same point group symmetry as the [o] and [pd]. Combining the previous example’s [o] structure [3S:6P:14X3] with its [pd] [14T:3X4:6X5] gives the [o-pd] structure [14T:3S:6P:21X4] (Fig. 5C). (The number of X4 vertices can be determined by totaling the corners of all polygons and dividing by four.) All three structures belong to point group D3h. The quasi-regular polyhedra comprise one of the five groups of Archimedean solids (2), which are generated by expanding the Platonic solids. The other four are the truncated polyhedra, the small rhombi-polyhedra, the great rhombi-polyhedra, and the snub polyhedra (10; 11, pp 51–65). In general, for any [o] closed structure whose faces are regular polygons, there exist a [pd] and six larger structures belonging to these five groups. (The snub structures have no mirror planes and occur in enantiomorphic pairs.) All of these structures (except the snub structures) will have the same point group symmetry as the [o] structure. To generate a truncated structure, all of a structure’s polygon faces are replaced by polygons with twice the number of edges (T→H, S→O = octagon, P→D = decagon, etc.) and the resulting truncated vertices are capped with appropriate polygons (X3 →T, X4 →S, etc.). The number of vertices for truncated structures (all X3) is always twice the number of vertices for the corresponding [o-pd] structure. As an example, the [o] structure [3S:6P:14X3] (Fig. 5B), when truncated, becomes [14T:3O:6D:42X3] (Fig. 5F). In the same manner, its [pd] structure [14T:3X4:6X5] (Fig. 5A), when truncated, becomes [3S:6P:14H:42X 3 ] (Fig. 5E). The [o-pd] is [14T:3S:6P:21X4] (Fig. 5C). All five of these structures have D3h symmetry. In a small rhombi-structure, the faces of an [o-pd] structure are separated by S’s. Since the edges of the polygons making up the [o-pd] structure only touch edges of the S’s, and vice versa, the number of new S’s can be determined by summing the edges of all polygons in the [o-pd] and dividing by four. Small rhombi-structures have the same number of vertices as their truncated-[o] or truncated-[pd], but the vertices are X4. In our example, the [o-pd] [14T:3S:6P:21X4] (Fig. 5C) becomes [14T:3S:21S′:6P:42X4] (Fig. 5G), where S′ represents the new squares added. The great rhombi-structures have the same number of faces as the small rhombi-structures, but twice as many vertices. All of the faces of the [o-pd] structure are replaced with polygons having twice the number of edges, before being separated by S’s. All vertices of the great rhombi-structures are X3. In our example, the [o-pd] structure [14T:3S:6P:21X4] (Fig. 5C) becomes [21S:14H:3O:6D:84X3] (Fig. 5H). All of the structures derived thus far from the [o] structure [3S:6P:14X3] belong to the same point group, D3h. The snub structures are similar to the small rhombistructures, but instead of separating the faces of the [o-pd] structure with S’s, pairs of T’s (parallelograms) are used. The number of vertices (all X5) is the same as that of the small rhombi-structures. In our example, the [o-pd] structure
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Research: Science and Education
Table 5. Selected D-Symmetry Structures and Their Duals Source
[o]
[Dual]
[2S:8P:16X3]
Table 1, entry 4a
[16T:2X4:8X5]
3
Point Group D4d
Table 1, entry 5a
[3S:6P:14X ]
[14T:3X4:6X5]
D3h
Table 1, entry 6a
[4S:4P:12X3]
[12T:4X4:4X5]
D2d
Table 3, entry 1
[2T:3S:6X3]
[6T:2X3:3X4]
D3h D3h
3
4
Table 4, entry 2
[6T:2X :3X ]
[2T:3S:6X3]
Table 4, entry 5
[10T:5X4:2X5]
[5S:2P:10X3]
D5h
[4S:4P:12X3]
D2d
4
Table 4, entry 12
5
[12T:4X :4X ]
Table 6. Compositions and Point Groups of the 8 Deltahedra No. Composition
Point Group
1
[4T:4X3]
Td
2
[6T:2X3:3X4]
D3h
3
[8T:6X4]
Oh
4
5
4
[10T:5X :2X ]
D5h
5
[12T:4X4:4X5]
D2d
4
5
6
[14T:3X :6X ]
D3h
7
[16T:2X4:8X5]
D4d
8
[20T:12X5]
Ih
[14T:3S:6P:21X4] (Fig. 5C) becomes [14T:42T′:3S:6P:42X5]. One of the snub enantiomers (point group D3) is shown in Figure 5D. Curved Faces and Strain All of the structures we have described here begin with faces that are planar polygons. However, in many cases the closed shapes that are generated are strained; that is, the faces must be flexed or bent to accommodate the curvature of the geometric structure. Nature does not seem to be concerned about this, as verified by the fullerenes, many of which have strained geometrical structures. A geometrical structure may be strain free (for example, the all-T structures of Table 4) even though its [pd] is highly strained. It is also true that a highly-strained [o] structure may evolve into larger structures that have considerably less strain. The Deltahedra All of the shapes listed in Tables 1, 3, and 4 may be used to describe submicroscopic structures, and any of those shapes may evolve into additional shapes as described above. The shapes most likely to find applications in describing real molecules, ions, or clusters, however, are those having the highest symmetry. Each of the Platonic solids and Archimedean solids have icosahedral, octahedral, or tetrahedral symmetry (their point groups have multiple high-order rotation axes). The next level of high symmetry belongs to the D point groups, which also have multiple rotation axes (but not multiple high-order axes). In Tables 1, 3, and 4 are a number of structures with D symmetries. To limit the sizes of evolving structures, we 1022
Figure 6. Photographs of models showing evolution of closed structures from the deltahedron [14T:3X4:6X5]. Each structure is shown down its C3 axis and down a C2 axis. (a) The [o] structure [14T:3X4:6X5]. (b) The [pd] structure [3S:6P:14X3]. (c) The [o-pd] structure [14T:3S: 6P:21X4]. (d) The snub structure [14T:42T’:3S:6P:42X5]. (e) The truncated [o] structure [3S:6P:14H:42X3]. (f) The truncated [pd] structure [14T:3O:6D:42X 3 ]. (g) The small rhombi-structure [14T:3S: 21S’:6P:42X4]. (h) The great rhombi-structure [21S:14H:3O:6D:84X3]. The same structures are shown in different orientations in Figure 5.
choose for further examination structures consisting of T’s, S’s, and P’s only. We exclude entries 4 and 7 in Table 3 and entry 11 in Table 4 because they are severely strained, and entries 6 and 13 in Table 3 because their evolved structures are quite awkward. The remaining D-symmetry structures and their duals are collected in Table 5. The two D2d dual pairs in Table 5 are identical, as are two of the D3h dual pairs. This leaves five distinct dual pairs in Table 5. In each of these, one of the structures consists of all T’s. These five all-T structures belong to a unique mathematical group known as the deltahedra, eight convex polyhedra whose faces consist entirely of equilateral T’s (2; 11, p 67). Their point groups and compositions are given in Table 6. All of the Archimedean solids evolve from the Platonic solids, three of which (the tetrahedron, Td; the octahedron, Oh; and the icosahedron, Ih) are deltahedra. The five non-Platonic deltahedra collected in Table 5 and the structures that evolve from them all have D symmetry. Figure 6 illustrates the evolution of structures from deltahedron 6 ([14T:3X4:6X5], Fig. 6A). Note that the structures of Figure 6 are identical to those of Figure 5, which were generated from [3S:6P:14X3] (Fig. 5B), the [pd] of deltahedron 6. In Figure 6, each structure is shown down its C3 axis and down a C2 axis. Concluding Remarks We have described a way to generate closed solid shapes from planar tilings, to generate additional pseudo-dual shapes
Journal of Chemical Education • Vol. 79 No. 8 August 2002 • JChemEd.chem.wisc.edu
Research: Science and Education
from those original closed shapes, and, from these, to generate additional larger truncated, rhombi, and snub shapes. The rules we have presented provide some insight into the realm of possible geometrical shapes and a road map for identifying many of the more symmetrical ones. Many of the geometrical structures found in Tables 1, 3, and 4 have already been used to describe the structures of real molecules and clusters (12). There is little doubt that many of these shapes will become useful to molecular and macromolecular scientists in both the laboratory and the classroom. For the nonspecialist, they provide unprecedented opportunities to demonstrate mathematics– science connections as well as the hidden beauty in nature.
10.
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