A Convergent Approach to Models of Trigonal Bipyramid, Cube, and Triakis Octahedron S. Y. Lam-Leung, Albert W. M. Lee, S. K. Tsang, and H. M. Wong Hong Kong Baptlst College, 224, Waterloo Road, Kowloon, Hong Kong The construction of paper models of useful molecular structures had been discussed recently in several articles ( I 5) published in the Journal. We now report a convergent approach to the construction of the model of trigonal bipyramid, cube, and triakis octahedron, all from one common basic folding unit. Our paper-folding approach requires no paper cutting or gluing in constructing of these models. The trigonal bipyramid model is useful in illustrating the structure of some carboranes, such as 1,5-dicarba-closo-pentaborane, CzB3Hs (6).The two carbon atoms are situated on the axial vertices of the bippamid, and the three boron atoms are located on the equatorial positions (Fig. 1).
Besides the crystal lattice structure of some inorganic salts, the cube model is useful in illustrating the atomic arrangement of the tetramers of alkylthiotricarbonyl manand alkylselenotricarbonyl manganese, [Mn(C0)3SR]a ganese or rhenium, [Mn(C0)3SeRI4 or [Re(CO)3SeR]a (8). The metal and nonmetal atoms are situated in the alternative corners of the cube (Fig. 2a and Fig. 2b). Finally, the triakis octahedral structure can he found in the structure of MsCla4+-likeclusters (9). The highly symmetrical triakis octahedron can be seen as an octahedron having all its eight exterior surfaces topped with a trigonal pyramid (Fig. 3a). The triply bridging halogen atoms are located on the top of the trigonal pyramids, while the six metal atoms are located a t the six vertices of the covered octahedron (Fig. 3h).
(n,
Concltructlon ot the Baslc Foldlng Unn (I) STEP 1 Starting with a piece of square paper, fold along the dotted Line with two opposite edges superimposed to each other (Fig. 4 4 . STEP 2 Fold the loose edges on top of the middle Line as shown in Fig. 4b and 4c. A four-folded rectangle with 1:4 length ratio in edges (EG':AE) is obtained. STEP 3 Three foldingtracks. two troughsand one ridge, are creat-
Flgure 1. I.SDl&a-closo-pemborane,
C2BsHr.
Flgure 2. (a)Al~imlonicaroonylmanganess,(Mn(CObSeR1. (btA kylselanonlcaroony manganese or rhenium. (M(COhSeR],.where M = Mn or Re.
Figure 3. (a) Triakls octahedron. (b) Structure of M6Cla4+. Volume 88 Number 8 August 1991
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Figure 4. Configurationsrelated to the construction of the basic folding unit. (I) Basic folding unit.
Figure 5. (a) Construction of submo6el A. (b) Top view of submodel A.
STEP 4
STEP 5
STEP 6 STEP 7
ed when the paper is opened up. Now, fold up along the dotted lines. AH and DG. as shown in Fieures 4d and 4e. Refold the &cave tracks, namely AE &d DF. The ahtained configurationis shown in Figure 4f. Fold up along the AB and CD lines. Then, place the ABFtriangle under the AHG'E-sheet and also the CDE-trianple under the DGH'F-sheet (Fig. 4g). The ohtained canfiguration is shown in Fieure 4hI old up along BC line i f the configuration obtained in step 5. Fold up upward along the CH (dotted), and fold down along BG (hidden) lines as shown in Figure 4i. A fourfolded triangle with the stacking sequence of ACH (on top), CBH, BCG, and BDG (at bottom) is obtained (Fig. 43.
Figures 4k and 41 show a side view of the loose basic folding unit (I) having four pockets BGO, CGO, CHO, and HHO, each with an opening, namely GO, CO, HO, and BO, rerrpectively. A mirror image of unit I might be obtained if the dotted lines in Figure 4d were situated on the alternative pair of corners, namely G' and H'. But in the construction of the following models, a single set of mirror images should be used. Construcllon ol Submodel A Submodel A which is the basic building block of all the three models can be constructed by joining three basic fold-
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d
Figure 7. (a) Submadel A'. (b) Submodel 8. (c) Construction of (d) cubic model.
cube model.
ing units (I) together. Tail-D of each basic folding unit is inserted to the GO-opening of the other unit in an alternative manner as shown in Figure 5a. The top view of a suhmodel A is given in Figure 5b. Condructlon of a Trlgonal Blpyramldal Model
submodel B in the manner shown in Figure 7c. Figure 7d shows the resulting cuhic model. Condructlon of a Trlakls Octahedral Modal
Submodel C is first constructed by joining three submodel A's together. One of three tail-A's of each submodel A is
Each tail-A of the submodel A is inserted into the neighboring HO-opening after changing the three BC troughs to three ridges. A trigonal bipyramidal model results (Fig. 6). Condrucllon of a Cublc Model
A cubic model can be built by using two half-cube structures, namely submodel A' (Fig. la) and submodel B (Fig. 7b). Submodel A' is resulted by flattening all the troughs of submodel A, whereas submodel B is constructed as follows: connect two basic folding units, I1 and 12,by inserting the tail-A of 1 2 into the GO-opening of I,. Now, a third basic foldingunit, 13, is introduced by inserting tail-A of IQinto the GO-opening of the I2unit and also inserting the tail-A of 4 into the GO-opening of 13. The obtained configuration is given in Figure 7(h). A cuhe model can then be built by joining all taillpocket linkages between submodel A' and
Figure 9. Triakis octahedral model.
Figure 8. (a) Construction of submodel C. (b) Top view of submodel C. (c)Top view of configuration 0.(d) Side view of configuration D. (e) Construction of triakis octahedra1model.
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inserted to the HO-opening of the other submodel A in an alternative manner as shown in Figure 8a. The top view of the resulting submodel C is shown in Figure 8b. Then the alternate tails are inserted to the neighboring pockets, as indicated by the arrows in Figure 8b. The resulting configuration D is given in Figures 8c and 8c. Finally, one additional submodel A is added to close up the open bottom of configuration D by joining all the taillpocket linkages between them (Fig. 8e). A triakis octahedral model results (Fig. 9).
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Literature Cited 1. Yamans, S.; Kawlsguchi, M. J. Chem. Educ. 1984,61,1054. 2. Yamsns.S.J. ChemEduc. 1984,bl. 1055. 3. Ysm8na.S. J.Chem.Educ. 1984,61.1058-1059. 4. Yamana, S. J Cham. Edue. 1987,64.1033-1034. 5. Yamana, S. J . Chem. Educ. 1987,64.1040. 6. Wade, K. Electron Deficient Compounds; Nelson: Landan, 1971; p 151. 7. Jahn8on.B.F. G:PoUick, P.J.:Williams, 1. 0.; Wojeicki, A.Inorg Cham. 1968, 7,831""
.
OJ*.
8. Abel, E. W.;Cross,B.C.;Hutaan,G. V.J.Chem, Soe. (A1 1967,2014-2017. 9. Cotton, F. A,: Wilkinaon. G.Aduoncrdhorgonic Chemistry. 5fhed.; Wiley: New York, 1988: p 1079.
