Interconversion among Polyhedra A Novel, Dynamic, and Inexpensive Model S. K. Ramaiingam, M. Raman, and K. Paulraj Madurai Kamaraj University, Madurai-625
021, India
diagonals of the four vertical faces using the 6-cm springs. The top In p m u a n c e of our work on models1 for improving teaching face will now turn 4 5 O to the bottom one. (Note that the number of we have now devised a novel dynamic and inexpensive model edges increases from 12 to 16 in the cube. Incidentally, the number for demonstrating the ready interconvertibility of related of edges for any plantsnoid can be calculated using the formulaE = polyhedra like cube, square antiprism, triangular dodecaheF C - 2 where E is the number of edges, F is the number of faces, dron. cuboctahedron. and icosahedron. and C is the number of corners.) Now, to produce a dodecahedron ~ b soft the related polyhedra, in general, are readily conconnect the like-colored (black)halls in the two remaining faces (top vertible into one another. not onlv in terms of enerw -.but also and hottom) using short springs (Fig. 1).Alternatively, if the white by spatial rearrangements. For example, a cube can be dishalls are connected in the top and bottom faces, a dodecahedron still torted to Droduce a sauare a n t i ~ r i s mand then a trianeular results but with interchanged vertices; each white hall now has five dodecahedron. These interconvekions shown in t e ~ t b o o ~ ' ~ 3 ~ ~ are somewhat intricate. T o obviate this we have develo~ed ball-and-spring models. A student who performs these distortions on our models as directed in the experimental part will easily comprehend the otherwise complex transformations. Likewise, in the case of carboranegeometries, the 1,2- and 1,l-icosahedra are swiftly interconvertible. The generation of 1,12-isomer requires a lot of positional rearrangements. These are apparent in the model.
+
Experimental
Figure 2. (a) 1,7disubsWMed icasahedron, (b) i n t d i a t e cuboctahedmn,and
(c) 1.Zdi~~bstiMed icosahedron.
Materials A minimum of 20 wooden spheres (2.5-em d i m ) . suitablv . minted . (low white, four blark, cen red, and two green). Curtam spring ahout fi m long and 2 mm diam. This may be rut into several pieces of different lengths (40 8 rm; 40.6 rm; and a few 7 rm). Abouc 50 hooks. Procedure In each ball, drill six holes (2 mm d i m ) to a depth of5 mm, all at right angles to each other. Construct a mhe choosing four white balls and four black balls, alternately positioned, using 8-cm springs. Fix hooks in the required positions in each ball. To generate the square antiprism connect the hooks of the balls of the same color along the
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Ramalingam. S. K.. and Anandan. C. R.. J. CKM.EDUC.. 51,681 (1974);Anandan, C. R., Ganesan, L, R., and Ramalingam, S. K., Indian J. Chem. Educ., 6, 30 (1979). Cotton, F. A.. and Wilkinson. G., "Ahrand Inorganic Chemisky" 3rd ed.. Wilev Eastern. New Delhi. 1972. DO. 28. 44. and 249. ~uheey,i.E., "inciganic~hemistly,';2nd ed:,Harper & Row, New York, 1978, p. 465. Cotton, F. A,, "Chemical Applications of Group Thewy." 2nd ed., Wiley Eastern, New Delhi, 1971. p. 57.
Figure 3. A cube over a n o m cube sharing a face.
Figwe 1. Distortion of a cube (a) to a square antiprism (b)andto adodecahedrm (C).
Volume 61
Number 12 December 1984
1049
neighbors, and the black four. The process ean be r e v e d to get back the cube. To demonstrate the 1,2- and 1,l-isomerization of icosahedron construct first a cuboctahedron as follows. Use 12 balls of which ten are red and two are green. Construct a cube over another cube sharing a face (Fig. 3). The green balls occupy A and I positions. Fasten hooks a t the appropriate positions. Connect the diagonal corners as fallows: A-H, B-E, C-F, D-G, e3, F-K, G-L, and H-I (while counting the number of edges, do not count EF, FG, GH, and HE as real edges). To distort the cuboetahedron into the 1,l-isornet, conned K-I, C-A, BJ,and D-L (overlook edges E F and GH). Todistort the cuboctahedron into the 1,2-isomer, conned A-I, B-D, J-L, and K-C (as before. overlook EH and FG). 'fhr unique fraturrs d o u r models are that they are inexpenrive, easily a w m b l r d from readily available materials. m d are easily rhanged toorhrr geometries. Also the pnxesr is reversible. Such dg-
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Journal of Chemical Education
namic models have not been previously described and are superior to juxtaposing static models. These models can also be used to illustrate the transformations encountered in caged boron compounds and the like. I hear (lectures) and I forget, I see (demonstrations) and I remember. I do (models, and I underqtand (and appreciat~) -a Chinrsr proverh rparcnthrtival addir~onsare ours) Acknowledgment W e t h a n k N. R. Suhharatnam, Co-ordinator, School of Chemistry, for t h e facilities, M. Muthirulappan of USIC for workshop assistance, and S. Krishnamurthy, of Tilak Vidyalaya, Kallidaikurichi, for t h e wooden balls. One of us (K.P.) t h a n k s U.G.C. for a fellowship.
