An Easily Constructed Model of Twin Cubes Having a ' Common Line (or Plane) Shuklchl Y a m a n a Faculty of
General Education, Kinkl University. Kowakae. Higashi Osaka 577, Japan
A model of twin cubes having a common line (or plane) that is useful for teaching stereochemistry (especially that of inorganic compounds) can be m a d e easily b y using a sealed e m p t y envelope. T h e steps a r e illustrated i n the figure a n d given below. 1. The envelope is folded down the center lengthwise, and the middle point of the hase AB is marked as C. The center line passing the point C is marked as the (C)-line. 2. The lower part of the envelope is folded up at the A so that the corner B falls on the (C)-line, and the new corner on the righthand side of the envelope is marked as D. 3. A horizontal line DE, perpendicular to the right-hand side of the envelope at D, is drawn. The intersection of the line DE and the hase AB is marked as F, the corresponding point on the observe of the envelope being marked as G. 4. The lower part of the envelope is unfolded. 5. Two lines parallel with hoth sides of the envelope, (GIand (F), are drawn so that they pass the points G and F, respectively. 6. The intersectionof the (GI-line and the base AB is markedasH. 7. The right lower part of the envelope is folded up a t the A so that the hase AB falls on the left-hand side of the envelope. Three points corresponding to the H, F, and B, are marked as I, J , and K, respectively. 8. The lower part of the envelope is unfolded. 9. Three lines parallel with the hase AB are drawn so that they pass the points K, J, and I, respectively. 10. The intersections of the right-hand side of the envelope and each of the three parallel lines mentioned above, are marked as L, M, and N, respectively. 11. The intersections of the (G)-lineandeach of the three lines KL, JM, and IN, are marked as 0 , P , and Q, respectively. 12. The intersections of the @-line and each of the three lines KL, JM, and IN, are marked as R, S, and T , respectively. 13. The lower part of the envelope is folded up along the line KL, and the points on the obverse of the envelope, corresponding to the points J , P , S, M, Q, and T, are marked as U,V, W, X, Y, and Z, respectively. 14. Thelower oart of the envelooe is unfolded. and the uooer oart of the envelope is cut off along the zigzag course UVYZWX eomposed of the five lines UV, VY, YZ, ZW, and WX.
.. .
15. On the back of the envelope, the points corresponding to the Y, Z, V, W, 0 , R, P, S, Q, and T, are marked as Y*, Z*, V*, W*, 0*, R*, Pa,S*,Q*, and T*, respectively. 16. On the front of the envelone. the sauare YVWZ is cut off alone the line VW. 17. The envelope is folded hoth backward and forward along the lines VH (i.e., V'H), WF (i.e., W*F),V'WS, KL, JM,and IN. 18. On the front of the envelope, 12 squares UKOV, KJPO, JIQP, IAHQ, VORW, OPSR, PQTS, QHFT, WRLX, RSML, STNM, and TFBN, are marked as bz, az, d z cz, a,, b,, c,, dl, bs, as, ds, and "*, 19. On the hack of the envelope, nine squares Z*W*V*Y*, XLR*We, LMSaR*, MNT'S*, NBFT*, V*O*KU, O'P'JK, P*Q*IJ, andQ'HA1, are marked asso, hz, h ~fi,, fi, gz, g,, el, and e.. resoectivelv. 20. The e&elopeuis cut along the six lines KO (i.e., KO"), J P (i.e., JP*), IQ (i.e., IQ*), LR (i.e., LR*), MS (i.e., MS'), and N T (i.e., .uvru-"
,.
NT*!
A .
21. The points K, J, I, L, M, andN, are rewritten as (K,and Kd), (Ju and (Nu and Nd), and Jd), (Iu and Id,) (Luand Ld), (Mu and h), respectively, according to the latitude (up or down) an which they are situated. 22. The front of the envelope is cut along the lines VH and WF. The points V, 0,P , Q, W, R, S, and T , are rewritten as (VI and V,) (OI,' 01,and Or),(Pi, PI, and PA (Qi, $1, and QA (WI and W,), (RI,R.', andR,), (SI,S.',andS,),and (TI,T.', andT.1, respectively, according to the side and latitude on which they are situated. 23. The base AB is cut along the lines HA and FB. The right lower corner of the square c2 is marked as HI and the left lower corner of the sauare ca is marked as F.. 24. While kbepingthe two lines P,SI and P'S' touching each other hark to hark, the line HF is fgxced to a p p n m h the line P.SI lie.. ~''S'JSUthat the four line* (PC(),,Q.H. HO', and Q'P') iorma square. 25. Faces marked with letters of the alphabet are brought together and stacked in the order given by their numerals to produce a cube. Thus, the three faces dl, dn,and ds are piled on each other in numerical order so that the face havine the lareest numeral. d?, is outside and the others. d l and d., arc fddedynside. 2fi. Similnrlg, three face%r , , c?, and r , are pilcd rnch other in numerical order so that the c3 i3 outside, folding el and c? inside. In
Volume 64
Number 9
September 1987
761
these processes, the faces ez and fi are forced to overlao the faces el and fi, respectively. 21. In steps 25 and 26, six new peaks I, 11,111, IV, [Q'], and [T*], are produced by the conjunction of (or by only) the three points H, I,, and T:; the three points F,Qi, and Nu;the four points Qr, Jd, S,, and A; the four points TI,PI, Md, and B;the Q*; and the S*, respectively. These new peaks are fixed with tape to construct the lower half of the model given in the title. 28. Similar procedures are applied to the other squares marked with series of a, b, g, and h. This produces six mare peaks V, VI, VII, VIII, [O*], and [R*] by the conjunction of (or by only) the
four points V', V., Kd,and R,; the four points W*, WI, 01, and Ld; the five points Y*, Or, J., S,', and U;the five points Z*, RI! PI', M., and X, the 0'; and the R*, respectively, which are all fixed with tape to construct a model of twin cubes having a common line. 29. When the two cubes obtained are pushed together so that the two faces ha and c3 are forced to approach and meet, a model of twin cubes having a common plane is produced.
Reference Yarnana,S. J. ChwnEduc. 1984,61,1053.
Gt
I
I
I
H F B
K JI
,1
A
762
1
I1
H
F
B
Journal of Chemical Education
