An Easily Constructed Model of a Distorted Tetrahedron Penetrated by two Trigonal Pyramids Shukichi Yamana Department of Kyoushoku Education, Kinki University, Kowakae, Higashi Osaka 577, Japan
A model of a distorted tetrahedron penetrated by two trigonal pyramids is useful for teaching stereochemistry (especially of rotational isomers of a propane derivative). This model can be made easily by using three sealed, empty envelopes of the same shape and magnitude as each other. The principle of the method is (a) to arrange first the hase-width of an original envelope in order to make the side that corresponds t o the same kind of interatomic distance equallength even in various kinds of unit models, (b) to produce models of a distorted di-touless tetrahedron and two triaonal uvramids by envelope-folding, and (c) last to combinethemwith each other to ~ i e l dthe model desired. The steps (illustrated in the figure) are given below. ( I ) An pnvrlope is placed lenyhu,iae and thr middle point of its
left-handrrde U.i marked i.'l'he middlr point afthe line.I.?is markcd 5. Further, the middle point ol the line 45 is marked 6, and the piddle point of the line 46 is marked 7. (2) A point 8 is settled on the left-hand side of the envelope so that the length of the line 08 is equal to that of the line 47. (3) A line parallel with the line 61 is drawn so that it passes the 8 and crosses the hase 01 at 8'. (4) A line parallel with theline 68' is drawn so that it passes the 4 and rmssr? the hast. 61 at 4'. ( 5 , A vertical line perpendicular to the haw 61 at 4' is drawn so that it crosqss rhe tupot'theenvelop~32at 9. On the hack of the envelope, the point corresponding to 9 is marked 9.. (6) The !eft part of the envelope is cut off along the line 94' be., 9.4').
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Journal of Chemical Education
(7) Two lines 94' and 9*4' are brought together and taped to
produce a new envelope having 92%width of that of the original envelope. This new slender envelope is used for making a distorted di-topless tetrahedron. But all marks used above are renewed as shown in the following steps. (8) The envelope is folded down the center lengthwise. The middle
point of the hase AK is marked C, and the intersection of the center line and the top of the envelope is marked C'. (9) The middle point of the line CB is marked D, and the middle point of the line CD is marked E. (10) Two points F and G are settled on the center line so that lengths of two lines CF and CG are equal to those of two lines AE and AC, respectively. (11) A line parallel with the line FD is drawn so that it passes the G and crosses the base AB at H. (12) The middle point of the line HD is marked I. Further, the middle point of the line HI is marked J. (131 A mint K is settled on the center line so that the leneth of the l i k f ' ~is equal to~thatof the line A>. On the h k k oi t h i envelope, the points correapmuing to C' and K are marked C'* and K*, respectively. (14) The reverse (not ohverse) of the envelope is cut along three lines C'*K*, K*A, andK*B. Two points C'* and K* are rewritten as (C'*i and C*,) and (K*!,K',, and K*,) according to the sides on which they are situated. (1-left. c-entral, and r-
' Yamana. S. J. Chem. Educ. WSO, 67,1029.
v
A-plane
K% (15) The left, right, and lower parts of the reverse of the envelope are folded outward along the left- and right-hand sides of the envelooe and the base A6 to make all orieinal surfaces h.. . , originil obverse and reverse) lie on a flat Gane. This plane is Inheled -. ..- - - "A-nlnne". - - ----~(16) On the right half of the A-plane, apoint L is settled so that the length of the line KL is equal to that of the line AI and the length of the line BL is equal to that of the line BK. (17) The middle paint of the line KL is marked M. (18) The line BMis extended to a point N s o that the length of the line M N is equal to that of the line BM. (19) On the line NB, two paints 00and 01are settled so that lengths of two lines MOOand MOl are equal to the length of the line ~
.MK ....
(20) A line parallel with the line LNis drawn so that it passes the 00 and crosses the line K N at Po. (21) Four points Qo, Q,,P,, and P,* are settled on four IinesLN, LB, KB, and P C B ,respectively, so that lengths of four lines LQo, LQ,,KP,, and K',P,* are all equal to thelength ofthe line KPo. (22) On the line AB, a point R, is settled so that the Length of the line BRI is equal to that of the line CJ. (23) A lineRISi parallel with the IineBK is drawn so that it passes the RI and its length is equal to the length of the line CD. (24) The triangle AK*,B is folded up along the line AB,and the point corresponding to Si is marked Sl*. (25) The triangle AK*,B is unfolded. (26) The right part of the A-plane is folded leftward along the line BOOand its extension so that the L falls on K. On the orieinal right-hand side of the A-plane, the intersection of this sideand the line KA is marked T. The right part of the A-plane is unfolded. The right half of the A-plane is folded leftward along the line C'K*, so that the B falls on A. The points corresponding to L, T, 01, SI,PI, QI, RI, SI*, and PI* are marked L', T , 02, SZ,Pa. Q2, Rz, Sp*, and P2*. respectively. The right half of the A-plane is unfolded. The upper part of the A-plane is cut off along the zigzag course
TL'KPaOoQoLTeomposed of seven lines TL', L'K, KPo,PaOo, OoQo, QoL, and LT. (30) The rieht art of the A-olane is cut off alone the ziezae course " ' TQIOI>I&RISI*PI*composed of seven lines TQ,, Q,O,, OiP,, PIS,, SIR,,R,S,*, and SI*PI*. (31) The left part of the A-plane is cut off along the zigzag course TQ2O2P&R2S1*Pf* composed of seven lines TQa Q202. OzPz,P2Ss S2Rz,R&*, and S2*P2*. (32) T w o t r i a n g l e s L Q I T a n d T'QZL' a n d a h a r p o o n R2S9*Pz*K*,P~*S1*Rlare labeled a,, a$, and ad, respectively. (33) Two topless triangles L'QrO2PzKand PoKLQoOoare labeled bl and bl, respectively. (34) The A-plane is folded both backward and forward along the lines L'Q2, KPs, KL, KPI, LQI, and R2R1. (35) The left, right, and lower parts of the A-plane are folded along five lines L'Qz, KPZ,KPI, LQI, and R&Llso that three points L, L', and K*, are forced to approach and meet to make one new top where faces marked with letter a are brought together and stacked in the order given by their numerals. Thus, three faces or, as and ad are piled on each other so that the face having the largest numeral, aj, is outside and the others a, and asare folded inside. In this step, the Q1 is overlaid with PI*. One side (cambination of three lines LQ,, L'T, and K*,P,*) is fixed with tape. (36) Similarly, two faces br and b~arepiledoneaeh other in numerical order so thatthe b2is outside,folding bi inside. In this step, the Pzand Pa* are overlaid with the Poand Qo,respectively. (37) The sides are fixed with tape to constructa distorted di-topless tetrahedron model. This model is labeled "central unit". (38) Next, hy steps 1-25 of a previously published article,' two trigonal pyramids are produced from two sealed, empty envelopes of the same shape and magnitude as the unnarrawed one used in step 1of this article. (39) Eachof two sharp tops of these two trigonal pyramids is inserted into each of the two hales of the central unit so that each side of trigonal pyramids touches the valley (01, Si, S,', Oo, Sg,or SS*).This configuration is taped to yield themodelgivenin the title.
Volume 68
Number 6
June 1991
485
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