An Easily Constructed Model of Twin Trigonal Pyramids Penetrating Each Other
Shukichi Yamana Department of Kyoushoku Education, Kinki University, Kowakae. Higashi Osaka 577. Japan
A model of twin trigonal pyramids penetrating each other is useful for teaching stereochemistry (especially of rotational isomers of a n ethane derivative). This model can he made easily by using two sealed, empty envelopes. T h e s t e p s (illustrated i n t h e figure) a r e given below.
( 5 ) The intersection of two lines K C and FI is marked L. ( 6 ) The right lower part of the envelope is folded leftward at J s o
that the corner B falls on the lower part of the line (D). The corresponding point is marked M. (7) A horizontal line parallel with the base ilB is drawn so that it passes the M and crosses the left-hand side of the envelope a t
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(1)The first figure shows the starting configuration, which is produced hy steps 1-8 of the previous article.' However, for this model, marks of all points and lines except the two corners A and B are rewritten as shown in the figure. (2)Two horizontal lines parallel with the base AB are drawn so that oneof them passes the G and crosses the right-hand side of the envelope a t H and the other passes the F and crosses the line (C)and the right-hand side of the envelope at I and J , resoeetivelv. , (3) The lower part of the envelope ia folded up along the line GH nnd the point un the ohverse of the envelope, corresponding tu E is marked K. (4) The Lower part of the envelope is unfolded.
(8)The right lower pan irf the envelope is unfolded. (9) The lower part oftheenvelope is rut oiialong the ziglag couwe NI-MJ. comoosed of the rhree l r n e ~A'/. lhf.,and AlJ. (10)The Gwer part of the envelope is faded up at J so that the point L falls on the upper part of the line (D). The corresponding point is marked 0 . (11)The lower part of the envelope is unfolded. (12)A horizontal line parallel with the line FJ is drawn so that it passes the Oand crosses theleft-hand sideof theenvelopeatP. (13)The upper part of the envelope is cut off alone- the line PO and its extiision. (14)The right lower part of the envelope is folded leftward along the line 01. On the line PI, the interseetion of two lines, PI and OJ.. is marked 0. (15) The right lower pan of the envelope is unfold~d. (161 'The intersection oi t t e lines QO and (C)is marked R. ~~~~~
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' Yamana. S. J. Chem. Educ. 1987, 64. 761.
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Volume 67
Number 12
December 1990
1029
Next. the second envelove of the s a m e shave and m a m i t u d e a s t h e first o n e used above is employed t h r o d u c e a new tooless triaonal vvramid, as follows: (But here, all m a r k s in thk secondenveiobe are identified b y their symbol '.)
(1:)
A hurilontal linp parallel with the line F J is drawn so that it passes rhr It and crosses the right-hand ride of the e n d u p e at
S. (18) The upper part of the envelope is cut off along the zigzag course PROS, composed of the three lines PR, RO, and OS. (19) On the back of the envelope, the points corresponding t o R, 0 , I, and M, are marked R*, 0*, I*, and Me, respectively. (20) Oblique lines are drawn t o shade three harpoons (PIOR on the front; PI'O'R* on the back; and OJO'S on both sides) and two triangles (I*MaJ on the hack; and INI* on both sides). (21) The envelope is folded both backward and forward along the Lines PI (i.e., PI'), OI (i.e., O*I*), O J (i.e., 0 ' 4 , FI (i.e., FI*), IJ(i.e., I'J), and RI (i.e.,R*I*l. (22) The right- and left-hand sides of the envelope are pushed together so that the three lines ( I J ,JI*, and II*) f o n n a regular triangle. (23) The three points N, M, and M*,are pushed together and the shaded triangles (INI* and I*M*J) are folded inside. The sides are taped together to form the bottom of a trigonal pyramid
mnad ..........
(24) Inorder to make the next rtepsmooth, the reverseof the three
creased harpoons 10JO'S. U'I'PH', and PIOR) are fuldrd in n n d a r e ~ w h e dnaaimt the insidc~vnllofthemodelufatrironal pvrarnid. 125) Thrrr pointsO.O',and P,are pushed togetherand r h ~ r h n d e d crmsed harpoons. mentioned alwve, are folded inride. The sides are taped topetner to produce a rrigonnl psramrd.
1030
Journal of Chemical Education
(26) Bythe similar steps to 1-12 mentioned above, the first configuration is obtained where the intersection of the lines G'H' and (C') (or D') is marked T (or U'). (27) The middle point of the line P G ' is marked V'. (28) A horizontal line ~ e r ~ e n d i c u l taor the left-hand side of the envelope a t V' is drawn so that i t crosses the lines PI', O T , and O'J', a t W', X', and Y', respectively. (29) The upper part of the envelope is cut off along the zigzag course G'W'TX'U'Y'H' composed of the six lines G'W', W'T', TX', X'U', U'Y', and Y'H'. (30) On the back of the envelope, the points corresponding to W', T , X', U', Y', P,and M' are marked W", T * , X'", U", Y * ,P*, and M'*, respectively. (31) Oblique lines are drawn t o shade three harpoons (W'I'X'T on the front; Wr*I'*X'*T* on the back; and Y'J'Y8*H' on both sides) and two triangles (P!M'*J') on the back; and I'N'P* on aidesl~ hoih -- .... .-. . ,. (32) The envelope is folded both backward and forward along the lines W'I' (i.e., W'*Pa), X'I' (i.e. X'*P*), Y'J' (i.e., Y'*J'),PI' (i.e.,FP'), I'J' (i.e.,It*J'), and T'P(i.e., T'*I'*l. (33) The right- and left-hand sides of the envelope are pushed toeether so that the three lines (I'J', J'I'*, and I'Ir*) form a regular triangle. (34) The three points N', M', and M'*, are pushed together so that the two shaded triangles (18*M'*J' and I'N'I'*) are folded inside. The sides are taped together to form the bottom of a topless trigonal pyramid model. (35) In order t o make the next step smooth, the reverse of the three creased harpoons (Y'JrY"H', Xr*P*W'*T*, and W'I'X'T) are folded in and are pushed against the inside wall of this model of a topless trigonal pyramid. (36) Each of the three sets of corners (X' and W'; Y' and Y*; and X'* and W'*), is oushed toeether and the three shaded creased " harpoons, mentioned above, are folded inside. (37) The sides are taped together to produce a topless trigonal pyramid. (38) When the top (combination of three points O', P, and 0 ) of the trigonal pyramid obtained already by the steps (1-25), is inserted into the hole of this topless trigonal pyramid so that eachside of the former touches the valley (U', U", or G'). This configuration is taped t o produce the model given in the title.
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