An Easily Constructed Icosahedron Model Shukichi Yamana' Kinki University, Kowakoe, Higashi Osaka 577, Japan Makoto Kawaguchi Ritsumeikan University, Kita-ku, Kyoto 603, Japan
A model of a n icosahedron which i s necessarv for teachine stereochemistry (especially of carhorane) can de m a d e easil; b y using a sealed, e m p t y envelope. T h e steps a r e illustrated in t h e figure a n d below. 1) The envelope is folded down the center lengthwise.
11) 12)
2) The lower part of the envelope is folded u p so that the corner B
falls on the center line. The new corner on the right-hand side of the envelope is marked as C . 3) A horizontal line CD, perpendicular to the right-hand side of the envelope a t C , is drawn and the intersection of the line CD and the base AR is marked as E whose corresponding point on the front of the envelope is marked as F. 4) The lower part of the envelope is unfolded. 5 ) Two vertical lines, perpendicular to the base AB, are drawn, so that one of them passes the F, and the other passes the E. 6 ) The intersection of the line AC and the first (or second) vertical line mentioned above is marked as G~.~(or HI. . 7) The inkrrertiun uf the rxtrnsivn uf the line EG and the left-hand side ottltr twelupa i s marked as I. 'The slupcc of rhr two liner AC and I I : will he used as thr stnndnrds for drawing the other line* i n the folloaing steps. 8) 'l'herxtrusion of thr line 1.11 is drnwnsorhat it rnmser the leftand right-hand sidrs o i t h c . m w l c ~ n~tethe twuouinrad and K. respectively. 9) The extension of the line IFis drawn so that i t crossestheseeond vertical line and the right-hand side of the envelope a t the two points L and M , respectively. 10) Two lines J P and QU are drawn so that they are parallel with the ~
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13) 14) 15) 16)
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17)
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standard line AC, and the other three lines QC, T M , and SP are drawn so that they are parallel with the other standard line IE. All the intersections are marked as shown in the figure. Cut off the upper part of the envelope along the zigzag course TRSP composed of the three lines T R , RS, and SP. Cut off the lower part of the envelope along the zigzag course IGHK composed of the three lines IG, GH, and HK Cut the remainingportion along the lines TQ, RN, S O , P M , I J , QF, HL, and KC. The two corners R and S of the upper zigzag course TRSP are rewritten as (RIandR,) and (SIand S,), respectively, according to the side on which they are situated. Similarly, the two corners G and H of the lower zigzag course IGHK are rewritten as (GI and G,) and (HI and H,), respectively. The corresponding points on the reverse, of the T , N , 0 ,P, I , F, L , K , R1, R,, S I ,S,, G I ,G,, HI,and H, are marked as T*, N*, 01, P*, I*, F*, L*, K*, R : , R i , S : , S i , G:, G i , H ; , and H i , respeetivelv. ~ w e f v regular e triangles on the front, TQRI,R,OS,, S,OP, QNRl, R,NO, OMP, JIF, FHIL, LH,C, IGIF, FG,Hl, and H,KC are , aa dl, d2, b ~h, ~g,, , g2, el. e2. and ea respecmarked as a ~as, tivalv~ .. .. .. ~welw regular trianylson the revem-. InO*S:. S:O'R:, R N P . P M O ' , U'N'R:, X'N'Q. CII:l.', 1..11:1.". l.'laJ. C K ' H : . HiGiF*, and P G i l * aremarked a8 ah as, as, bz,el, cr; f l , f2, hi; ec es, and es, respectively.
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19) The envelope is folded hoth backward and forward along all the lines on its surface. 20) The nght- and left-hand sides of the envelope are pushed together so that the SIX lines ( Q J , NF, OL, MC, O*L*, and N'F*) form
of the icosahedron. ~ h u isix , faces a ~0 ,2 , as, a*, as, and a6 are piled on each other in numerical order so that the face having the largest numeral, as, is outside and the others a,-oa are folded inside. In this process, the faces bz, ea, and d z are forced to overlap the faces b ~c,,, and d l , respectively. 221 In the step 21, three new peaks I, 11, and 111are produced by the conjunction of the five points T,SI,P, 0*,and Ri; the five points Q, R,, S,, S:,and R:; and the five points Rt, 0 ,F*,S:, and T*,
respectively. These new peaks are fixed with tape to construct the upper half of the icosahedron. 23) Similar procedures are applied to the other regular triangles marked with series of e , f , 8,and h. This produces three more peaks IV, V, and VI by the conjunction of the five points I, G,, K, H;,and F*;the five points G I , HI, C, H i , and Gj; and the five points F, H,, K*, G:, and I*, respectively,whiehare all fixed with tape to construct the lower half of the icosahedron.
References Yamana,S.,d.CHEM. EDUC.,I5.245 (1968). Ysmana, S.,and Kawaguchi, M.. Ann. Rep. Rss. Sci. Edue., Kyoto Univ. Educ., 10.15.25 (1980). Ysmsna.S.,sndKswaguehi. M.,J.CHEM. EDUC..57,134 (1980). Yamans. %and Kawaguchi, M.,J.CHRM.EDUC..(1981) 59.196.197 (1982).
Volume 59
Number 7
July 1982
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