An Easily Constructed Model of a Square Antiprism Shukichi Yamana Faculty Of General Education, Kinki University, Kowakae, Higashi Osaka 577, Japan
A model of a square antiprism which is necessary for teachine stereochemistrv (for e x a m ~ l of e t h e octafluorotant a l a t e i&, TaFs3-)' can b e m a d e easily by using a sealed, e m p t y envelope. The s t e p s a r e illustrated in t h e figure and given below. 1) ?he envelope is folded down the center lengthwise, and the middle m i n t of the hase AR is marked as C. 2) The middle puint d t h e line CH ir n~arkcdas I ) , and a line perpcnd:culartothe bwt At3 at l i , i%drdun.'l'hislineismarked as the second center line (ii). 3) The lower part of the right half of the envelope is folded up at C so that the corner R falls on the second center line (ii) mentioned above. The new corner on the ripht-hand side of the envelope is . marked as E. 4) A horizontal line EF, perpendicular to the right-hand side of the envelope at E, is drawn so that it crosses the line CR at F, whose corresponding point on the front of the envelope is marked as G. 5) The lower part of the envelope is unfolded. 6) A lineisdrawn from the G so that it is perpendicular to the hase AB at H .
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Hoard, J. L., Martin, W. J.. Smiih, M. E.. and Whitney, J. F., J. Amer. Chem. SOC.,76,3820(1954).
7, The middle point of rhe line AH is marked na I, and a line perpendicular to the base AH at I, is drawn. This line is marked as
the third center line (iii). 8) The lower part of the envelope is folded up at H, so that the corner A falls on the third center line (iii), mentioned above. The new corner on the left-hand side of the envelope is marked as J . 9) A horizontal line, perpendicular to the left-hand side of the envelope at J , is drawn so that it crosses the line AH at K, whose corresponding point on the front of the envelope is marked as
L. 10) The lower part of the envelope is unfolded. 11) The middle point of the line J L is marked as M. 12) A line is d r a m from the M so that it is parallel with the left-hand side of the envelope. The intersection of this line and the top of the envelope is marked asN. Notice that the extension of this line passes the K on the base AB. 13) The lower part of the envelope is folded up along the line JL, and the points corresponding to the two points A and K on the obverse are marked as O and P , respectively. The position of the P i s adjusted so that the lengths of the lines L J and LP are equal to each other. 14) The lower part of the envelopeis unfolded. 15) The lower art of the envelooe is folded un at P so that the line 1'0 fallsw'the line I'N. Thr'nea corner on thr left-hand side of the e n d u p e iamarkedas ($and thc point rorrespunding ru J is marked as R.The position ofR is a d j ~ s t c dso rhat the line OR is perpendicular to the left-hand side of the envelope at Q. 16) The lower part of the envelooe is unfolded. Notice that the lengths of the two lines PJ and PR are equal to each other. 17) A line is drawn from the R so that it is parallel with the line QPand crosses the right-hand side of the envelope at S . 18) The left half of the envelope is folded rightward at R so that the line RP falls on the line RS, and the point corresponding to P is marked as T
191 The envelope is unfdded. AII The right half of thr envelope IS foldpd lrrtwanl at R i o that the line RS falls an the line RL, and the point corresponding to T is marked as 11 -.
21) The envelope is unfolded. Notice that the lengths of the three lines RP, RU, and RT are all the same. 22, A line ir drawn trom the L'so rhnt it is parallel uith the line SIf and crosses the line Rl'at
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23) The lower half of the envelope is folded up along theline QR and the points corresponding to 0, J , L , P , V , U , and T o n the obverse are marked as 0', J', L', P', V', U', and T', respectively.
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24) The lower half of the envelope is unfolded. 25) Cut off the right part of the envelope along the line TT' and its extensions. 26) The right-hand side of the envelope are taped along the cut edge to produce a new envelope. 27) Cut off the upper part of the envelope along the zigzag course J'L'P'V'U'T' composed of the five lines J'L', L'P', P'V', V'LI', and U'T'. 28) Cut off the lower part of the envelope along the zigzag course JLPVUT composed of the five lines J L , LP, PV, VU, and UT. 29) The points on the back corresponding toL',P', V', U', R, U , V, P. and L. are marked as Lr*.P'*. . . V'*.. Ur*.R*. . . U*.. V*,. P*, and L*, respectively. 30) Five regular and two scalene triangles on the front, (P'PR, RUT, PJL, U'RT', and J'P'L') and (VUR and V'RLI'), are markedas (a,, b ~bz, , f l , and f z ) and (a2 and a d , respectively.
31) Five regular and two scalene triangles on the reverse (R*TU*, P*L*J, Pf*R*P*, T'R*U'*, and L'*P'*J') and (R*U*V* and U'*R*V'*), are marked as (CI,cz, dl, el, and ez) and (dz and d d , respectively. 32) Cut the envelope along the line VR (i.e., V*R*). 33) The envelope is folded both backward and forward along all the lines forming all the regular triangles on its surfaze. 34) The right- and left-hand sides of the envelope are pushed together so that the set of four lines (PP', P'P'*, P'*P*, and P P ) and the other set of four lines (T'R, RT, TR*, and R*T') form two squares, respectively. 35) Faces marked with a letter of Ihr alphaber are t,n,uyl~r 1"grther and staikcd in the rewr.e wdrr g i v m by their numerals to produce a face of the square antiprism. Thus, two faces, a , and az, are piled on each other so that the face having the small numeral, a,, is outside and the other, a?, is folded inside. In this process, the faces bl, cl, and d l overlap the faces bz, cz, and dz, respectively. 36) In the step 35, three new peaks, I, 11, and 111are produced by the conjunction of the two points P and U, the two points T and J , and the twopointsp* and U*, respectively. Thesenew peaks are fixed with tape to construct the half of the square antiprism. Similarly, the other square antiprism moiety can h e m a d e
by the following procedure. 37) ('ut the envrlvpe along the lines V'R nnd \'"I?' :is) The faces d,. en, i?,and osare pushed inlo nmacr with rhe b a r k of the facesd, (lined with d ? l , r l , f l , s n da , dined nith "21. rcspectively. 39) In step 38, three new peaks, IV, V, and VI, are produced by the conjunction of the two points P'* and U'*;the two points T'and J': and the two mints P' and U,!. res~ectivelv.These new peaks are fixed with tape to construct thk madei mentioned k the title.
Volume 61
Number 12
December 1984
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