0. H. Freeland and R. J. O'Brien Lash Miller Chemical Laboratories University of Toronto Toronto 181, Ontario, Canada
I I
Easily Constructed Paper Stereomodels
A
system of low cost paper stereomodels for personal use by students of organic chemistry has been outlined by Larsen.' The student of coordination chemistry has a similar need, in introductory courses, for three dimensional models of the standard %olidn geometrical figures. Using these models he is more easily able to appreciate stereochemistries of metal complexes and their symmetry elements for group theoretical considerations. Yamana2 has provided such a low cost model of the tetrahedron by a simple construction from a sealed empty envelope. I n an extension of this technique, sealed, empty envelopes are found to be particularly well-suited to the construction of three-dimensional figures of the octahedron, the pentagonal bipyramid, the trigonal bipyramid, and the square pyramid. These four figures may all be reduced to tetrahedral 'L.~RsEN, G . O.,J. CHEM. EDUC.,42,274 (1965). YAM AN.^, S., J. CHISM.EDUC.,45, 245 (1968). Those not familiar with paper construction will mare readily appreciate thesignificance of using envelopes after a trial of Yamana's canstruction of the tet.rahedron.
subunits, either regular or distorted, generated from three basic rectangular envelope shapes. The steps outlining their construction are illustrated in the accompanying illustrations (Figs. 14), and are described in stepwise fashion below. Folds must he made as accurately as possible. I n the illustrations, positions are indicated with the aid of letters, a superscript prime mark indicating a position on the reverse side of the envelope which corresponds to the position on the face with the same letter of designation. Due to the low cost of materials, even for the large legal size mailing envelopes (which are particularly easily folded), construction of one or two figures by each member of a class, with an additional figure as an assignment, is a worthwhile undertaking. The figures are robust and when prepared in a large size make suitable models for classroom demonstration. Octahedron 1. The envelope is folded lengthwise down the center (Fig. I(*)). 2. Thelower part of the envelopeis folded up suoh thst corner B falla on the center line. This point is marked as E, and a line C&D, perpendicular to the center line a t B, is then drawn. This fold is then creased back and forth (Fig. l(b)). 3. Corner A is folded uo i n s like mknner and creased back and forth also.
Volume 48, Number 1 I , November 1971
/
771
Figure 1.
Construction of the odahedron.
4. The upper part of the envelope is out off slong the horizontal line. CED. The remaining portion is folded hack and forth along the lines A E and B E (Fig. I(c1). 5. The ~ o i n t P s and E %re reversed and all the previous folda are repehted, SO t h a t the envelope appears as i n Figure l(d1. 6. The envelope ia folded up such t h a t E meets F and is creased back and forth along OH. 7. Edge D B is folded in t o line E F and oreased baok and forth d o n g J K . Edge AC is folded in t o line EI" and creased hack snd forth along LM (Fig. l ( 4 ) . 8. The sealed end of the envelope is cut with aciasors slong L N and JO. 9. The figureis opened completely, i.e.. such t h a t theinterior of theenvelope is opened, and panels LNFN' and JOFO' are folded inside under F
NOTE:
Figure 3. Con~tructionof the trigonal hipyramid.
Severd of the folds described here (e.p.. AX) are not required for the construction of the octahedron. However sinoe they are used in the eame standard oonstruotion for the ~ e n t a g o n a lbipyrarnid and sinoe they lend flexibility t o the figure they sre included here. The unneoessary folds are omitted from Figure l ( 0 .
Pentagonal Bipyramid I. Stsps 1 through 7 for the octahedron are performed.
L N and JO are NOT out. 2. The figure is opened and folded i n the opposite senae t o the aothhedron, i.e., the faoea hio oh form the exterior of the oompleted octahedron, vir. NOE and NOF, are now folded inside the figure while those faces which ware folded inside the octahedron, uir. CEN hnd EOD, now become Lhe exterior faces of the figure. IIenee panel ENFO is depressed 8 s are faocs CNA and DOB (Fig. 2s)))).Paint N is then puahod t o meet point 0 , and N' is pushed t o meet A thus leaving D and B far w a r t but with panel DOBO' depressed (Fig. 2(b)). 3. A tetrahedron is made, by the method of Yamansz, of the same side length as the pentagons,l figure, i.e.. one half the wider of the envelope, and is slippedinto thevaoant position. 4. The figure is secured with t a w
Figure 4.
Construction of the square pyramid.
3. The points where these folds meet a t the edges of the envelope are msrked C and D , respectively, and the top part of the envelope is out off along CED (Fig. 3(cIl. 4. A fold is made from A t o E and from B to E and creased back and forth (Fig. 3(d)I. 5. The envelope is opened oornpletely, i.e., sueh t h a t the interior of the env e l o ~ is e opened, then A is pushed to meet B depressing the AFBF' panel (Fig. 3 (0)). 6. The figure is secured with tape.
Square Pyramid
(01
Figure 2.
( bl
Construction of the pentagon01 hipyramid.
Trigonal Bipyramid 1. The envelopeis folded down the oenter lengthwise (Fig. 3(a)). 2. The lower part of the envelope is folded up such t h a t corner B falls on the oentsr line and the fold is creased hack and forth (Fig. 3(11)). Then oorner A is folded in a like manner.
772 / Journal o f Chemical Education
1. The envelooe is folded down the oenter lenethwise and is left folded . (Fig. 4(*)). 2. The lower part of the envelope is folded uo so t h a t oorner A falls on the opposite side (Fig. 4(b)). 3. The top part of the envelope is cut off along AB and the Lower portion is kept folded ha i n steo 2. 4. Corner B is folded over t o AC sueh t h a t B X forms a perpendicular t o AC (Fig. 4(e)). 5. The point t h a t B makes with edge AC is msrked sa point N. 6. The envelopeis o ~ e n e dalongthe dihgonal BCand point Cisfoldedup the cemtezlineBCsuch t h a t theline joiningN and Nnmhkes a perpendicular to BC (Fig. 4(dll. 7. The envelope is folded back and forth along N B and NnB (Fig. 4(e)). 8. The dosed oorner of the envelope is out with scissors from Nn to some point Q part way along A B (Fig. 4(eI). 9. The envelope is opened out eornpletely and the out panels NnQB and Nn'Q'B'are folded inside the figure (Fig. 4(fl). 10. The figure is opened and point B ia poshed t o meet B' depressing panels BAB'N and BA'B'N' (Fig. 4(g)1. 11. The figureis secured with tape.
