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Roo /, Rhombohedra Are Projected in the Same Way: All That Need Be Given Is the Angle VW/~, Which Is the Complement I /I

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Roo /, Rhombohedra Are Projected in the Same Way: All That Need Be Given Is the Angle VW/~, Which Is the Complement I /I 62 On some twins of calcite ; and on a simple method of drawing crystals of calcite and other rhombohedral crystals, and of deducing the relations of their symbols. BY W. J. LgwIs, M.A. Professor of ~[ineralogy in the University of Cambridge. [Read January 21, 1908.J 1. Having had occasion recently to draw the twins of calcite, I noticed the advantage of the twin-axes being all placed in the plane of the paper. The representation of the rhombohedron which results from this condition is so unsatisfactory that, obvious as the idea must have been to all crystal draughtsmen, few, if any, seem to have tested the utility of the projection. I desire to point out a few of its advantages. 2. The 9"hombohedron. Fig. 1 shows a rhombohedron in two positions--the upper one being , :a /•,, a plan on the central horizontal (equa- torial) plane, and the lower one an o'o', :/ ,% elevation with the principal axis VW, the polar edge V/~, and the polar diagonal W/~ all in the paper. All roo /, rhombohedra are projected in the same way: all that need be given is the angle VW/~, which is the complement I /I . of the angle made by a face with the I I, j base. Thus, for the fundamental rhom- i ~,.V" I bohedron of calcite, shown in fig. 1 and i [, by interrupted lines in figs. 2 and 8, the angle VV'g is 45 ~ 28.4'. The figure is easily drawn. A ver- tical llne being drawn for the principal axis, three equal lengths, V't', t't, and t F, are marked off on it. Through 17' V~ a second line is drawn at the known Fig. 1. angle FlPg, meeting horizontal lines through t' and t at the points /~', g. Then clearly W/x'=g'g; and A SIMPLE METHOD OF DRAWING RItOMBOHEDRAL CRY S'TALS. 63 ~t:2~'t~2tz,~. The figure is then completed by drawing the large parallelogram WIZV, and bisecting it by the median edge tZ'lZ,. The points A, /~, A" in the plan are projected in the same point in the elevation--midway between Iz and Iz'. They are important points in the geometry of rhombohedral crystals, and the distinction must be borne in mind. From the method of drawing, it is clear that the median edge of the two faces shown is in the case of every rhombohedron parallel to the paper, and is projected in a line through the centre, which may be taken as origin. After drawing the fundamental rhombohedron it is advan- tageous to draw through the median coigns tz faint vertical lines, for the similar coigns of every derived rhombohedron will lie on them, if it is drawn to the same horizontal scale. The figure being taken to represent the fundamental rhombohedron {100} of Miller----R of Naumann, the ~illerian axis OX coincides with with 0]z', and the parameter on it may be taken equal to Vtz ; whilst the axes 0 Y and OZ and the parameters on them are all projected in a line parallel, and equal, to V~,. The face (100) is projected in the diagonal F'/z; the upper parallelogram is (001), and the lower (010). Corollary 1. Now, tan VPOX = tan V'Vbt ---- /zt 2/~,t ~=Vt" Also, tan D -- tan V~,t -~ --Vt o .'. tan V'OX tan D -----2 ft,t x --Vt_-- 2. (1) Vt t~,t This is the well-known equation which gives the angle made with the principal axis by the polar edge which }rP. is taken as axis of reference, when D, the inclination of the face to the base, 7 is known. Corollary 2. {ll0}=- fig. 2. ~-. ~ je,,//., \ Each face of the rhombohedron {110} truncates a polar edge of {100} =R. One of its faces may be taken to pass through the polar edge V't2 of R, and Fig. 2. may be projected in any convenient length on this edge. If the length is taken equal to V~/2, then t' is the rower point of trisection of the length between the apices, and 0 must be the lower apex of {110}. Since, to the same horizontal scale, 64 W. J. LEWIS OI~" the length between the apices is one-half of VV', and the face per- pendicular to the paper and projected in 0/~', meets the principM axis at the lower apex, the rhornbohedron is Nau- mann's --~-R. The completion of the drawing needs no further explanation. Corollary 3. {Ill}-----2R; fig. 8. This rhombohedron has each of its polar edges truncated by a face of {100}. Hence the polar diagonals V'g, V'g,may be taken for two of the polar edges of {Ill} ; and the point t is then the uF/~er point of trisection of the length between the apices. The lower apex is at V,, where VV~=VV'. The com- pletion of the figure affords no difficulty. The Naumannian symbol is clearly --2R; Fig. 8. for, to the same horizontal scale, all lengths on the principal axis are doubled, and the faces are turned the opposite way to those of {100}. Its Millerian symbol is found from the fact that two of its faces are in a zone with (100) ; hence it is {1il} or {Ill}. Corollary'4. Similar constructions will give us further rhombohedra, p,~ the faces and edges of which ~M either truncate, or are trun- cated by, those already drawn; tile sign being Changed with each trunca- k .s'. tion, and the length inter- ~"l&h/HhO~.1. X~.,.,// eepted on the principal axis being successively halved or doubled. Thus we can draw -~>'" ~R,--~-R,&c.and4R,--8R,&e. Corollary 5. mR={hll}. ~\\'.~1" " " '- .[-~-" " J " If it is desired to draw this rhombohedron to the same scale as {100}, lengths ~w are marked off from 0 on \77K ~" both sides of the principal Fig. ~. axis to give the apices pn, lrm. The length lrmlrm is trisected at r and r', and horizontal lines are drawn through them to meet at h, h', &c. the vertical lines through A SIMPLE METSOD OF DRAWING RHOMBOHEDRAL CRYSTALS. 65 /% /~', &e. The figure is then drawn as in the eases already given. Such a figure is shown by interrupted and continuous lines kk', &c. in fig. 4. If m is negative, as for instance --~R, the points A, h p, &e. on the horizontal lines through r and r' must be taken on the opposite sides of the principal axis to those in the figure. We thus have for each value of ~n two rhombohedra with equal angles, but in which the edges and faces are turned opposite ways. They are sometimes distinguished as Tositive and negative ; and sometimes as direct and inverse; the first designations in each case indicating those rhombohedra in which the faces and edges are directed in the same way as those of R----{100}. If qb is the angle made by the faces of mR with the base, then, from fig. 4, 2 mc 2 me tanqb----_ V~r+r ~ +TA~ ---~- +~t. 2c But, from fig 1, tan D ---- Vq +/zt = --~ + Izt. .'. tan qb -- m tan D. (2) It is clear that, if m and D are known, the angle qb can be computed. The rhombohedron can then be drawn to any arbitrary scale by the same process as was followed in drawing the elevation in fig. 1. h--l We also have (see section 13, corollary 1), m ----2-------~"h + (3) 8. The scalenohedron, turn--= {hkl} ; fig. 4. When the Millerian indices are given, m and n have to be found by the relations proved in section 13 :-- O--Sk h--1 m- ~ , n_0_ak, (4) where 8----h+k+~, and k is that index which occupies the same place in the face-symbol as zero does in the symbol of that face of the prism {10I} which truncates the median edge of the face. Now the prism-face which truncates the rhombohedral edge through 0 is parallel to the paper, and therefore to the axis of X. Hence zero and therefore k occupy the first place in the symbols of the prism*face and the two scalenohedral faces which intersect in this median edge. The sign of m has to be carefully attended to; for ttle diagonal YmA lies to Y 0(J W. J. LEWIS O~~ the right or left of the principal axis according as m is positive or negative. The sign of n is indifferent, for the length nmc has to be marked off on both sides of the origin, and the apices joined indifferently to each median eoign k. The auxiliary rhombohedron mR has to be first drawn in the way described in section 2, corollary 5. The scalenohedral apices P, V,, where OV'*----nmc, are then both joined to each median coign )~ of mR. The scalenohedron is said to be direct or inverse according as m is positive or negative. It is clear that the scalenohedron --tuRn must have the same angles as tuRn; for it only differs in the points X occu- pying positions on the opposite sides of the principal axis, but each at the same distance respectively on the lines through r and r' as in the case represented in the figure. 4. The trapezohedron, a{hlk} ; fig. 5. It is clear that figures in which the horizontal planes are projected iu lines will not be satlstactory, when the base and a number of horizontal lines in, or parallel to, it have to be shown. Hence this method and :Naumann's with six equal vertical spaces --both being orthogonal projections with the axis in the paper--are not suitable for hematite, quartz, and similar crystals.
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