268 Penfield — Solution of ProUems in Crystallography 



70° 32^, taken from scale J^o. 2 of the printed sheets. To find 

 the mterfacial angle from 221 : 212, make use of the great 

 circle protractor for finding the great circle between the poles, 

 construct the great circle and measure with the small circle 

 protractor as indicated on page 253. From 221 to 212 was 

 found to be 27° 35', and from ^221 to 122, 27° 15', the calcu- 

 lated distance being 27° 16^ 



The poles of the trapezohedron are found on the great circles 

 between octahedron and cube. With garnet as an example, 

 measurement may be made over an axis, and the angle is 70° 

 32', one-half of which is the angle of cube on trapezohedron. 

 The pole between 001 and 111 may then be located by means 

 of the stereographic scale, at 35° 16' from 001, and the slope 

 plotted from the base-line Oy' goes to -| on the vertical axis ; 

 the indices are therefore 112. Bj applying the great circle 

 protractor it is found that 112 is in the zone of the dodeca- 

 hedrons 101 and Oil, and this relation may be made use of for 

 locating other poles of the trapezohedron, 211 and 121. Other 

 interf acial angles may then be measured ; for example, 112 A 112 

 and 211 A2ll were found to be, respectively, 48° 30' and 48° 

 10', theory 48° 11', and 211A121 and 121A112, 33° 45' and 

 33° 15', theory 33° 34'. 



The hexoctahedron most frequently met with on garnet 

 bevels the edges of the dodecahedron, the angle between the 

 bevelling faces being 21° 47', or 10° 54' from the trapezo- 

 hedrons. To locate the pole between 101 and 112, place t_he 

 small circle protractor over the projection with its zero at 110, 

 then with dividers space off 10"" 54' from 112 and transfer to 

 the paper. A radius drawn through the point thus found in- 

 tersects the divided circle at a little over 26-|° from 100. A 

 line from 010, drawn at right angles to this radius, intersects 

 the first axis (radius) at |-, and establishes the relation ^a : a 

 on the first and second axes and also the base-line Ox. The 

 distance from the hexoctahedron to 001, measured with the 

 stereographic scale, was found to be just short of 37°, and this 

 slope plotted from x' intersects the vertical radius at ^. The 

 form is therefore ^a \ a \ -§-(/, the indices being 213. The poles 

 of tlie hexoctahedron under consideration are especially easy 

 to locate by means of zones, as indicated in figure 16, and to 

 show the accuracy w4th which the plotting was done some dis- 

 tances measured on the projection are given, as follows : 

 111A231 = 22° 5^, theory 22°'l2'; 010 A231 = 36° 45', theory 

 36° 42': 312A312-31° 10', theory 31° 00'; and 00lA231 = 

 74° 25',' theory 74° 30'. The method of finding the symbol 

 of the pole 231 {^a : a : Za) by means of the base-line Oz 

 and slope 74° 25' is indicated in figure 16, but needs no dis- 

 cussion. Attention is called to the six poles in the zone 



