THE SYMMETRIES OF CRYSTALS 581 



ADAMAS: OR THE SYMMETRIES OF 

 ISOMETRIC CRYSTALS 



By Trofessor B. K. EMERSON 



AMHERST COLLEGE 



THE number 3, the first to have a beginning, middle and end, has 

 always been sacred. We are all trinitarians. Four is the second 

 prominent number. It is the first square. The strong man stands four 

 square to all the winds of fortune. The combination of these in the 

 number 7 has always had a peculiar mythical significance. 



The triangle with the eye in its center is the symbol of Freemasonry. 

 And we may see how far this triangle will symbolize the three change- 

 less and four variable solids which together constitute the seven crystal 

 forms of the isometric system. 



The triangle has three points which are unique and three and only 

 three unique forms — the octahedron, cube and dodecahedron find place 

 in the three corners of this triangle at a, b and c (as shown in the plate), 

 forms made of eight triangles : of six squares, or twelve diamonds, and 

 these numbers are twice the number three, or twice the number four or 

 the product of three and four. 



We may refer the planes of all crystals to three equal rectangular 

 axes and only three permutations can be made from the only non-vari- 

 able parameters 1 and x, viz., 1 : 1 : 1 for the octahedron, 1 : 1 : » for the 

 •dodecahedron, 1 : » : vo for the cube — and so we reach the same result 

 •algebraically. 



The three corners of the triangle are joined by three lines, each line 

 made of a series of points which should symbolize each a linear series of 

 ■cognate forms, and we have these three forms in the trapezohedron d, 

 the three-faced octahedron e, and the four-faced cube f, each linking by 

 a single unbroken series the three corner forms. These are placed each 

 on its proper line on the diagram. They are each twenty-four-sided 

 figures. In two, each side is an isosceles triangle, in one, a trapezium, a 

 combination of two isosceles triangles. 



The three-faced octahedron starts as a 3 X 8-faced figure and ends 

 tts a 2 X 12-faced figure. The 4-faced cube starts as 2 X 12-faced 

 form and ends as 4 X 6-faced form. The trapezohedren starts as a 

 -1 X 6-faced form and ends as a 3 X 8-faced form, the three-faced 

 octahedron with which we began. This is expressed algebraically by the 

 three formula? 1:1 \m, l:m:m, l:m:=°, which have a single variable 

 parameter, and no additional similar formulae can exist. 



There remains the space of the triangle made up of points arranged 

 in two dimensions, or in lines connecting any of the forms represented 

 'by the previous positions or formulas with the center of the triangle. 



There is left a group of forms of a single type to occupy each point 

 •of this surface — the hexoctahedrons, and a sample of these is numbered 



