# Page:Popular Science Monthly Volume 79.djvu/585

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THE SYMMETRIES OF CRYSTALS

 ADAMAS: OR THE SYMMETRIES OF ISOMETRIC CRYSTALS
By Professor 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 changeless 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-variable parameters 1 and x, viz., ${\displaystyle 1:1:1}$ for the octahedron, ${\displaystyle 1:1:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,}$ for the dodecahedron, ${\displaystyle 1:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,}$ 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 ${\displaystyle \times }$ 8-faced figure and ends as a 2 ${\displaystyle \times }$ 12-faced figure. The 4-faced cube starts as 2 ${\displaystyle \times }$ 12-faced form and ends as 4 ${\displaystyle \times }$ 6-faced form. The trapezohedren starts as a 4 ${\displaystyle \times }$ 6-faced form and ends as a 3 ${\displaystyle \times }$ 8-faced form, the three-faced octahedron with which we began. This is expressed algebraically by the three formulæ ${\displaystyle 1:1:m,\ \,1:m:m,\ \,1:m:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,}$, which have a single variable parameter, and no additional similar formulæ 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