Popular Science Monthly/Volume 79/December 1911/Adamas: Or the Symmetries of Isometric Crystals

by B. K. Emerson

Layout 4

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., $1:1:1$ for the octahedron, $1:1:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,$ for the dodecahedron, $1:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,$ 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 $\times$ 8-faced figure and ends as a 2 $\times$ 12-faced figure. The 4-faced cube starts as 2 $\times$ 12-faced form and ends as 4 $\times$ 6-faced form. The trapezohedren starts as a 4 $\times$ 6-faced form and ends as a 3 $\times$ 8-faced form, the three-faced octahedron with which we began. This is expressed algebraically by the three formulæ $1:1:m,\ \,1:m:m,\ \,1:m:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,$, 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 g on the diagram. As the space has two dimensions this has two variables, 1:n:m. The first group at the corners of the triangle had faces with like sides, the second at the sides of the triangle had faces with two like sides, this third and last group has faces with unlike sides; they are scaline triangles. They have forty-eight sides, a number that can factor into 24, 12, 8, 6, 4 and 3. As they are arranged over the surface of the triangle, these nearest the octahedron are six-faced octahedrons in appearance, those nearest the cube are eight-faced cubes, those nearest the trapezohedrons are two-faced trapezohedrons, and so on.

For many years I have shown this symmetrical passage of these seven forms into each other by using three colors, red for the octahedral lines, blue for the dodecahedral, and green for the cubical, a device my old pupil, Geo. H. Williams, used in his "Elements of Crystallography." The upper corner is all octahedral, the middle horizontal band is half so r the base not at all octahedral, and so of the other corners symmetrically.

The law of symmetry permits any symmetrical half of these faces to appear independently on the crystal, and the crystal fulfils the law of symmetry, and this may be done in three ways. (1) We may take all the faces in half the octants, or half the faces in each octant, and in the second case we may begin in the second octant (2) with the face adjacent to the initial face, or (3) with a face not adjacent. In accordance with the first law the half of the faces of the octahedron forms the tetrahedron which we naturally place in the figure, as (h) directly beneath the octahedron from which it is derived.

In the same way the tetragonal dodecahedron (i), the half form of the three-faced octahedron, and the trigonal dodecahedron (j), that of

the trapezahedron, are placed beside their parent forms, and the hexatetrahedron (k), the half form of the hexoctahedron, is placed necessarily above the latter, from its relation to the tetrahedron.

In accord with the second law, the four-faced cube gives rise to the pentagonal dodecahedron (l) which is placed above it, and the central figure, the hexoctahedron (g) gives rise to the diploid (n) which is placed naturally just above its associate the pentagonal dodecahedron.

There remains the single gyroidal form (n) obtained by the third law, which is placed directly beneath the central figure (g) from which it was derived.

An inspection of the figure will show that the triangle with which we began, the mason's symbol of the trinity, has most naturally developed itself into the form of a cross. Isolated on either side stand the cube and dodecahedron, two unique forms not capable of change or conversion into any other form, like the two thieves beside the cross. But said one of my friends, who is a good crystallographer, as I called her attention to this similitude, one of the thieves was converted.

This would seem to throw doubt on the record, I replied, and yet there are infinite possibilities present, as one sees, in the formulæ, $1:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,,\;\,1:1:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,$

One observes next that the five Platonic forms find symmetrical place on the figure: two at the top, two at the lower corners and one—the icosahedron—by evenly balanced combination of the top and bottom of the figure. $111,$${12x} \over {2}$.

The cross may be a cross of gold or of any other of the noble metals, and an inspection of the figure shows further that it culminates in an upper triangle placed like a crescent above the cross which contains the perfect forms attained by the perfect mineral, the diamond. At the center of this triangle is the tetrahedron (h) which gives the model of the atom of carbon and the hexatetradon (k) the most typical form of the diamond itself.

So again in a new arrangement of the elements in accord with the periodic law, proposed by the writer,^[1] carbon is the culmination of the first octave and the very center and omphalos around which all the elements circle in their grand evolution. It has four-fold valence and threefold allotrophism and stands as the center of the seven elements of the first octant. And as the diamond is brought down from the heavens in the meteorites and brought up from the depths of the earth with the deepest rocks, and as it is endowed with the greatest power over light and over all solid bodies, so it presents in its almost spherical hexatetrahedron a mean around which the earth seems many times to have oscillated, as Arldt has shown,^[2] now varying slightly toward the tetrahedron; now almost recovering again the spheroidal form.

1. "Helix Chemica," Am. Chem. Jour., Vol. XLV., p. 160, 1911.
2. Dr. Theodor Arldt, "Die Entwickelung der Kontinente," Leipzig, 1907.