# Page:Popular Science Monthly Volume 79.djvu/587

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THE SYMMETRIES OF CRYSTALS

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, ${\displaystyle 1:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,,\;\,1:1:\,\infty \!\!\!\!\!\!\!\!\!{\color {white}\blacksquare }\;\;\,}$

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. ${\displaystyle 111,}$${\displaystyle {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.