Note 490 — Chapter IX · ON CONCRETIONS, SPICULES, AND SPICULAR SKELETONS (continued 2)

If the equation of any plane face of a crystal be written in the form h x + k y + l z = 1, then h, k, l are the indices of which we are speaking. They are the reciprocals of the parameters, or reciprocals of the distances from the origin at which the plane meets the several axes. In the case of the regular or pentagonal dodecahedron these indices are 2, 1 + √5, 0. Kepler described as follows, briefly but adequately, the common char­ac­teris­tics of the dodecahedron and icosahedron: “Duo sunt corpora regularia, dodecaedron et icosaedron, quorum illud quinquangulis figuratur expresse, hoc triangulis quidem sed in quinquanguli formam coaptatis. Utriusque horum corporum ipsiusque adeo quinquanguli structura perfici non potest sine proportione illa, quam hodierni geometrae divinam appellant” (De nive sexangula (1611), Opera, ed. Frisch, VII, p. 723). Here Kepler was dealing, somewhat after the manner of Sir Thomas Browne, with the mysteries of the quincunx, and also of the hexagon; and was seeking for an explanation of the mysterious or even mystical beauty of the 5-petalled or 3-petalled flower,—pulchritudinis aut proprietatis figurae, quae animam harum plantarum characterisavit.