German mineralogist who made important studies of crystallography. His first major publication, Index der Kristallformen (3 vol., 1886–91; “Index of Crystal Forms”), was a catalog of the known forms of crystals of all minerals. New tables of crystal angles to meet his new needs were devised and published as Krystallographische Winkeltabellen (1897; “Crystallographic Table of Angles”). The compilation of all published figures of mineral crystals appeared in Atlas der Kristallformen (9 vol., 1912–23; “Atlas of Crystal Forms”). From his interest in the number series appearing in crystal symbols, he developed a philosophic theory of number and harmony, which led to an analysis of musical harmony, colour, and the development of the colour sense in man and, finally, of the spacing of the planets about the Sun.