Proceedings of the American Academy of Arts and Sciences. VOL. 61. No. 7. JUNE, 1926. A MATHEMATICAL STUDY OF CRYSTAL SYMMETRY By AUSTIN F. ROGERS A MATHEMATICAL STUDY OF CRYSTAL SYMMETRY. BY AUSTIN F. ROGERS. CONTENTS Introduction.. Symmetry Operations. Symmetry Elements. List of Operations Corresponding to Faces of the General Form.. Group Theory and Crystal Systems. Table of the 32 Crystal Classes with Subgroups. Summary Acknowledgments. PAGE 161 162 170 173 174 178 198 198 200 201 203 INTRODUCTION. The modern classification of crystals is based primarily upon symmetry. The types of symmetry possible in crystals are now firmly established. As early as 1830, Hessel predicted the existence of the 32 crystal classes when representatives of only 17 of the 32 were known. Hessel's work was overlooked for many years, but was confirmed by later investigators. At the present time examples of all but one (the trigonal bipyramidal class) of the 32 possible crystal classes have been found, either among minerals or products of the laboratory. Although the various types of crystal symmetry are now well established, there is a decided lack of uniformity in the manner of expressing the symmetry. Some authors emphasize planes of symmetry while others emphasize axes of symmetry. Many authorities disregard the center of symmetry. Some writers use rotatory-reflections as symmetry operations, while a few employ rotatory-inversions instead. There is, indeed, a marked difference of opinion as to what constitutes the true elements of symmetry. This paper is presented with the conviction that a mathematical treatment of the subject will settle the disputed points and enable us |