Symmetry

HERMANN WEYL
Copyright Date: 1980
Pages: 176
https://www.jstor.org/stable/j.ctt155jm66

Table of Contents

1. Front Matter
(pp. [i]-[iv])
2. PREFACE AND BIBLIOGRAPHICAL REMARKS
(pp. [v]-[vi])
Hermann Weyl
3. Table of Contents
(pp. [vii]-2)
4. BILATERAL SYMMETRY
(pp. 3-38)

If i am not mistaken the wordsymmetryis used in our everyday language in two meanings. In the one sense symmetric means something like well-proportioned, well-balanced, and symmetry denotes that sort of concordance of several parts by which they integrate into a whole.Beautyis bound up with symmetry. Thus Polykleitos, who wrote a book on proportion and whom the ancients praised for the harmonious perfection of his sculptures, uses the word, and Dürer follows him in setting down a canon of proportions for the human figure.¹ In this sense the idea is by no means restricted to spatial...

5. TRANSLATORY, ROTATIONAL, AND RELATED SYMMETRIES
(pp. 41-80)

From bilateral we shall now turn to other kinds of geometric symmetry. Even in discussing the bilateral type I could not help drawing in now and then such other symmetries as the cylindrical or the spherical ones. It seems best to fix the underlying general concept with some precision beforehand, and to that end a little mathematics is needed, for which I ask your patience. I have spoken of transformations. A mappingSof space associates with every space pointpa pointp′as its image. A special such mapping is the identityIcarrying every pointpinto...

6. ORNAMENTAL SYMMETRY
(pp. 83-116)

This lecture will have a more systematic character than the preceding one, in as much as it will be dedicated to one special kind of geometric symmetry, the most complicated but also the most interesting from every angle. In two dimensions the art of surface ornaments deals with it, in three dimensions it characterizes the arrangement of atoms in a crystal. We shall therefore call it the ornamental or crystallographic symmetry.

Let us begin with an ornamental pattern in two dimensions which probably occurs more frequently than any other, both in art and nature: the hexagonal pattern so often used...

7. CRYSTALS. THE GENERAL MATHEMATICAL IDEA OF SYMMETRY
(pp. 119-146)

In the last lecture we Considered for two dimensions the problem of making up a complete list (i) of all orthogonally inequivalent finite groups of homogeneous orthogonal transformations, (ii) of all such groups as have invariant lattices, (iii) of all unimodularly inequivalent finite groups of homogenous transformations with integral coefficients, (iv) of all unimodularly inequivalent discontinuous groups of non-homogeneous linear transformations which contain the translations with integral coordinates but no other translations.

Problem (i) was answered by Leonardo’s list

Cn,Dn(n= 1, 2, 3, ⋅ ⋅ ⋅),

(ii) by limiting the indexnin it to the values...

8. APPENDIX A DETERMINATION OF ALL FINITE GROUPS OF PROPER ROTATIONS IN 3-SPACE (cf. p. 77).
(pp. 149-154)
9. APPENDIX B INCLUSION OF IMPROPER ROTATIONS (cf. p. 78).
(pp. 155-156)
10. ACKNOWLEDGMENTS
(pp. 157-160)
11. INDEX
(pp. 161-169)