The Analytic Theory of Point Systems, by J. D. Bernal

Introductory note

R. L. E. Schwarzenberger
Science Education Department
University of Warwick

One way to assess the novelty of Bernal's manuscript is to place it side by side with the two books which he himself used:
  • H. HILTON.    Mathematical Crystallography and the theory of groups of movement, Clarendon Press, Oxford 1903.
  • P. NIGGLI.     Geometrische Kristallographie des Diskontinuums, Borntraeger, Leipzig 1919.

The two books have in common that they include the listing of the 230 space groups as it was published by Schönflies* in 1891 (Niggli copies Schönflies exactly whereas Hilton changes some nomenclature in translation and alters the order in which the crystal systems occur. Bernal follows Hilton). They differ in that Hilton attempts to give a full proof that there are precisely 230 possibilities whereas Niggli gives proofs of general results, listings of groups and much additional information about each group (along the lines of the future International Tables). Hilton is more interested in group theory than in crystallography and uses geometrical methods which stem directly from those of Schönflies, whereas Niggli uses algebra more and deals also with crystal form and with deformations of structure.

It is worth recalling that neither Schönflies nor Fedorov nor Barlow succeeded in getting the number of distinct groups correct at their first attempt: the list of 230 space groups was achieved only as a result of mutual checking between Schönflies and Fedorov. I do not believe that Hilton would have done better, had he not been writing with the list of Schönflies in front of him, because his method is not guaranteed to catch every possible special case of existence or equivalence. Thus Bernal was inspired by Niggli's more analytic approach to try to make more precise the "simple and geometric" qualitative proof contained in Hilton's book. In this approach Bernal was following very closely in the footsteps of Fedorov who had written over 30 years earlier:

" ... here, for the first time, the symmetry of figures is expressed in analytical terms and in this way the theory of symmetry is itself introduced into the realm of analytic geometry. Originally I intended to find analytic terms for the symmetry of finite regular systems. I was prompted to do this by the difficulty of interpreting Sohncke's derivations ... an error which remained unnoticed for a long time by the author himself ... this could hardly have happened if these systems had been expressed in analytical terms"
(E. S. Fedorov "Symmetry of Finite Figures" 1889 translated by D. Harker 1971)

Similarly Bernal realised that more precise and analytic methods were required for the qualitative listing (Sohncke, Schönflies, Barlow, Hilton) to become useful in the new crystallographic applications. In the remainder of this note I would like to draw attention to three ways in which Bernal notably achieved this aim.

The first important improvement made to Hilton's proof is the decision to work in vector space (i.e. with fixed origin) rather than in affine space. Hilton follows Schönflies in writing, for example, A(α) to denote a rotation through angle α about an axis A, or S(t) to denote a glide transformation with translation component t lying in the plane of reflection S. But the axis A and the plane S need not pass through the origin, which is good for nice qualitative pictures of crystals but bad for precise analytic proofs of the mathematics. Bernal chooses an origin and assigns each affine transformation

X → ρ X + c

its linear part ρ and its translation component c. The importance of this change for the development of mathematical crystallography is in no way diminished by the fact that, under the steadily increasing influence of modern algebra, many other mathematicians did the same thing quite naturally and most crystallographers do so today.

This leads to the most striking aspects of the manuscript: the influence of modern algebra in the discussion of quadratic forms (in Chapters II and III the symbol SXY is the scalarproduct of X and Y) and in the use of quaternions throughout. In crystallography the linear part of a symmetry is, of course, a 3 × 3 matrix determined by coefficients. The fact that ρ preserves lengths and angles imposes 6 conditions so only 3 degrees of freedom are left for ρ. Clearly a more precise analytic approach would benefit from a more economic method of presentation of 3 parameters. It was fairly well known among theoretical physicists and mathematicians that all such linear parts arise from transformations of the form

X → ± qXq-1

where q is a unit quaternion (the correspondence between unit quaternions and linear parts is actually not one-one, because -q determines the same transformation as q ; this does not cause any confusion in practice). Most readers of this note will be familiar with this fact but, for those who are not, a brief note on quaternions is appended which covers these facts assumed without comment by Bernal in Chapters II and III. Note that the quaternion notation - unlike matrix notation - allows angles of rotation, axis of rotation, plane of rotation to be read off immediately. This is a great advantage when trying to decide whether two groups are or are not equivalent.

The third important improvement is the explicit mention of specialisation : no doubt influenced by the algebraic geometry of the time. This is merely the very simple observation that, for example, a tetragonal lattice is a special case of a orthorhombic lattice. This observation is present but less explicit in Hilton, and explains why Hilton (and so also Bernal) deals with the tetragonal system immediately after the orthorhombic system (Schönflies has the rhombohedral system in between, presumably on the ground that 12 is between 8 and 16).

The effect of these three improvenents is that far more facts can be established as general theorems applicable to many crystal systems. Here Bernal is following the example set by Niggli in contrast to Hilton's insistence on a separate discussion for each crystal system. It follows too that the listing of space groups becomes essentially a listing of the relevant translation components which can be handled algebraically. In modern language, Bernal is listing the relevant cocycles to determine a first cohomology group.

In summary, Bernal has rewritten the proof (Sohncke, Schönflies) which he found in Hilton but has brought to it the more analytic attitudes of a quite different crystallographic tradition (Möbius, Fedorov, Niggli). His excellent knowledge of current mathematics yields an improvement upon the treatment of Hilton, 20 years earlier, and comes close to foreshadowing the cohomological work of Zassenhaus, 20 years later.

[Rolph Schwarzenberger, April 1981] 



* Kristallsysteme und Kristallstruktur, B. G. Teubner, Leipzig, 1891. The second revised edition appeared as Theorie der Kristallstruktur - ein Lehrbuch, Borntraeger, Berlin, 1923. There is no evidence in the manuscript of Bernal having consulted Schönflies directly - in any case Hilton is considerably clearer.

Based on the Emmanuel College, Cambridge, Prize Essay of 1923
Published as Occasional Paper No. 1 from the Department of Crystallography, Birkbeck College, University of London 1981
Published on the Web 2009 for the IUCr Commission on Mathematical and Theoretical Crystallography of the International Union of Crystallography with the kind permission of Birkbeck College