The Architecture of
Crystals

Recognition of the design simplicity leads to the discovery of universal interrelationships . . . .

T E X T
. . . . when any snowflake is turned 60 degrees around an axis perpendicular to its plane, its original appearance is reproduced. The reason for its constancy lies in the arrangement of the atoms in the crystals, while the variety in external shapes is caused by environmental conditions governing the growth of the crystals . . . .

In investigating the nature and properties of the world around us, we tend to alternate between the collection of data and the organization of the results of experiments. This alternation is very important in the progress of science; the organization process would be futile without experiment. The alternation of these two methods motivates further investigations and reveals universal principles linking very distinct physical phenomena. Abilities of scientists vary: some are ingenious in the design of experiments, while others are adept at discovery of patterns and relationships. The discovery of order is thus quite subjective, and the structure of matter as it is known to us is partly of our own making; we tend to structure our perceptions and to create frames of reference suitable for relating various observations to each other. [p. 38]

Patterns and Simplicity
The search for order and structural regularity in the world around us has been a powerful motivation in human evolution. Pythagoras' discovery of the simple ratios in the frequencies of musical notes, Leonardo da Vinci's sectio aurea in architecture and design, Johannes Kepler's Harmonica Mundi, and Mendeleev's Periodic Table of the Elements have all eventually had a strong effect in guiding fundamental research and eventually on our daily lives. [Pyythagoras' discovery was ultimately traced to the modes of vibration of air in pipes or of strings under tension; Mendeleev's order of the elements was explained on the basis of the orderly structure of electrons inside the atom.' Occasionally these attempts have led to occultism and cabalism, and even some of Kepler's statements represent unwarranted extrapolations into mysticism. Notable among recent discoveries of hidden regularities in man-made creations was M. van Crevel's analysis of Jacob Obrecht's Missa Sub Tuum Praesiduum. Using a new system of transcribing the music of this fifteenth-century Mass into modern notation, making the space occupied by each not proportional to its duration, Dr. van Crevel discovered simple relations between the durations of the various portions of the Mass, and between the pitch and duration of the individual notes.

It is notable that some of the greatest masterpieces of art possess a simplicity and regularity of structure. This regularity may be at once obvious, or hidden, as it was in Obrecht's Mass. Equally notable is the fact that the structures required or generated by the laws of physics are also generally simple, with the result that we often find such structures pleasing. This makes us wonder whether simplicity is a subjective norm, or whether simplicity can be evaluated quantitatively . . . . [p. 39]

An example of a simple curve occurring in nature is the shell of the nautilus. The mathematical expression for this shell is a logarithmic spiral, not because the nautilus is magically expert in the use of logarithms, but because the rate of growth of the shell of the nautilus obeys physical laws that happen to generate a logarithmic spiral. The growth of the spiral is mathematically similar to the growth of capital by interest compounded continuously: whenever an entity grows at a rate proportional to its own size, this same growth law occurs . . . . The logarithmic spiral is completely defined by two parameters: its initial value and its growth rate. It is therefore a simple curve . . . . [p. 40]

Summarizing, then, we maintain that simplicity of structure can be expressed quantitatively, and that both present-day aesthetic judgment and physical laws favor simplicity. This simplicity is not always obvious, and requires the choice of a proper frame of reference. Dr. van Crevel's discovery in Obrecht's Mass illustrates a simplicity of structure of which the listener might have been instinctively aware, but its technical design remained hidden for centuries. The simplicity of structure derived from physical laws is related to the fact that the differential equations relating forces and motion in physics are seldom of an order higher than two, so that the number of parameters describing this motion is usually small. Only when complex obstacles of human design are placed in the lines of flow are geometrical complexities created in the flow lines. It is this realization that has led to the streamline of fast-moving objects, affecting in turn the representation of motion in art . . . . [p. 42]

Modules and Generating Functions . . . . infinite patterns were generated [in the carbon and diamond structures] by using a building block or "module" [triangle for graphite, tetrahedron for diamond], and the condition that all identical atoms have identical environment. The rule determining the environment of each atom is called a "generating function."

Fascinating planar patterns can be generated in this manner . . . . [pp. 44-45]

In the pictorial and musical arts there are nowadays many experiments with modules and generating functions. Sometimes the module is subjected to very rigorous generating functions, whereas some artists allow chance to generate aural and visual patterns from the module. In neither case does the creation of the pattern from the module constitute in itself a creative act, for creativity necessarily involves decisions by the artist. [Figs] were created mechanically from the module, but the dimensions of the module, colors and texture are to be freely chosen. It is important to recognize that these patterns produce a framework for endless variations to be chosen by the artist. Although generated from very simple modules by simple generating functions, these patterns are sufficiently complex that they might not have occurred to the artist by the use of free imagination only. They appear complex to the eye because of their unfamiliar framework, but because of their small number of parameters and generating functions they should really be considered quite simple. Rules such as Equation [2] [1/k+1/l+1/m=1] can assist the artist by telling him in general what combinations not to attempt . . . .

Thus in art the module and the generating function have a function in synthesis. We shall see presently that in science they have an analytic function, namely to assist in the recognition of patterns occurring in nature and the interrelationships of these patterns . . . . [p. 48]

Occupancy of Interstices: Crystal Building Blocks
Most metals can be represented by closely packed spheres: copper, silver, gold, and platinum are cubically close packed, while magnesium and zinc are hexagonal. Cobalt, nickel and lanthanum exist in either form. Nonmetallic crystals usually contain several different elements, each represented by a sphere of characteristic radius. Such is the case of the spinel, which contains oxygen, aluminum, and magnesium . . . .

We have seen that the net formed by joining the centers of closely packed spheres divides space into tetrahedral and octahedral cells or interstices. It should therefore be possible to represent crystals by tetrahedral and octahedral modules, empty when the interstices are not occupied by small ions, containing a sphere when a small ion occupies the interstices . . . . [p. 57]

We can assemble these modules in various permutations and combinations to synthesize models of existing or impossible crystal structures . . . . [p. 68]

The crystal structures discussed here by no means cover the vast variety of naturally occurring minerals and man-made crystals. Yet they represent a large fraction of these crystals, and the analysis of the spinel problem does not essentially differ from the analysis being performed on yet more complex structures. The important conclusion in the present context is that the natural design turned out on proper analysis to be simpler than had been originally imaged. And the simpler the design, the fewer parameters were needed for a description of the structure. In turn this facilitates the communication of information regarding the structure among scientists themselves as well as between scientists and automatic information storage, processing, and retrieval systems. Most importantly, the recognition of the design simplicity leads to the discovery of universal interrelationships between vastly different materials. [p. 63]