Point Group Symmetry Research Papers (original) (raw)

Many people experience an inner delight when beholding regularly shaped crystals, a quiet reverence for their wonderfully regular forms. And thus the question soon arises regarding the nature of crystals, the lawfulness underlying their shapes and properties. In textbooks, popular writings and in museum exhibitions, mineralogists and crystallographers mostly emphasize the lattice structure of crystalline matter, which is said to underlie the vast variety of crystal phenomena. Yet a certain disappointment may be felt as a result of being unable to bring one’s inner experience into any sort of deeper relationship with these lattice structures. This essay will draw attention to the fact that the assumption of a lattice structure only expresses one aspect of a crystal’s nature, and consequently that it can and must be embedded in a more comprehensive relationship.
Phenomenological and experimental investigations of crystals reveal that the majority of their physical and geometrical properties are dependent on entirely distinct spatial directions, immanent in the individual crystals themselves. For example, the level of hardness and the formation of planar cleavages are not the same in all directions; they have maxima and minima in different directions, thus laying the foundation for an initial (inner) orientation of the crystal. Similar properties are exhibited in the thermal expansion, the electrical conductivity, the piezoelectricity, the elastic vibrations, as well as in the refraction of light and polarization phenomena of non-cubic crystallizing minerals.
Here the so-called Neumann principle holds: the symmetries of the physical properties of a crystal at least contain the geometrical symmetries of the corresponding crystal polyhedron. This means that a geometrical examination of the shapes and symmetries of a crystal within the context of geometric crystal morphology furnishes the higher order symmetries that underlie all the symmetries of the specific physical properties. Hence symmetries (especially axes and planes of symmetry) are naturally occurring properties, which may indeed be discovered and ascertained in a specific, finite crystal polyhedron, yet point beyond the boundaries of the individual crystal. For on the one hand the same symmetries may be attributed to different crystals, while on the other the axes and planes of symmetry point beyond the finite crystal body to the entire surrounding space. This opens up the conceptual possibility of no longer simply comprehending crystals in a local-additive manner, as structures solely arising from the reciprocal effects of elementary particles. Therefore crystals may also be understood as distinctly unified forms, having their origin in non-localized formative principles, which, considered geometrically, correspond to configurations spanning the whole of space.
Among other things the present investigation is a case study of a central problem – how to undertake an anthroposophical/spiritual-scientific extension of traditional natural-scientific knowledge. Using crystallography, it will be shown how out of a “sound instinct for knowledge” (Steiner (1918), p. 107), crystal physicists have concentrated on the inner, objectively conditioned necessity of one of the fundamental assumptions of scientific crystallography. This at once paves the way for an extension into the conceptual possibility of spiritual forces  yet without coming into conflict with any of the fundamental laws of physics. This fundamental assumption concerns the infinite nature of a crystal lattice (see section 6). It also opens the door to so-called reciprocal space, which plays such a vital role in solid-state physics (section 7).
To begin with, the present study provides a summary of the most important morphological viewpoints (section 2) and the classic crystallographic laws (section 3), as well as outlining the systematic construction of the theory of crystal forms (section 4). Section 5 on morphology and structure theory examines the two complementary interpretations of the fundamental law of crystallography, the lattice structure hypothesis, and the crystal morphological hypothesis. The relationship between the two will be discussed in greater detail in section 6. This latter hypothesis will require some projective geometry, which will be considered in section 5. The article concludes with a brief look at a number of physical considerations (section 7), as well as the structure and morphogenesis of crystals and their synthesis (section 8).