Chapter I: Crystalline Form (2)

Scalenohedron (fig. 68), bounded by twelve scalene triangles, and with
the general indices {hkl}. The zig-zag lateral edges coincide with the
similar edges of a rhombohedron, as shown in fig. 69; if the indices
of the inscribed rhombohedron be {100}, the indices of the
scalenohedron represented in the figure are {201'}. The scalenohedron
{201'} is a characteristic form of calcite, which for this reason is
sometimes called "dog-tooth-spar." The angles over the three edges of
a face of a scalenohedron are all different; the angles over three
alternate polar edges are more obtuse than over the other three polar
edges. Like the two sets of rhombohedra, there are also direct and
inverse scalenohedra, which may be similar in form and angles, but
different in orientation and indices.

Hexagonal bipyramid (fig. 70), bounded by twelve isosceles triangles
each of which are equally inclined to two planes of symmetry. The
indices are {210}, {412'}, &c., or in general (_hkl_), where h - 2k +
l = 0.

Hexagonal prism of the first order (21'1'), consisting of six faces
parallel to the principal axis and perpendicular to the planes of
symmetry; the angles between (the normals to) the faces are 60 deg.

Hexagonal prism of the second order (101'), consisting of six faces
parallel to the principal axis and parallel to the planes of symmetry.
The faces of this prism are inclined to 30 deg. to those of the last
prism.

Dihexagonal prism, consisting of twelve faces parallel to the
principal axis and inclined to the planes of symmetry. There are two
sets of angles between the faces. The indices are {32'1'}, {53'2'} ...
{hk'l}, where h + k + l = 0.

Basal pinacoid {111}, consisting of a pair of parallel faces
perpendicular to the principal axis.

Fig. 71 shows a combination of a hexagonal prism (m) with the basal
pinacoid (c). For figures of other combinations see CALCITE and
CORUNDUM. The relation between rhombohedral forms and their indices
are best studied with the aid of a stereographic projection (fig. 72);
in this figure the thicker lines are the projections of the three
planes of symmetry, and on these lie the poles of the rhombohedra (six
of which are indicated).

Numerous substances, both natural and artificial, crystallize in this
class; for example, calcite, chalybite, calamine, corundum (ruby and
sapphire), haematite, chabazite; the elements arsenic, antimony,
bismuth, selenium, tellurium and perhaps graphite; also ice, sodium
nitrate, thymol, &c.

DITRIGONAL PYRAMIDAL CLASS

(Hemimorphic-hemihedral).

Here there are three similar planes of symmetry intersecting in the
triad axis; there are no dyad axes and no centre of symmetry. The
triad axis is uniterminal and polar, and the crystals are differently
developed at the two ends; crystals of this class are therefore
pyro-electric. The forms are all open forms:--

Trigonal pyramid {hkk}, consisting of the three faces which correspond
to the three upper or the three lower faces of a rhombohedron of the
holosymmetric class.

Ditrigonal pyramid {hkl}, of six faces, corresponding to the six upper
or lower faces of the scalenohedron.

Hexagonal pyramid (hkl) where (h - 2k + l = 0), of six faces,
corresponding to the six upper or lower faces of the hexagonal
bipyramid.

Trigonal prism {21'1'} or {2'11}, two forms each consisting of three
faces parallel to principal axis and perpendicular to the planes of
symmetry.

Hexagonal prism {101'}, which is geometrically the same as in the last
class.

Ditrigonal prism {hk'l'} (where h + k + l = 0), of six faces parallel
to the principal axis, and with two sets of angles between them.

Basal pedion (111) or (1'1'1'), each consisting of a single plane
perpendicular to the principal axis.

Fig. 73 represents a crystal of tourmaline with the trigonal prism
(21'1'), hexagonal prism (101'), and a trigonal pyramid at each end.
Other substances crystallizing in this class are pyrargyrite,
proustite, iodyrite (AgI), greenockite, zincite, spangolite, sodium
lithium sulphate, tolylphenylketone.

TRAPEZOHEDRAL CLASS

(Trapezohedral-hemihedral).

Here there are three similar dyad axes inclined to one another at 60
deg. and perpendicular to the triad axis. There are no planes or
centre of symmetry. The dyad axes are uniterminal, and are
pyro-electric axes. Crystals of most substances of this class rotate
the plane of polarization of a beam of light.

FIG. 74.--Trigonal Trapezohedron.

FIG. 75.--Trigonal Bipyramid.

In this class the rhombohedra {hkk}, the hexagonal prism {21'1'}, and
the basal pinacoid {111} are geometrically the same as in the
holosymmetric class; the trigonal prism {101'} and the ditrigonal
prisms are as in the ditrigonal pyramidal class. The remaining simple
forms are:--

Trigonal trapezohedron (fig. 74), bounded by six trapezoidal faces.
There are two complementary and enantiomorphous trapezohedra, {hkl}
and {hlk}, derivable from the scalenohedron.

Trigonal bipyramid (fig. 75), bounded by six isosceles triangles; the
indices are {hkl}, where h - 2k + l = 0, as in the hexagonal
bipyramid.

The only minerals crystallizing in this class are quartz (q.v.) and
cinnabar, both of which rotate the plane of a beam of polarized light
transmitted along the triad axis. Other examples are dithionates of
lead (PbS2O6.4H2O), calcium and strontium, and of potassium (K2S2O6),
benzil, matico-stearoptene.

RHOMBOHEDRAL CLASS

(Parallel-faced hemihedral).

The only elements of symmetry are the triad axis and a centre of
symmetry. The general form {hkl} is a rhombohedron, and is a
hemihedral form, with parallel faces, of the scalenohedron. The form
{hkl}, where h - 2k + l = 0, is also a rhombohedron, being the
hemihedral form of the hexagonal bipyramid. The dihexagonal prism
{hk'l'} of the holosymmetric class becomes here a hexagonal prism. The
rhombohedra (hkk), hexagonal prisms {21'1'} and {101'}, and the basal
pinacoid {111} are geometrically the same in this class as in the
holosymmetric class.

Fig. 76 represents a crystal of dioptase with the fundamental
rhombohedron r {100} and the hexagonal prism of the second order m
{101'} combined with the rhombohedron s {031'}.

Examples of minerals which crystallize in this class are phenacite,
dioptase, willemite, dolomite, ilmenite and pyrophanite: amongst
artificial substances is ammonium periodate ((NH4)4I2O9.3H2O).

TRIGONAL PYRAMIDAL CLASS

(Hemimorphic-tetartohedral).

Here there is only the triad axis of symmetry, which is uniterminal.
The general form {hkl} is a trigonal pyramid consisting of three faces
at one end of the crystal. All other forms, in which the faces are
neither parallel nor perpendicular to the triad axis, are trigonal
pyramids. All the prisms are trigonal prisms; and perpendicular to
these are two pedions.

The only substance known to crystallize in this class is sodium
periodate (NaIO4.3H2O), the crystals of which are circularly
polarizing.

TRIGONAL BIPYRAMIDAL CLASS

Here there is a plane of symmetry perpendicular to the triad axis. The
trigonal pyramids of the last class are here trigonal bipyramids (fig.
75); the prisms are all trigonal prisms, and parallel to the plane of
symmetry is the basal pinacoid. No example is known for this class.

DITRIGONAL BIPYRAMIDAL CLASS

Here there are three similar planes of symmetry intersecting in the
triad axis, and perpendicular to them is a fourth plane of symmetry;
at the intersection of the three vertical planes with the horizontal
plane are three similar dyad axes; there is no centre of symmetry.

The general form is bounded by twelve scalene triangles and is a
ditrigonal bipyramid. Like the general form of the last class, this
has two sets of indices {hkl, p'q'r'}, (hkl) for faces above the
equatorial plane of symmetry and (p'q'r') for faces below: with
hexagonal axes there would be only one set of indices. The hexagonal
bipyramids, the hexagonal prism {101'} and the basal pinacoid {111}
are geometrically the same in this class as in the holosymmetric
class. The trigonal prism {21'1'} and ditrigonal prisms {hkl} are the
same as in the ditrigonal pyramidal class.

The only representative of this type of symmetry is the mineral
benitoite (q.v.).

_Hexagonal Division._

In crystals of this division of the hexagonal system the principal
axis is a hexad axis of symmetry. Hexagonal axes of reference are
used: if rhombohedral axes be used many of the simple forms will have
two sets of indices.

HOLOSYMMETRIC CLASS

(Holohedral; Dihexagonal bipyramidal).

Intersecting in the hexad axis are six planes of symmetry of two
kinds, and perpendicular to them is an equatorial plane of symmetry.
Perpendicular to the hexad axis are six dyad axes of two kinds and
each perpendicular to a vertical plane of symmetry. The seven simple
forms are:--

Dihexagonal bipyramid, bounded by twenty-four scalene triangles (fig.
77; v in fig. 80). The indices are {213'1}, &c., or in general {hikl}.
This form may be considered as a combination of two scalenohedra, a
direct and an inverse.

Combinations of Hexagonal forms.]

Hexagonal bipyramid of the first order, bounded by twelve isosceles
triangles (fig. 70; p and u in fig. 80); indices {101'1}, {202'1} ...
(hoh'l). The hexagonal bipyramid so common in quartz is geometrically
similar to this form, but it really is a combination of two
rhombohedra, a direct and an inverse, the faces of which differ in
surface characters and often also in size.

Hexagonal bipyramid of the second order, bounded by twelve faces (s in
figs. 79 and 80); indices {112'1}, {112'2} ... {h.h.2'h'.l}.

Dihexagonal prism, consisting of twelve faces parallel to the hexad
axis and inclined to the vertical planes of symmetry; indices {hiko}.

Hexagonal prism of the first order {1010}, consisting of six faces
parallel to the hexad axis and perpendicular to one set of three
vertical planes of symmetry (m in figs. 71, 78-80).

Hexagonal prism of the second order {112'0}, consisting of six faces
also parallel to the hexad axis, but perpendicular to the other set of
three vertical planes of symmetry (a in fig. 78).

Basal pinacoid {0001}, consisting of a pair of parallel planes
perpendicular to the hexad axis (c in figs. 71, 78-80).

Beryl (emerald), connellite, zinc, magnesium and beryllium crystallize
in this class.

BIPYRAMIDAL CLASS

(Parallel-faced hemihedral).

Here there is a plane of symmetry perpendicular to the hexad axis;
there is also a centre of symmetry. All the closed forms are hexagonal
bipyramids; the open forms are hexagonal prisms or the basal pinacoid.
The general form {hikl} is hemihedral with parallel faces with respect
to the general form of the holosymmetric class.

Apatite (q.v.), pyromorphite, mimetite and vanadinite possess this
degree of symmetry.

DIHEXAGONAL PYRAMIDAL CLASS

(Hemimorphic-hemihedral).

Six planes of symmetry of two kinds intersect in the hexad axis. The
hexad axis is uniterminal and all the forms are open forms. The
general form {hikl} consists of twelve faces at one end of the
crystal, and is a dihexagonal pyramid. The hexagonal pyramids {hoh'l}
and (h.h.2'h'.l) each consist of six faces at one end of the crystal.
The prisms are geometrically the same as in the holosymmetric class.
Perpendicular to the hexad axis are the pedions (0001) and (0001').

Iodyrite (AgI), greenockite (CdS), wurtzite (ZnS) and zincite (ZnO)
are often placed in this class, but they more probably belong to the
hemimorphic-hemihedral class of the rhombohedral division of this
system.

TRAPEZOHEDRAL CLASS

(Trapezohedral-hemihedral).

Six dyad axes of two kinds are perpendicular to the hexad axis. The
general form {hikl} is the hexagonal trapezohedron bounded by twelve
trapezoidal faces. The other simple forms are geometrically the same
as in the holosymmetric class. Barium-anti-monyldextro-tartrate +
potassium nitrate (Ba(SbO)2(C4H4O6)2.KNO3) and the corresponding lead
salt crystallize in this class.

HEXAGONAL PYRAMIDAL CLASS

(Hemimorphic-tetartohedral).

No other element is here associated with the hexad axis, which is
uniterminal. The pyramids all consist of six faces at one end of the
crystal, and prisms are all hexagonal prisms; perpendicular to the
hexad axis are the pedions.

Lithium potassium sulphate, strontium-antimonyl dextro-tartrate, and
lead-antimonyl dextro-tartrate are examples of this type of symmetry.
The mineral nepheline is placed in this class because of the absence
of symmetry in the etched figures on the prism faces (fig. 92).

(g) _Regular Grouping of Crystals._

Crystals of the same kind when occurring together may sometimes be grouped in parallel position and so give rise to special structures, of which the dendritic (from [Greek: dendrou], a tree) or branch-like aggregations of native copper or of magnetite and the fibrous structures of many minerals furnish examples. Sometimes, owing to changes in the surrounding conditions, the crystal may continue its growth with a different external form or colour, e.g. sceptre-quartz.

Regular intergrowths of crystals of totally different substances such as staurolite with cyanite, rutile with haematite, blende with chalcopyrite, calcite with sodium nitrate, are not uncommon. In these cases certain planes and edges of the two crystals are parallel. (See O. Mugge, "Die regelmassigen Verwachsungen von Mineralien verschiedener Art," _Neues Jahrbuch fur Mineralogie_, 1903, vol. xvi. pp. 335-475).

But by far the most important kind of regular conjunction of crystals is that known as "twinning." Here two crystals or individuals of the same kind have grown together in a certain symmetrical manner, such that one portion of the twin may be brought into the position of the other by reflection across a plane or by rotation about an axis. The plane of reflection is called the twin-plane, and is parallel to one of the faces, or to a possible face, of the crystal: the axis of rotation, called the twin-axis, is parallel to one of the edges or perpendicular to a face of the crystal.

In the twinned crystal of gypsum represented in fig. 81 the two portions are symmetrical with respect to a plane parallel to the ortho-pinacoid (100), i.e. a vertical plane perpendicular to the face b. Or we may consider the simple crystal (fig. 82) to be cut in half by this plane and one portion to be rotated through 180 deg. about the normal to the same plane. Such a crystal (fig. 81) is therefore described as being twinned on the plane (100).

An octahedron (fig. 83) twinned on an octahedral face (111) has the two portions symmetrical with respect to a plane parallel to this face (the large triangular face in the figure); and either portion may be brought into the position of the other by a rotation through 180 deg. about the triad axis of symmetry which is perpendicular to this face. This kind of twinning is especially frequent in crystals of spinel, and is consequently often referred to as the "spinel twin-law."

In these two examples the surface of the union, or composition-plane, of the two portions is a regular surface coinciding with the twin-plane; such twins are called "juxtaposition-twins." In other juxtaposed twins the plane of composition is, however, not necessarily the twin-plane. Another type of twin is the "interpenetration twin," an example of which is shown in fig. 84. Here one cube may be brought into the position of the other by a rotation of 180 deg. about a triad axis, or by reflection across the octahedral plane which is perpendicular to this axis; the twin-plane is therefore (111).

Since in many cases twinned crystals may be explained by the rotation of one portion through two right angles, R. J. Hauy introduced the term "hemitrope" (from the Gr. [Greek: hemi]-, half, and [Greek: tropos], a turn); the word "macle" had been earlier used by Rome d'Isle. There are, however, some rare types of twins which cannot be explained by rotation about an axis, but only by reflection across a plane; these are known as "symmetric twins," a good example of which is furnished by one of the twin-laws of chalcopyrite.

Twinned crystals may often be recognized by the presence of re-entrant angles between the faces of the two portions, as may be seen from the above figures. In some twinned crystals (e.g. quartz) there are, however, no re-entrant angles. On the other hand, two crystals accidentally grown together without any symmetrical relation between them will usually show some re-entrant angles, but this must not be taken to indicate the presence of twinning.

Twinning may be several times repeated on the same plane or on other similar planes of the crystal, giving rise to triplets, quartets and other complex groupings. When often repeated on the same plane, the twinning is said to be "polysynthetic," and gives rise to a laminated structure in the crystal. Sometimes such a crystal (e.g. of corundum or pyroxene) may be readily broken in this direction, which is thus a "plane of parting," often closely resembling a true cleavage in character. In calcite and some other substances this lamellar twinning may be produced artificially by pressure (see below, Sect. II. (a), _Glide-plane_).

Another curious result of twinning is the production of forms which apparently display a higher degree of symmetry than that actually possessed by the substance. Twins of this kind are known as "mimetic-twins or pseudo-symmetric twins." Two hemihedral or hemimorphic crystals (e.g. of diamond or of hemimorphite) are often united in twinned position to produce a group with apparently the same degree of symmetry as the holosymmetric class of the same system. Or again, a substance crystallizing in, say, the orthorhombic system (e.g. aragonite) may, by twinning, give rise to pseudo-hexagonal forms: and pseudo-cubic forms often result by the complex twinning of crystals (e.g. stannite, phillipsite, &c.) belonging to other systems. Many of the so-called "optical anomalies" of crystals may be explained by this pseudo-symmetric twinning.

(h) _Irregularities of Growth of Crystals; Character of Faces._

Only rarely do actual crystals present the symmetrical appearance shown in the figures given above, in which similar faces are all represented as of equal size. It frequently happens that the crystal is so placed with respect to the liquid in which it grows that there will be a more rapid deposition of material on one part than on another; for instance, if the crystal be attached to some other solid it cannot grow in that direction. Only when a crystal is freely suspended in the mother-liquid and material for growth is supplied at the same rate on all sides does an equably developed form result.

Misshappen Octahedra.]

Two misshapen or distorted octahedra are represented in figs. 85 and 86; the former is elongated in the direction of one of the edges of the octahedron, and the latter is flattened parallel to one pair of faces. It will be noticed in these figures that the edges in which the faces intersect have the same directions as before, though here there are additional edges not present in fig. 3. The angles (70 deg. 32' or 109 deg. 28') between the faces also remain the same; and the faces have the same inclinations to the axes and planes of symmetry as in the equably developed form. Although from a geometrical point of view these figures are no longer symmetrical with respect to the axes and planes of symmetry, yet crystallographically they are just as symmetrical as the ideally developed form, and, however much their irregularity of development, they still are regular (cubic) octahedra of crystallography. A remarkable case of irregular development is presented by the mineral cuprite, which is often found as well-developed octahedra; but in the variety known as chalcotrichite it occurs as a matted aggregate of delicate hairs, each of which is an individual crystal enormously elongated in the direction of an edge or diagonal of the cube.

The symmetry of actual crystals is sometimes so obscured by irregularities of growth that it can only be determined by measurement of the angles. An extreme case, where several of the planes have not been developed at all, is illustrated in fig. 87, which shows the actual shape of a crystal of zircon from Ceylon; the ideally developed form (fig. 88) is placed at the side for comparison, and the parallelism of the edges between corresponding faces will be noticed. This crystal is a combination of five simple forms, viz. two tetragonal prisms (a and m,) two tetragonal bipyramids (e and p), and one ditetragonal bipyramid (x, with 16 faces).

Crystal of Zircon (clinographic drawings and plans).]

The actual form, or "habit," of crystals may vary widely in different crystals of the same substance, these differences depending largely on the conditions under which the growth has taken place. The material may have crystallized from a fused mass or from a solution; and in the latter case the solvent may be of different kinds and contain other substances in solution, or the temperature may vary. Calcite (q.v.) affords a good example of a substance crystallizing in widely different habits, but all crystals are referable to the same type of symmetry and may be reduced to the same fundamental form.

When crystals are aggregated together, and so interfere with each other's growth, special structures and external shapes often result, which are sometimes characteristic of certain substances, especially amongst minerals.

Incipient crystals, the development of which has been arrested owing to unfavourable conditions of growth, are known as crystallites (q.v.). They are met with in imperfectly crystallized substances and in glassy rocks (obsidian and pitchstone), or may be obtained artificially from a solution of sulphur in carbon disulphide rendered viscous by the addition of Canada-balsam. To the various forms H. Vogelsang gave, in 1875, the names "globulites," "margarites" (from [Greek: margarites], a pearl), "longulites," &c. At a more advanced stage of growth these bodies react on polarized light, thus possessing the internal structure of true crystals; they are then called "microlites." These have the form of minute rods, needles or hairs, and are aggregated into feathery and spherulitic forms or skeletal crystals. They are common constituents of microcrystalline igneous rocks, and often occur as inclusions in larger crystals of other substances.

Inclusions of foreign matter, accidentally caught up during growth, are frequently present in crystals. Inclusions of other minerals are specially frequent and conspicuous in crystals of quartz, and crystals of calcite may contain as much as 60% of included sand. Cavities, either with rounded boundaries or with the same shape ("negative crystals") as the surrounding crystal, are often to be seen; they may be empty or enclose a liquid with a movable bubble of gas.

The faces of crystals are rarely perfectly plane and smooth, but are usually striated, studded with small angular elevations, pitted or cavernous, and sometimes curved or twisted. These irregularities, however, conform with the symmetry of the crystal, and much may be learnt by their study. The parallel grooves or furrows, called "striae," are the result of oscillatory combination between adjacent faces, narrow strips of first one face and then another being alternately developed. Sometimes the striae on crystal-faces are due to repeated lamellar twinning, as in the plagioclase felspars. The directions of the striations are very characteristic features of many crystals: e.g. the faces of the hexagonal prism of quartz are always striated horizontally, whilst in beryl they are striated vertically. Cubes of pyrites (fig. 89) are striated parallel to one edge, the striae on adjacent faces being at right angles, and due to oscillatory combination of the cube and the pentagonal dodecahedron (compare fig. 36); whilst cubes of blende (fig. 90) are striated parallel to one diagonal of each face, i.e. parallel to the tetrahedron faces (compare fig. 31). These striated cubes thus possess different degrees of symmetry and belong to different symmetry-classes. Oscillatory combination of faces gives rise also to curved surfaces. Crystals with twisted surfaces (see DOLOMITE) are, however, built up of smaller crystals arranged in nearly parallel position. Sometimes a face is entirely replaced by small faces of other forms, giving rise to a drusy surface; an example of this is shown by some octahedral crystals of fluorspar (fig. 2) which are built up of minute cubes.

The faces of crystals are sometimes partly or completely replaced by smooth bright surfaces inclined at only a few minutes of arc from the true position of the face; such surfaces are called "vicinal faces," and their indices can be expressed only by very high numbers. In apparently perfectly developed crystals of alum the octahedral face, with the simple indices (111), is usually replaced by faces of very low triakis-octahedra, with indices such as (251.251.250); the angles measured on such crystals will therefore deviate slightly from the true octahedral angle. Vicinal faces of this character are formed during the growth of crystals, and have been studied by H. A. Miers (_Phil. Trans._, 1903, Ser. A. vol. 202). Other faces with high indices, viz. "prerosion faces" and the minute faces forming the sides of etched figures (see below), as well as rounded edges and other surface irregularities, may, however, result from the corrosion of a crystal subsequent to its growth. The pitted and cavernous faces of artificially grown crystals of sodium chloride and of bismuth are, on the other hand, a result of rapid growth, more material being supplied at the edges and corners of the crystal than at the centres of the faces.

(i) _Theories of Crystal Structure._

The ultimate aim of crystallographic research is to determine the internal structure of crystals from both physical and chemical data. The problem is essentially twofold: in the first place it is necessary to formulate a theory as to the disposition of the molecules, which conforms with the observed types of symmetry--this is really a mathematical problem; in the second place, it is necessary to determine the orientation of the atoms (or groups of atoms) composing the molecules with regard to the crystal axes--this involves a knowledge of the atomic structure of the molecule. As appendages to the second part of our problem, there have to be considered: (1) the possibility of the existence of the same substance in two or more distinct crystalline forms--polymorphism, and (2) the relations between the chemical structure of compounds which affect nearly identical or related crystal habits--isomorphism and morphotropy. Here we shall discuss the modern theory of crystal structure; the relations between chemical composition and crystallographical form are discussed in Part III. of this article; reference should also be made to the article CHEMISTRY: _Physical_.

Hauy.

The earliest theory of crystal structure of any moment is that of Hauy, in which, as explained above, he conceived a crystal as composed of elements bounded by the cleavage planes of the crystal, the elements being arranged contiguously and along parallel lines. There is, however, no reason to suppose that matter is continuous throughout a crystalline body; in fact, it has been shown that space does separate the molecules, and we may therefore replace the contiguous elements of Hauy by particles equidistantly distributed along parallel lines; by this artifice we retain the reticulated or net-like structure, but avoid the continuity of matter which characterizes Hauy's theory; the permanence of crystal form being due to equilibrium between the intermolecular (and interatomic) forces. The crystal is thus conjectured as a "space-lattice," composed of three sets of parallel planes which enclose parallelopipeda, at the corners of which are placed the constituent molecules (or groups of molecules) of the crystal.

Frankenheim; Bravais.

The geometrical theory of crystal structure (i.e. the determination of the varieties of crystal symmetry) is thus reduced to the mathematical problem: "in how many ways can space be partitioned?" M. L. Frankenheim, in 1835, determined this number as fifteen, but A. Bravais, in 1850, proved the identity of two of Frankenheim's forms, and showed how the remaining fourteen coalesced by pairs, so that really these forms only corresponded to seven distinct systems and fourteen classes of crystal symmetry. These systems, however, only represented holohedral forms, leaving the hemihedral and tetartohedral classes to be explained. Bravais attempted an explanation by attributing differences in the symmetry of the crystal elements, or, what comes to the same thing, he assumed the crystals to exhibit polar differences along any member of the lattice; for instance, assume the particles to be (say) pear-shaped, then the sharp ends point in one direction, the blunt ends in the opposite direction.

Sohncke.

A different view was adopted by L. Sohncke in 1879, who, by developing certain considerations published by Camille Jordan in 1869 on the possible types of regular repetition in space of identical parts, showed that the lattice-structure of Bravais was unnecessary, it being sufficient that each molecule of an indefinitely extended crystal, represented by its "point" (or centre of gravity), was identically situated with respect to the molecules surrounding it. The problem then resolves itself into the determination of the number of "point-systems" possible; Sohncke derived sixty-five such arrangements, which may also be obtained from the fourteen space-lattices of Bravais, by interpenetrating any one space-lattice with one or more identical lattices, with the condition that the resulting structure should conform with the homogeneity characteristic of crystals. But the sixty-five arrangements derived by Sohncke, of which Bravais' lattices are particular cases, did not complete the solution, for certain of the known types of crystal symmetry still remained unrepresented. These missing forms are characterized as being enantiomorphs consequently, with the introduction of this principle of repetition over a plane, i.e. mirror images. E. S. Fedorov (1890), A. Schoenflies (1891), and W. Barlow (1894), independently and by different methods, showed how Sohncke's theory of regular point-systems explained the whole thirty-two classes of crystal symmetry, 230 distinct types of crystal structure falling into these classes.

By considering the atoms instead of the centres of gravity of the molecules, Sohncke (_Zeits. Kryst. Min._, 1888, 14, p. 431) has generalized his theory, and propounded the structure of a crystal in the following terms: "A crystal consists of a finite number of interpenetrating regular point-systems, which all possess like and like-directed coincidence movements. Each separate point-system is occupied by similar material particles, but these may be different for the different interpenetrating partial systems which form the complex system." Or we may quote the words of P. von Groth (_British Assoc. Rep._, 1904): "A crystal--considered as indefinitely extended--consists of n interpenetrating regular point-systems, each of which is formed of similar atoms; each of these point-systems is built up from a number of interpenetrating space-lattices, each of the latter being formed from similar atoms occupying parallel positions. All the space-lattices of the combined system are geometrically identical, or are characterized by the same elementary parallelopipedon."

A complete resume, with references to the literature, will be found in
"Report on the Development of the Geometrical Theories of Crystal
Structure, 1666-1901" (_British Assoc. Rep._, 1901).