Chapter 15

1. CUBIC SYSTEM(Isometric; Regular; Octahedral; Tesseral).In this system the three crystallographic axes of reference are all at right angles to each other and are equal in length. They are parallel to the edges of the cube, and in the different classes coincide either with tetrad or dyad axes of symmetry. Five classes are included in this system, in all of which there are, besides other elements of symmetry, four triad axes.In crystals of this system the angle between any two facesPandQwith the indices (hkl) and (pqr) is given by the equationCOS PQ =hp + kq + lr.√(h² + k² + l²) (p² + q² + r²)The angles between faces with the same indices are thus the same in all substances which crystallize in the cubic system: in other systems the angles vary with the substance and are characteristic of it.Holosymmetric Class(Holohedral (ὅλος, whole); Hexakis-octahedral).Crystals of this class possess the full number of elements of symmetry already mentioned above for the octahedron and the cube, viz. three cubic planes of symmetry, six dodecahedral planes, three tetrad axes of symmetry, four triad axes, six dyad axes, and a centre of symmetry.Fig. 13.—Rhombic Dodecahedron.Fig. 14.—Combination ofRhombic Dodecahedron andOctahedron.There are seven kinds of simple forms, viz.:—Cube (fig. 5). This is bounded by six square faces parallel to the cubic planes of symmetry; it is known also as the hexahedron. The angles between the faces are 90°, and the indices of the form are {100}. Salt, fluorspar and galena crystallize in simple cubes.Fig. 15.—Triakis-octahedron.Fig. 16.—Combination ofTriakis-octahedron and Cube.Octahedron (fig. 3). Bounded by eight equilateral triangular faces perpendicular to the triad axes of symmetry. The angles between the faces are 70° 32′ and 109° 28′, and the indices are {111}. Spinel, magnetite and gold crystallize in simple octahedra. Combinations of the cube and octahedron are shown in figs. 6-8.Rhombic dodecahedron (fig. 13). Bounded by twelve rhomb-shaped faces parallel to the six dodecahedral planes of symmetry. The angles between the normals to adjacent faces are 60°, and between other pairs of faces 90°; the indices are {110}. Garnet frequently crystallizes in this form. Fig. 14 shows the rhombic dodecahedron in combination with the octahedron.Fig. 17.—Icositetrahedron.Fig. 18.—Combination ofIcositetrahedron and Cube.In these three simple forms of the cubic system (which are shown in combination in fig. 11) the angles between the faces and the indices are fixed and are the same in all crystals; in the four remaining simple forms they are variable.Fig. 19.—Combination ofIcositetrahedron and Octahedron.Fig. 20.—Combination ofIcositetrahedron {211} andRhombic Dodecahedron.Triakis-octahedron (three-faced octahedron) (fig. 15). This solid is bounded by twenty-four isosceles triangles, and may be considered as an octahedron with a low triangular pyramid on each of its faces. As the inclinations of the faces may vary there is a series of these forms with the indices {221}, {331}, {332}, &c. or in general {hhk}.Fig. 21.—Tetrakis-hexahedron.Fig. 22.—Tetrakis-hexahedron.Icositetrahedron (fig. 17). Bounded by twenty-four trapezoidal faces, and hence sometimes called a “trapezohedron.” The indices are {211}, {311}, {322}, &c., or in general {hkk}. Analcite, leucite and garnet often crystallize in the simple form {211}. Combinations are shown in figs. 18-20. The planeABein fig. 9 is one face (112) of an icositetrahedron; the indices of the remaining faces in this octant being (211) and (121).Fig. 23.—Combination of Tetrakis-hexahedron and Cube.Tetrakis-hexahedron (four-faced cube) (figs. 21 and 22). Like the triakis-octahedron this solid is also bounded by twenty-four isosceles triangles, but here grouped in fours over the cubic faces. The two figures show how, with different inclinations of the faces, the form may vary, approximating in fig. 21 to the cube and in fig. 22 to the rhombic dodecahedron. The angles over the edges letteredAare different from the angles over the edges letteredC. Each face is parallel to one of the crystallographic axes and intercepts the two others in different lengths; the indices are therefore {210}, {310}, {320}, &c., in general {hko}. Fluorspar sometimes crystallizes in the simple form {310}; more usually, however, in combination with the cube (fig. 23).Hexakis-octahedron (fig. 24). Here each face of the octahedron is replaced by six scalene triangles, so that altogether there areforty-eight faces. This is the greatest number of faces possible for any simple form in crystals. The faces are all oblique to the planes and axes of symmetry, and they intercept the three crystallographic axes in different lengths, hence the indices are all unequal, being in general {hkl}, or in particular cases {321}, {421}, {432}, &c. Such a form is known as the “general form” of the class. The interfacial angles over the three edges of each triangle are all different. These forms usually exist only in combination with other cubic forms (for example, fig. 25), but {421} has been observed as a simple form on fluorspar.Fig. 24.—Hexakis-octahedron.Fig. 25.—Combination ofHexakis-octahedron andCube.Several examples of substances which crystallize in this class have been mentioned above under the different forms; many others might be cited—for instance, the metals iron, copper, silver, gold, platinum, lead, mercury, and the non-metallic elements silicon and phosphorus.Tetrahedral Class(Tetrahedral-hemihedral; Hexakis-tetrahedral).In this class there is no centre of symmetry nor cubic planes of symmetry; the three tetrad axes become dyad axes of symmetry, and the four triad axes are polar,i.e.they are associated with different faces at their two ends. The other elements of symmetry (six dodecahedral planes and six dyad axes) are the same as in the last class.Fig. 26.—Tetrahedron.Fig. 27.—Deltoid Dodecahedron.Of the seven simple forms, the cube, rhombic dodecahedron and tetrakis-hexahedron are geometrically the same as before, though on actual crystals the faces will have different surface characters. For instance, the cube faces will be striated parallel to only one of the diagonals (fig. 90), and etched figures on this face will be symmetrical with respect to two lines, instead of four as in the last class. The remaining simple forms have, however, only half the number of faces as the corresponding form in the last class, and are spoken of as “hemihedral with inclined faces.”Fig. 28.—Triakis-tetrahedron.Fig. 29.—Hexakis-tetrahedron.Tetrahedron (fig. 26). This is bounded by four equilateral triangles and is identical with the regular tetrahedron of geometry. The angles between the normals to the faces are 109° 28′. It may be derived from the octahedron by suppressing the alternate faces.Deltoid1dodecahedron (fig. 27). This is the hemihedral form of the triakis-octahedron; it has the indices {hhk} and is bounded by twelve trapezoidal faces.Triakis-tetrahedron (fig. 28). The hemihedral form {hkk} of the icositetrahedron; it is bounded by twelve isosceles triangles arranged in threes over the tetrahedron faces.Fig. 30.—Combination oftwo Tetrahedra.Fig. 31.—Combination ofTetrahedron and Cube.Hexakis-tetrahedron (fig. 29). The hemihedral form {hkl} of the hexakis-octahedron; it is bounded by twenty-four scalene triangles and is the general form of the class.Fig. 32.—Combination ofTetrahedron, Cube and RhombicDodecahedron.Fig. 33.—Combination ofTetrahedron and RhombicDodecahedron.Corresponding to each of these hemihedral forms there is another geometrically similar form, differing, however, not only in orientation, but also in actual crystals in the characters of the faces. Thus from the octahedron there may be derived two tetrahedra with the indices {111} and {111}, which may be distinguished as positive and negative respectively. Fig. 30 shows a combination of these two tetrahedra, and represents a crystal of blende, in which the four larger faces are dull and striated, whilst the four smaller are bright and smooth. Figs. 31-33 illustrate other tetrahedral combinations.Tetrahedrite, blende, diamond, boracite and pharmacosiderite are substances which crystallize in this class.Pyritohedral2Class(Parallel-faced hemihedral; Dyakis-dodecahedral).Crystals of this class possess three cubic planes of symmetry but no dodecahedral planes. There are only three dyad axes of symmetry, which coincide with the crystallographic axes; in addition there are three triad axes and a centre of symmetry.Fig. 34.Pentagonal Dodecahedron.Fig. 35.Dyakis-dodecahedron.Here the cube, octahedron, rhombic dodecahedron, triakis-octahedron and icositetrahedron are geometrically the same as in the first class. The characters of the faces will, however, be different; thus the cube faces will be striated parallel to one edge only (fig. 89), and triangular markings on the octahedron faces will be placed obliquely to the edges. The remaining simple forms are “hemihedral with parallel faces,” and from the corresponding holohedral forms two hemihedral forms, a positive and a negative, may be derived.Pentagonal dodecahedron (fig. 34). This is bounded by twelve pentagonal faces, but these are not regular pentagons, and the angles over the three sets of different edges are different. The regular dodecahedron of geometry, contained by twelve regular pentagons, is not a possible form in crystals. The indices are {hko}: as a simple form {210} is of very common occurrence in pyrites.Dyakis-dodecahedron (fig. 35). This is the hemihedral form ofthe hexakis-octahedron and has the indices {hkl}; it is bounded by twenty-four faces. As a simple form {321} is met with in pyrites.Fig. 36.—Combination ofPentagonal Dodecahedronand Cube.Fig. 37.—Combination ofPentagonal Dodecahedronand Octahedron.Combinations (figs. 36-39) of these forms with the cube and the octahedron are common in pyrites. Fig. 37 resembles in general appearance the regular icosahedron of geometry, but only eight of the faces are equilateral triangles. Cobaltite, smaltite and other sulphides and sulpharsenides of the pyrites group of minerals crystallize in these forms. The alums also belong to this class; from an aqueous solution they crystallize as simple octahedra, sometimes with subordinate faces of the cube and rhombic dodecahedron, but from an acid solution as octahedra combined with the pentagonal dodecahedron {210}.Fig. 38.—Combination ofPentagonal Dodecahedron, Cubeand Octahedron.Fig. 39.—Combination ofPentagonal Dodecahedrone{210}, Dyakis-dodecahedronf{321}, and Octahedrond{111}.Plagihedral3Class(Plagihedral-hemihedral; Pentagonal icositetrahedral; Gyroidal4).In this class there are the full number of axes of symmetry (three tetrad, four triad and six dyad), but no planes of symmetry and no centre of symmetry.Fig. 40.—PentagonalIcositetrahedron.Fig. 41.—Tetrahedral PentagonalDodecahedron.Pentagonal icositetrahedron (fig. 40). This is the only simple form in this class which differs geometrically from those of the holosymmetric class. By suppressing either one or other set of alternate faces of the hexakis-octahedron two pentagonal icositetrahedra {hkl} and {khl} are derived. These are each bounded by twenty-four irregular pentagons, and although similar to each other they are respectively right- and left-handed, one being the mirror image of the other; such similar but nonsuperposable forms are said to be enantiomorphous (ἐναντίος, opposite, andμορφή, form), and crystals showing such forms sometimes rotate the plane of polarization of plane-polarized light. Faces of a pentagonal icositetrahedron with high indices have been very rarely observed on crystals of cuprite, potassium chloride and ammonium chloride, but none of these are circular polarizing.Tetartohedral Class(Tetrahedral pentagonal dodecahedral).Here, in addition to four polar triad axes, the only other elements of symmetry are three dyad axes, which coincide with the crystallographic axes. Six of the simple forms, the cube, tetrahedron, rhombic dodecahedron, deltoid dodecahedron, triakis-tetrahedron and pentagonal dodecahedron, are geometrically the same in this class as in either the tetrahedral or pyritohedral classes. The general form is the Tetrahedral pentagonal dodecahedron (fig. 41). This is bounded by twelve irregular pentagons, and is a tetartohedral or quarter-faced form of the hexakis-octahedron. Four such forms may be derived, the indices of which are {hkl}, {khl}, {hkl} and {khl}; the first pair are enantiomorphous with respect to one another, and so are the last pair. Barium nitrate, lead nitrate, sodium chlorate and sodium bromate crystallize in this class, as also do the minerals ullmannite (NiSbS) and langbeinite (K2Mg2(SO4)3).2. TETRAGONAL SYSTEM(Pyramidal; Quadratic; Dimetric).In this system the three crystallographic axes are all at right angles, but while two are equal in length and interchangeable the third is of a different length. The unequal axis is spoken of as the principal axis or morphological axis of the crystal, and it is always placed in a vertical position; in five of the seven classes of this system it coincides with the single tetrad axis of symmetry.Fig. 42.Fig. 43.Tetragonal Bipyramids.The parameters area:a:c, where a refers to the two equal horizontal axes, andcto the vertical axis;cmay be either shorter (as in fig. 42) or longer (fig. 43) thana. The ratioa:cis spoken of as the axial ratio of a crystal, and it is dependent on the angles between the faces. In all crystals of the same substance this ratio is constant, and is characteristic of the substance; for other substances crystallizing in the tetragonal system it will be different. For example, in cassiterite it is given asa:c= 1 : 0.67232 or simply asc= 0.67232,abeing unity; and in anatase asc= 1.7771.Holosymmetric Class(Holohedral; Ditetragonal bipyramidal).Crystals of this class are symmetrical with respect to five planes, which are of three kinds; one is perpendicular to the principal axis, and the other four intersect in it; of the latter, two are perpendicular to the equal crystallographic axes, while the two others bisect the angles between them. There are five axes of symmetry, one tetrad and two pairs of dyad, each perpendicular to a plane of symmetry. Finally, there is a centre of symmetry.There are seven kinds of simple forms, viz.:—Tetragonal bipyramid of the first order (figs. 42 and 43). This is bounded by eight equal isosceles triangles. Equal lengths are intercepted on the two horizontal axes, and the indices are {111}, {221}, {112}, &c., or in general {hhl}. The parametral plane with the interceptsa:a:cis a face of the bipyramid {111}.Fig. 44.Fig. 45.Tetragonal Bipyramids of the first and second orders.Tetragonal bipyramid of the second order. This is also bounded by eight equal isosceles triangles, but differs from the last form in its position, four of the faces being parallel to each of the horizontal axes; the indices are therefore {101}, {201}, {102}, &c., or {hol}.Fig. 44 shows the relation between the tetragonal bipyramidsof the first and second orders when the indices are {111} and {101} respectively:ABBis the face (111), andACCis (101). A combination of these two forms is shown in fig. 45.Fig. 46.—Ditetragonal Bipyramid.Ditetragonal bipyramid (fig. 46). This is the general form; it is bounded by sixteen scalene triangles, and all the indices are unequal, being {321}, &c., or {hkl}.Tetragonal prism of the first order. The four faces intersect the horizontal axes in equal lengths and are parallel to the principal axis; the indices are therefore {110}. This form does not enclose space, and is therefore called an “open form” to distinguish it from a “closed form” like the tetragonal bipyramids and all the forms of the cubic system. An open form can exist only in combination with other forms; thus fig. 47 is a combination of the tetragonal prism {110} with the basal pinacoid {001}. If the faces (110) and (001) are of equal size such a figure will be geometrically a cube, since all the angles are right angles; the variety of apophyllite known as tesselite crystallizes in this form.Tetragonal prism of the second order. This has the same number of faces as the last prism, but differs in position; each face being parallel to the vertical axis and one of the horizontal axes; the indices are {100}.Ditetragonal prism. This consists of eight faces all parallel to the principal axis and intercepting the horizontal axes in different lengths; the indices are {210}, {320}, &c., or {hko}.Basal pinacoid (fromπίναξ, a tablet). This consists of a single pair of parallel faces perpendicular to the principal axis. It is therefore an open form and can exist only in combination (fig. 47).Fig. 47.Combination ofTetragonal Prismand Basal Pinacoid.Fig. 48.Fig. 49.Combinations of Tetragonal Prisms and Pyramids.Combinations of holohedral tetragonal forms are shown in figs. 47-49; fig. 48 is a combination of a bipyramid of the first order with one of the second order and the prism of the first order; fig. 49 a combination of a bipyramid of the first order with a ditetragonal bipyramid and the prism of the second order. Compare also figs. 87 and 88.Examples of substances which crystallize in this class are cassiterite, rutile, anatase, zircon, thorite, vesuvianite, apophyllite, phosgenite, also boron, tin, mercuric iodide.Scalenohedral Class(Bisphenoidal-hemihedral).Here there are only three dyad axes and two planes of symmetry, the former coinciding with the crystallographic axes and the latter bisecting the angles between the horizontal pair. The dyad axis of symmetry, which in this class coincides with the principal axis of the crystal, has certain of the characters of a tetrad axis, and is sometimes called a tetrad axis of “alternating symmetry”; a face on the upper half of the crystal if rotated through 90° about this axis and reflected across the equatorial plane falls into the position of a face on the lower half of the crystal. This kind of symmetry, with simultaneous rotation about an axis and reflection across a plane, is also called “composite symmetry.”In this class all except two of the simple forms are geometrically the same as in the holosymmetric class.Bisphenoid (σφήν, a wedge) (fig. 50). This is a double wedge-shaped solid bounded by four equal isosceles triangles; it has the indices {111}, {211}, {112}, &c., or in general {hhl}. By suppressing either one or other set of alternate faces of the tetragonal bipyramid of the first order (fig. 42) two bisphenoids are derived, in the same way that two tetrahedra are derived from the regular octahedron.Tetragonal scalenohedron or ditetragonal bisphenoid (fig. 51). This is bounded by eight scalene triangles and has the indices {hkl}. It may be considered as the hemihedral form of the ditetragonal bipyramid.Fig. 50.—TetragonalBisphenoids.Fig. 51.—TetragonalScalenohedron.The crystal of chalcopyrite (CuFeS2) represented in fig. 52 is a combination of two bisphenoids (PandP′), two bipyramids of the second order (bandc), and the basal pinacoid (a). Stannite (Cu2FeSnS4), acid potassium phosphate (H2KPO4), mercuric cyanide, and urea (CO(NH2)2) also crystallize in this class.Bipyramidal Class(Parallel-faced hemihedral).The elements of symmetry are a tetrad axis with a plane perpendicular to it, and a centre of symmetry. The simple forms are the same here as in the holosymmetric class, except the prism {hko}, which has only four faces, and the bipyramid {hkl}, which has eight faces and is distinguished as a “tetragonal pyramid of the third order.”Fig. 52.—Crystal ofChalcopyrite.Fig. 53.—Crystal ofFergusonite.Fig. 53 shows a combination of a tetragonal prism of the first order with a tetragonal bipyramid of the third order and the basal pinacoid, and represents a crystal of fergusonite. Scheelite (q.v.), scapolite (q.v.), and erythrite (C4H10O4) also crystallize in this class.Pyramidal Class(Hemimorphic-tetartohedral).Here the only element of symmetry is the tetrad axis. The pyramids of the first {hhl}, second {hol} and third {hkl} orders have each only four faces at one or other end of the crystal, and are hemimorphic. All the simple forms are thus open forms.Examples are wulfenite (PbMoO4) and barium antimonyl dextro-tartrate (Ba(SbO)2(C4H4O6)·H2O).Ditetragonal Pyramidal Class(Hemimorphic-hemihedral).Here there are two pairs of vertical planes of symmetry intersecting in the tetrad axis. The pyramids {hhl} and {hol} and the bipyramid {hkl} are all hemimorphic.Examples are iodosuccimide (C4H4O2NI), silver fluoride (AgF·H2O), and penta-erythrite (C5H12O4). No examples are known amongst minerals.Trapezohedral Class(Trapezohedral-hemihedral).Here there are the full number of axes of symmetry, but no planes or centre of symmetry. The general form {hkl} is bounded by eight trapezoidal faces and is the tetragonal trapezohedron.Examples are nickel sulphate (NiSO4·6H2O), guanidine carbonate ((CH5N3)2H2CO3), strychnine sulphate ((C21H22N2O2)2·H2SO4·6H2O).Bisphenoidal Class(Bisphenoidal-tetartohedral).Here there is only a single dyad axis of symmetry, which coincides with the principal axis. All the forms, except the prisms and basal pinacoid, are sphenoids. Crystals possessing this type of symmetry have not yet been observed.3. ORTHORHOMBIC SYSTEM(Rhombic; Prismatic; Trimetric).In this system the three crystallographic axes are all at right angles, but they are of different lengths and not interchangeable. The parameters, or axial ratios, area:b:c, these referring to the axesOX,OYandOZrespectively. The choice of a vertical axis,OZ = c, is arbitrary, and it is customary to place the longer of the two horizontal axes from left to right (OY = b) and take it as unity: this is called the “macro-axis” or “macro-diagonal” (fromμακρός, long), whilst the shorter horizontal axis (OX = a) is called the “brachy-axis” or “brachy-diagonal” (fromβραχύς, short). The axial ratios are constant for crystals of any one substance and are characteristic of it; for example, in barytes (BaSO4),a:b:c= 0.8152 : 1 : 1.3136; in anglesite (PbSO4),a:b:c= 0.7852: 1 : 1.2894; in cerussite (PbCO3),a:b:c= 0.6100 : 1 : 0.7230.There are three symmetry-classes in this system:—Holohedral Class(Holohedral; Bipyramidal).Here there are three dissimilar dyad axes of symmetry, each coinciding with a crystallographic axis; perpendicular to them are three dissimilar planes of symmetry; there is also a centre of symmetry. There are seven kinds of simple forms:—Fig. 54.Fig. 55.Orthorhombic Bipyramids.Bipyramid (figs. 54 and 55). This is the general form and is bounded by eight scalene triangles; the indices are {111}, {211}, {221}, {112}, {321}, {123}, &c., or in general {hkl}. The crystallographic axes join opposite corners of these pyramids and in the fundamental bipyramid {111} the parametral plane has the interceptsa:b:c. This is the only closed form in this class; the others are open forms and can exist only in combination. Sulphur often crystallizes in simple bipyramids.Prism. This consists of four faces parallel to the vertical axis and intercepting the horizontal axes in the lengths a and b or in any multiples of these; the indices are therefore {110}, {210}, {120} or {hko}.Fig. 56.—Macro-prism andBrachy-pinacoid.Fig. 57.—Brachy-prism andMacro-pinacoid.Macro-prism. This consists of four faces parallel to the macro-axis, and has the indices {101}, {201} ... or {hol}.Brachy-prism. This consists of four faces parallel to the brachy-axis, and has the indices {011}, {021} ... {okl}. The macro- and brachy-prisms are often called “domes.”Basal pinacoid, consisting of a pair of parallel faces perpendicular to the vertical axis; the indices are {001}. The macro-pinacoid {100} and the brachy-pinacoid {010} each consist of a pair of parallel faces respectively parallel to the macro- and the brachy-axis.Figs. 56-58 show combinations of these six open forms, and fig. 59 a combination of the macro-pinacoid (a), brachy-pinacoid (b), a prism (m), a macro-prism (d), a brachy-prism (k), and a bipyramid (u).Fig. 58.—Prism and BasalPinacoid.Fig. 59.—Crystal ofHypersthene.Holohedral Orthorhombic Combinations.Examples of substances crystallizing in this class are extremely numerous; amongst minerals are sulphur, stibnite, cerussite, chrysoberyl, topaz, olivine, nitre, barytes, columbite and many others; and amongst artificial products iodine, potassium permanganate, potassium sulphate, benzene, barium formate, &c.Pyramidal Class(Hemimorphic).Here there is only one dyad axis in which two planes of symmetry intersect. The crystals are usually so placed that the dyad axis coincides with the vertical crystallographic axis, and the planes of symmetry are also vertical.The pyramid {hkl} has only four faces at one end or other of the crystal. The macro-prism and the brachy-prism of the last class are here represented by the macro-dome and brachy-dome respectively, so called because of the resemblance of the pair of equally sloped faces to the roof of a house. The form {001} is a single plane at the top of the crystal, and is called a “pedion”; the parallel pedion {001}, if present at the lower end of the crystal, constitutes a different form. The prisms {hko} and the macro- and brachy-pinacoids are geometrically the same in this class as in the last. Crystals of this class are therefore differently developed at the two ends and are said to be “hemimorphic.”Fig. 60.—Crystal ofHemimorphite.Fig. 61.—OrthorhombicBisphenoid.Fig. 60 shows a crystal of the mineral hemimorphite (H2Zn2SiO5) which is a combination of the brachy-pinacoid {010} and a prism, with the pedion (001), two brachy-domes and two macro-domes at the upper end, and a pyramid at the lower end. Examples of other substances belonging to this class are struvite (NH4MgPO4·6H2O), bertrandite (H2Be4Si2O9), resorcin, and picric acid.Bisphenoidal Class(Hemihedral).Here there are three dyad axes, but no planes of symmetry and no centre of symmetry. The general form {hkl} is a bisphenoid (fig. 61) bounded by four scalene triangles. The other simple forms are geometrically the same as in the holosymmetric class.Examples: epsomite (Epsom salts, MgSO4·7H2O), goslarite (ZnSO4·7H2O), silver nitrate, sodium potassium dextro-tartrate (seignette salt, NaKC4H4O6·4H2O), potassium antimonyl dextro-tartrate (tartar-emetic, K(SbO)C4H4O6), and asparagine (C4H8N2O8·H2O).4. MONOCLINIC5SYSTEM(Oblique; Monosymmetric).In this system two of the angles between the crystallographic axes are right angles, but the third angle is oblique, and the axes are of unequal lengths. The axis which is perpendicular to the other two is taken asOY = b(fig. 62) and is called the ortho-axis or ortho-diagonal. The choice of the other two axes is arbitrary; the vertical axis (OZ = c) is usually taken parallel to the edges of a prominently developed prismatic zone, and the clino-axis or clino-diagonal (OX = a) parallel to the zone-axis of some other prominent zone on the crystal. The acute angle between the axesOXandOZis usually denoted as β, and it is necessary to know its magnitude, in addition to the axial ratiosa:b:c, before the crystal is completely determined. As in other systems, except the cubic, these elements,a:b:cand β, are characteristic of the substance. Thus for gypsuma:b:c= 0.6899 : 1 : 0.4124; β = 80° 42′; for orthoclasea:b:c= 0.6585 : 1 : 0.5554; β = 63° 57′; and for cane-sugara:b:c= 1.2595 : 1 : 0.8782; β = 76° 30′.Holosymmetric Class(Holohedral; Prismatic).Here there is a single plane of symmetry perpendicular to which is a dyad axis; there is also a centre of symmetry. The dyad axis coincides with the ortho-axisOY, and the vertical axisOZand the clino-axisOXlie in the plane of symmetry.Fig. 62.—Monoclinic Axes andHemi-pyramid.Fig. 63.—Crystal of Augite.All the forms are open, being either pinacoids or prisms; the former consisting of a pair of parallel faces, and the latter of four faces intersecting in parallel edges and with a rhombic cross-section. The pair of faces parallel to the plane of symmetry is distinguished as the “clino-pinacoid” and has the indices {010}. The other pinacoids are all perpendicular to the plane of symmetry (and parallel to the ortho-axis); the one parallel to the vertical axis is called the “ortho-pinacoid” {100}, whilst that parallel to the clino-axis is the “basal pinacoid” {001}; pinacoids not parallel to the arbitrarily chosen clino- and vertical axes may have the indices {101}, {201}, {102} ... {hol} or {101}, {201}, {102} ... {hol}, according to whether they lie in the obtuse or the acute axial angle. Of the prisms, those with edges (zone-axis) parallel to the clino-axis, and having indices {011}, {021}, {012} ... {okl}, are called “clino-prisms”; those with edges parallel to the vertical axis, and with the indices {110}, {210}, {120} ... {hko}, are called simply “prisms.” Prisms with edges parallel to neither of the axesOXandOYhave the indices {111}, {221}, {211}, {321} ... {hkl} or {111} ... {hkl}, and are usually called “hemi-pyramids” (fig. 62); they are distinguished as negative or positive according to whether they lie in the obtuse or the acute axial angle β.Fig. 63 represents a crystal of augite bounded by the clino-pinacoid (l), the ortho-pinacoid (r), a prism (M), and a hemi-pyramid (s).The substances which crystallize in this class are extremely numerous: amongst minerals are gypsum, orthoclase, the amphiboles, pyroxenes and micas, epidote, monazite, realgar, borax, mirabilite (Na2SO4·10H2O), melanterite (FeSO4·7H2O) and many others; amongst artificial products are monoclinic sulphur, barium chloride (BaCl2·2H2O), potassium chlorate, potassium ferrocyanide (K4Fe(CN)6·3H2O), oxalic acid (C2O4H2·2H2O), sodium acetate (NaC2H3O2·3H2O) and naphthalene.Hemimorphic Class(Sphenoidal).In this class the only element of symmetry is a single dyad axis, which is polar in character, being dissimilar at the two ends.The form {010} perpendicular to the axis of symmetry consists of a single plane or pedion; the parallel face is dissimilar in character and belongs to the pedion {010}. The pinacoids {100}, {001}, {hol} and {hol} parallel to the axis of symmetry are geometrically the same in this class as in the holosymmetric class. The remaining forms consist each of only two planes on the same side of the axial planeXOZand equally inclined to the dyad axis (e.g.in fig. 62 the two planesXYZandXYZ); such a wedge-shaped form is sometimes called a sphenoid.Fig. 64.—Enantiomorphous Crystals of Tartaric Acid.Fig. 64 shows two crystals of tartaric acid,aa right-handed crystal of dextro-tartaric acid, andba left-handed crystal of laevo-tartaric acid. The two crystals are enantiomorphous,i.e.although they have the same interfacial angles they are not superposable, one being the mirror image of the other. Other examples are potassium dextro-tartrate, cane-sugar, milk-sugar, quercite, lithium sulphate (Li2SO4·H2O); amongst minerals the only example is the hydrocarbon fichtelite (C5H8).Clinohedral Class(Hemihedral; Domatic).Crystals of this class are symmetrical only with respect to a single plane. The only form which is here geometrically the same as in the holosymmetric class is the clino-pinacoid {010}. The forms perpendicular to the plane of symmetry are all pedions, consisting of single planes with the indices {100}, {100}, {001}, {001}, {hol}, &c. The remaining forms, {hko}, {okl} and {hkl}, are domes or “gonioids” (γωνία, an angle, andεἶδος, form), consisting of two planes equally inclined to the plane of symmetry.Examples are potassium tetrathionate (K2S4O6), hydrogen trisodium hypophosphate (HNa3P2O6·9H2O); and amongst minerals, clinohedrite (H2ZnCaSiO4) and scolectite.5. ANORTHIC SYSTEM(Triclinic).In the anorthic (fromἀν, privative, andὀρθός, right) or triclinic system none of the three crystallographic axes are at right angles, and they are all of unequal lengths. In addition to the parameters a : b : c, it is necessary to know the angles, α, β, and γ, between the axes. In anorthite, for example, these elements area:b:c= 0.6347 : 1 : 0.5501; α = 93° 13′, β = 115° 55′, γ = 91° 12′.Holosymmetric Class(Holohedral; Pinacoidal).Here there is only a centre of symmetry. All the forms are pinacoids, each consisting of only two parallel faces. The indices of the three pinacoids parallel to the axial planes are {100}, {010} and {001}; those of pinacoids parallel to only one axis are {hko}, {hol} and {okl}; and the general form is {hkl}.Fig. 65.—Crystal of Axinite.Several minerals crystallize in this class; for example, the plagioclastic felspars, microcline, axinite (fig. 65), cyanite, amblygonite, chalcanthite (CuSO4·5H2O), sassolite (H3BO3); among artificial substances are potassium bichromate, racemic acid (C4H6O6·2H2O), dibrom-para-nitrophenol, &c.Asymmetric Class(Hemihedral, Pediad).Crystals of this class are devoid of any elements of symmetry. All the forms are pedions, each consisting of a single plane; they are thus hemihedral with respect to crystals of the last class. Although there is a total absence of symmetry, yet the faces are arranged in zones on the crystals.Examples are calcium thiosulphate (CaS2O3·6H2O) and hydrogen strontium dextro-tartrate ((C4H4O6H)2Sr·5H2O); there is no example amongst minerals.6. HEXAGONAL SYSTEMCrystals of this system are characterized by the presence of a single axis of either triad or hexad symmetry, which is spoken of as the “principal” or “morphological” axis. Those with a triad axis are grouped together in the rhombohedral or trigonal division, and those with a hexad axis in the hexagonal division. By some authors these two divisions are treated as separate systems; or again the rhombohedral forms may be considered as hemihedral developmentsof the hexagonal. On the other hand, hexagonal forms may be considered as a combination of two rhombohedral forms.Owing to the peculiarities of symmetry associated with a single triad or hexad axis, the crystallographic axes of reference are different in this system from those used in the five other systems of crystals. Two methods of axial representation are in common use; rhombohedral axes being usually used for crystals of the rhombohedral division, and hexagonal axes for those of the hexagonal division; though sometimes either one or the other set is employed in both divisions.Rhomobohedral axes are taken parallel to the three sets of edges of a rhombohedron (fig. 66). They are inclined to one another at equal oblique angles, and they are all equally inclined to the principal axis; further, they are all of equal length and are interchangeable. With such a set of axes there can be no statement of an axial ratio, but the angle between the axes (or some other angle which may be calculated from this) may be given as a constant of the substance. Thus in calcite the rhombohedral angle (the angle between two faces of the fundamental rhombohedron) is 74° 55′, or the angle between the normal to a face of this rhombohedron and the principal axis is 44° 36½′.Hexagonal axes are four in number, viz. a vertical axis coinciding with the principal axis of the crystal, and three horizontal axes inclined to one another at 60° in a plane perpendicular to the principal axis. The three horizontal axes, which are taken either parallel or perpendicular to the faces of a hexagonal prism (fig. 71) or the edge of a hexagonal bipyramid (fig. 70), are equal in length (a) but the vertical axis is of a different length (c). The indices of planes referred to such a set of axes are four in number; they are written as {hikl}, the first three (h+i+k= 0) referring to the horizontal axes and the last to the vertical axis. The ratioa:cof the parameters, or the axial ratio, is characteristic of all the crystals of the same substance. Thus for beryl (including emerald)a:c= 1 : 0.4989 (often writtenc= 0.4989); for zincc= 1.3564.Rhombohedral Division.In the rhomobohedral or trigonal division of the hexagonal system there are seven symmetry-classes, all of which possess a single triad axis of symmetry.Holosymmetric Class(Holohedral; Ditrigonal scalenohedral).In this class, which presents the commonest type of symmetry of the hexagonal system, the triad axis is associated with three similar planes of symmetry inclined to one another at 60° and intersecting in the triad axis; there are also three similar dyad axes, each perpendicular to a plane of symmetry, and a centre of symmetry. The seven simple forms are:—Fig. 66.Fig. 67.Direct and Inverse Rhombohedra.Fig. 68.—Scalenohedron.Rhombohedron (figs. 66 and 67), consisting of six rhomb-shaped faces with the edges all of equal lengths: the faces are perpendicular to the planes of symmetry. There are two sets of rhombohedra, distinguished respectively as direct and inverse; those of one set (fig. 66) are brought into the orientation of the other set (fig. 67) by a rotation of 60° or 180° about the principal axis. For the fundamental rhombohedron, parallel to the edges of which are the crystallographic axes of reference, the indices are {100}. Other rhombohedra may have the indices {211}, {411}, {110}, {221}, {111}, &c., or in general {hkk}. (Compare fig. 72; for figures of other rhombohedra seeCalcite.)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), whereh− 2k+l= 0.Fig. 69.—Scalenohedron withinscribed Rhombohedron.Fig. 70.—HexagonalBipyramid.Fig. 71.—Hexagonal Prismand Basal Pinacoid.Hexagonal prism of the first order (211), 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°.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° 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 {321}, {532} ... {hkl}, whereh+k+l= 0.Basal pinacoid {111}, consisting of a pair of parallel faces perpendicular to the principal axis.Fig. 72.—Stereographic Projection of a Holosymmetric Rhombohedral Crystal.Fig. 71 shows a combination of a hexagonal prism (m) with the basal pinacoid (c). For figures of other combinations seeCalciteandCorundum. 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, crystallizein 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:—Fig. 73.—Crystal of Tourmaline.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 {211} or {211}, 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 {hkl} (whereh+k+l= 0), of six faces parallel to the principal axis, and with two sets of angles between them.Basal pedion (111) or (111), each consisting of a single plane perpendicular to the principal axis.Fig. 73 represents a crystal of tourmaline with the trigonal prism (211), 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° 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.—TrigonalTrapezohedron.Fig. 75.—TrigonalBipyramid.In this class the rhombohedra {hkk}, the hexagonal prism {211}, 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}, whereh− 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}, whereh− 2k+l= 0, is also a rhombohedron, being the hemihedral form of the hexagonal bipyramid. The dihexagonal prism {hkl} of the holosymmetric class becomes here a hexagonal prism. The rhombohedra (hkk), hexagonal prisms {211} 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 rhombohedronr{100} and the hexagonal prism of the second orderm{101} combined with the rhombohedrons{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).Fig.76.—Crystal of Dioptase.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 ClassHere 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 ClassHere 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.Fig.77.—Dihexagonal Bipyramid.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,pqr}, (hkl) for faces above the equatorial plane of symmetry and (pqr) 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 {211} 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;vin fig. 80). The indices are {2131}, &c., or in general {hikl}. This form may be considered as a combination of two scalenohedra, a direct and an inverse.Fig. 79.Fig. 80.Fig. 81.Combinations of Hexagonal forms.Hexagonal bipyramid of the first order, bounded by twelve isosceles triangles (fig. 70;panduin fig. 80); indices {1011}, {2021} ... (hohl). 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 (sin figs. 79 and 80); indices {1121}, {1122} ... {h.h.2h.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 (min figs. 71, 78-80).Hexagonal prism of the second order {1120}, consisting of six faces also parallel to the hexad axis, but perpendicular to the other set of three vertical planes of symmetry (ain fig. 78).Basal pinacoid {0001}, consisting of a pair of parallel planes perpendicular to the hexad axis (cin 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 {hohl} and (h.h.2h.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).

1. CUBIC SYSTEM

(Isometric; Regular; Octahedral; Tesseral).

In this system the three crystallographic axes of reference are all at right angles to each other and are equal in length. They are parallel to the edges of the cube, and in the different classes coincide either with tetrad or dyad axes of symmetry. Five classes are included in this system, in all of which there are, besides other elements of symmetry, four triad axes.

In crystals of this system the angle between any two facesPandQwith the indices (hkl) and (pqr) is given by the equation

The angles between faces with the same indices are thus the same in all substances which crystallize in the cubic system: in other systems the angles vary with the substance and are characteristic of it.

Holosymmetric Class

(Holohedral (ὅλος, whole); Hexakis-octahedral).

Crystals of this class possess the full number of elements of symmetry already mentioned above for the octahedron and the cube, viz. three cubic planes of symmetry, six dodecahedral planes, three tetrad axes of symmetry, four triad axes, six dyad axes, and a centre of symmetry.

There are seven kinds of simple forms, viz.:—

Cube (fig. 5). This is bounded by six square faces parallel to the cubic planes of symmetry; it is known also as the hexahedron. The angles between the faces are 90°, and the indices of the form are {100}. Salt, fluorspar and galena crystallize in simple cubes.

Octahedron (fig. 3). Bounded by eight equilateral triangular faces perpendicular to the triad axes of symmetry. The angles between the faces are 70° 32′ and 109° 28′, and the indices are {111}. Spinel, magnetite and gold crystallize in simple octahedra. Combinations of the cube and octahedron are shown in figs. 6-8.

Rhombic dodecahedron (fig. 13). Bounded by twelve rhomb-shaped faces parallel to the six dodecahedral planes of symmetry. The angles between the normals to adjacent faces are 60°, and between other pairs of faces 90°; the indices are {110}. Garnet frequently crystallizes in this form. Fig. 14 shows the rhombic dodecahedron in combination with the octahedron.

In these three simple forms of the cubic system (which are shown in combination in fig. 11) the angles between the faces and the indices are fixed and are the same in all crystals; in the four remaining simple forms they are variable.

Triakis-octahedron (three-faced octahedron) (fig. 15). This solid is bounded by twenty-four isosceles triangles, and may be considered as an octahedron with a low triangular pyramid on each of its faces. As the inclinations of the faces may vary there is a series of these forms with the indices {221}, {331}, {332}, &c. or in general {hhk}.

Icositetrahedron (fig. 17). Bounded by twenty-four trapezoidal faces, and hence sometimes called a “trapezohedron.” The indices are {211}, {311}, {322}, &c., or in general {hkk}. Analcite, leucite and garnet often crystallize in the simple form {211}. Combinations are shown in figs. 18-20. The planeABein fig. 9 is one face (112) of an icositetrahedron; the indices of the remaining faces in this octant being (211) and (121).

Tetrakis-hexahedron (four-faced cube) (figs. 21 and 22). Like the triakis-octahedron this solid is also bounded by twenty-four isosceles triangles, but here grouped in fours over the cubic faces. The two figures show how, with different inclinations of the faces, the form may vary, approximating in fig. 21 to the cube and in fig. 22 to the rhombic dodecahedron. The angles over the edges letteredAare different from the angles over the edges letteredC. Each face is parallel to one of the crystallographic axes and intercepts the two others in different lengths; the indices are therefore {210}, {310}, {320}, &c., in general {hko}. Fluorspar sometimes crystallizes in the simple form {310}; more usually, however, in combination with the cube (fig. 23).

Hexakis-octahedron (fig. 24). Here each face of the octahedron is replaced by six scalene triangles, so that altogether there areforty-eight faces. This is the greatest number of faces possible for any simple form in crystals. The faces are all oblique to the planes and axes of symmetry, and they intercept the three crystallographic axes in different lengths, hence the indices are all unequal, being in general {hkl}, or in particular cases {321}, {421}, {432}, &c. Such a form is known as the “general form” of the class. The interfacial angles over the three edges of each triangle are all different. These forms usually exist only in combination with other cubic forms (for example, fig. 25), but {421} has been observed as a simple form on fluorspar.

Several examples of substances which crystallize in this class have been mentioned above under the different forms; many others might be cited—for instance, the metals iron, copper, silver, gold, platinum, lead, mercury, and the non-metallic elements silicon and phosphorus.

Tetrahedral Class

(Tetrahedral-hemihedral; Hexakis-tetrahedral).

In this class there is no centre of symmetry nor cubic planes of symmetry; the three tetrad axes become dyad axes of symmetry, and the four triad axes are polar,i.e.they are associated with different faces at their two ends. The other elements of symmetry (six dodecahedral planes and six dyad axes) are the same as in the last class.

Of the seven simple forms, the cube, rhombic dodecahedron and tetrakis-hexahedron are geometrically the same as before, though on actual crystals the faces will have different surface characters. For instance, the cube faces will be striated parallel to only one of the diagonals (fig. 90), and etched figures on this face will be symmetrical with respect to two lines, instead of four as in the last class. The remaining simple forms have, however, only half the number of faces as the corresponding form in the last class, and are spoken of as “hemihedral with inclined faces.”

Tetrahedron (fig. 26). This is bounded by four equilateral triangles and is identical with the regular tetrahedron of geometry. The angles between the normals to the faces are 109° 28′. It may be derived from the octahedron by suppressing the alternate faces.

Deltoid1dodecahedron (fig. 27). This is the hemihedral form of the triakis-octahedron; it has the indices {hhk} and is bounded by twelve trapezoidal faces.

Triakis-tetrahedron (fig. 28). The hemihedral form {hkk} of the icositetrahedron; it is bounded by twelve isosceles triangles arranged in threes over the tetrahedron faces.

Hexakis-tetrahedron (fig. 29). The hemihedral form {hkl} of the hexakis-octahedron; it is bounded by twenty-four scalene triangles and is the general form of the class.

Corresponding to each of these hemihedral forms there is another geometrically similar form, differing, however, not only in orientation, but also in actual crystals in the characters of the faces. Thus from the octahedron there may be derived two tetrahedra with the indices {111} and {111}, which may be distinguished as positive and negative respectively. Fig. 30 shows a combination of these two tetrahedra, and represents a crystal of blende, in which the four larger faces are dull and striated, whilst the four smaller are bright and smooth. Figs. 31-33 illustrate other tetrahedral combinations.

Tetrahedrite, blende, diamond, boracite and pharmacosiderite are substances which crystallize in this class.

Pyritohedral2Class

(Parallel-faced hemihedral; Dyakis-dodecahedral).

Crystals of this class possess three cubic planes of symmetry but no dodecahedral planes. There are only three dyad axes of symmetry, which coincide with the crystallographic axes; in addition there are three triad axes and a centre of symmetry.

Here the cube, octahedron, rhombic dodecahedron, triakis-octahedron and icositetrahedron are geometrically the same as in the first class. The characters of the faces will, however, be different; thus the cube faces will be striated parallel to one edge only (fig. 89), and triangular markings on the octahedron faces will be placed obliquely to the edges. The remaining simple forms are “hemihedral with parallel faces,” and from the corresponding holohedral forms two hemihedral forms, a positive and a negative, may be derived.

Pentagonal dodecahedron (fig. 34). This is bounded by twelve pentagonal faces, but these are not regular pentagons, and the angles over the three sets of different edges are different. The regular dodecahedron of geometry, contained by twelve regular pentagons, is not a possible form in crystals. The indices are {hko}: as a simple form {210} is of very common occurrence in pyrites.

Dyakis-dodecahedron (fig. 35). This is the hemihedral form ofthe hexakis-octahedron and has the indices {hkl}; it is bounded by twenty-four faces. As a simple form {321} is met with in pyrites.

Combinations (figs. 36-39) of these forms with the cube and the octahedron are common in pyrites. Fig. 37 resembles in general appearance the regular icosahedron of geometry, but only eight of the faces are equilateral triangles. Cobaltite, smaltite and other sulphides and sulpharsenides of the pyrites group of minerals crystallize in these forms. The alums also belong to this class; from an aqueous solution they crystallize as simple octahedra, sometimes with subordinate faces of the cube and rhombic dodecahedron, but from an acid solution as octahedra combined with the pentagonal dodecahedron {210}.

Plagihedral3Class

(Plagihedral-hemihedral; Pentagonal icositetrahedral; Gyroidal4).

In this class there are the full number of axes of symmetry (three tetrad, four triad and six dyad), but no planes of symmetry and no centre of symmetry.

Pentagonal icositetrahedron (fig. 40). This is the only simple form in this class which differs geometrically from those of the holosymmetric class. By suppressing either one or other set of alternate faces of the hexakis-octahedron two pentagonal icositetrahedra {hkl} and {khl} are derived. These are each bounded by twenty-four irregular pentagons, and although similar to each other they are respectively right- and left-handed, one being the mirror image of the other; such similar but nonsuperposable forms are said to be enantiomorphous (ἐναντίος, opposite, andμορφή, form), and crystals showing such forms sometimes rotate the plane of polarization of plane-polarized light. Faces of a pentagonal icositetrahedron with high indices have been very rarely observed on crystals of cuprite, potassium chloride and ammonium chloride, but none of these are circular polarizing.

Tetartohedral Class

(Tetrahedral pentagonal dodecahedral).

Here, in addition to four polar triad axes, the only other elements of symmetry are three dyad axes, which coincide with the crystallographic axes. Six of the simple forms, the cube, tetrahedron, rhombic dodecahedron, deltoid dodecahedron, triakis-tetrahedron and pentagonal dodecahedron, are geometrically the same in this class as in either the tetrahedral or pyritohedral classes. The general form is the Tetrahedral pentagonal dodecahedron (fig. 41). This is bounded by twelve irregular pentagons, and is a tetartohedral or quarter-faced form of the hexakis-octahedron. Four such forms may be derived, the indices of which are {hkl}, {khl}, {hkl} and {khl}; the first pair are enantiomorphous with respect to one another, and so are the last pair. Barium nitrate, lead nitrate, sodium chlorate and sodium bromate crystallize in this class, as also do the minerals ullmannite (NiSbS) and langbeinite (K2Mg2(SO4)3).

2. TETRAGONAL SYSTEM

(Pyramidal; Quadratic; Dimetric).

In this system the three crystallographic axes are all at right angles, but while two are equal in length and interchangeable the third is of a different length. The unequal axis is spoken of as the principal axis or morphological axis of the crystal, and it is always placed in a vertical position; in five of the seven classes of this system it coincides with the single tetrad axis of symmetry.

The parameters area:a:c, where a refers to the two equal horizontal axes, andcto the vertical axis;cmay be either shorter (as in fig. 42) or longer (fig. 43) thana. The ratioa:cis spoken of as the axial ratio of a crystal, and it is dependent on the angles between the faces. In all crystals of the same substance this ratio is constant, and is characteristic of the substance; for other substances crystallizing in the tetragonal system it will be different. For example, in cassiterite it is given asa:c= 1 : 0.67232 or simply asc= 0.67232,abeing unity; and in anatase asc= 1.7771.

Holosymmetric Class

(Holohedral; Ditetragonal bipyramidal).

Crystals of this class are symmetrical with respect to five planes, which are of three kinds; one is perpendicular to the principal axis, and the other four intersect in it; of the latter, two are perpendicular to the equal crystallographic axes, while the two others bisect the angles between them. There are five axes of symmetry, one tetrad and two pairs of dyad, each perpendicular to a plane of symmetry. Finally, there is a centre of symmetry.

There are seven kinds of simple forms, viz.:—

Tetragonal bipyramid of the first order (figs. 42 and 43). This is bounded by eight equal isosceles triangles. Equal lengths are intercepted on the two horizontal axes, and the indices are {111}, {221}, {112}, &c., or in general {hhl}. The parametral plane with the interceptsa:a:cis a face of the bipyramid {111}.

Tetragonal bipyramid of the second order. This is also bounded by eight equal isosceles triangles, but differs from the last form in its position, four of the faces being parallel to each of the horizontal axes; the indices are therefore {101}, {201}, {102}, &c., or {hol}.

Fig. 44 shows the relation between the tetragonal bipyramidsof the first and second orders when the indices are {111} and {101} respectively:ABBis the face (111), andACCis (101). A combination of these two forms is shown in fig. 45.

Ditetragonal bipyramid (fig. 46). This is the general form; it is bounded by sixteen scalene triangles, and all the indices are unequal, being {321}, &c., or {hkl}.

Tetragonal prism of the first order. The four faces intersect the horizontal axes in equal lengths and are parallel to the principal axis; the indices are therefore {110}. This form does not enclose space, and is therefore called an “open form” to distinguish it from a “closed form” like the tetragonal bipyramids and all the forms of the cubic system. An open form can exist only in combination with other forms; thus fig. 47 is a combination of the tetragonal prism {110} with the basal pinacoid {001}. If the faces (110) and (001) are of equal size such a figure will be geometrically a cube, since all the angles are right angles; the variety of apophyllite known as tesselite crystallizes in this form.

Tetragonal prism of the second order. This has the same number of faces as the last prism, but differs in position; each face being parallel to the vertical axis and one of the horizontal axes; the indices are {100}.

Ditetragonal prism. This consists of eight faces all parallel to the principal axis and intercepting the horizontal axes in different lengths; the indices are {210}, {320}, &c., or {hko}.

Basal pinacoid (fromπίναξ, a tablet). This consists of a single pair of parallel faces perpendicular to the principal axis. It is therefore an open form and can exist only in combination (fig. 47).

Combinations of holohedral tetragonal forms are shown in figs. 47-49; fig. 48 is a combination of a bipyramid of the first order with one of the second order and the prism of the first order; fig. 49 a combination of a bipyramid of the first order with a ditetragonal bipyramid and the prism of the second order. Compare also figs. 87 and 88.

Examples of substances which crystallize in this class are cassiterite, rutile, anatase, zircon, thorite, vesuvianite, apophyllite, phosgenite, also boron, tin, mercuric iodide.

Scalenohedral Class

(Bisphenoidal-hemihedral).

Here there are only three dyad axes and two planes of symmetry, the former coinciding with the crystallographic axes and the latter bisecting the angles between the horizontal pair. The dyad axis of symmetry, which in this class coincides with the principal axis of the crystal, has certain of the characters of a tetrad axis, and is sometimes called a tetrad axis of “alternating symmetry”; a face on the upper half of the crystal if rotated through 90° about this axis and reflected across the equatorial plane falls into the position of a face on the lower half of the crystal. This kind of symmetry, with simultaneous rotation about an axis and reflection across a plane, is also called “composite symmetry.”

In this class all except two of the simple forms are geometrically the same as in the holosymmetric class.

Bisphenoid (σφήν, a wedge) (fig. 50). This is a double wedge-shaped solid bounded by four equal isosceles triangles; it has the indices {111}, {211}, {112}, &c., or in general {hhl}. By suppressing either one or other set of alternate faces of the tetragonal bipyramid of the first order (fig. 42) two bisphenoids are derived, in the same way that two tetrahedra are derived from the regular octahedron.

Tetragonal scalenohedron or ditetragonal bisphenoid (fig. 51). This is bounded by eight scalene triangles and has the indices {hkl}. It may be considered as the hemihedral form of the ditetragonal bipyramid.

The crystal of chalcopyrite (CuFeS2) represented in fig. 52 is a combination of two bisphenoids (PandP′), two bipyramids of the second order (bandc), and the basal pinacoid (a). Stannite (Cu2FeSnS4), acid potassium phosphate (H2KPO4), mercuric cyanide, and urea (CO(NH2)2) also crystallize in this class.

Bipyramidal Class

(Parallel-faced hemihedral).

The elements of symmetry are a tetrad axis with a plane perpendicular to it, and a centre of symmetry. The simple forms are the same here as in the holosymmetric class, except the prism {hko}, which has only four faces, and the bipyramid {hkl}, which has eight faces and is distinguished as a “tetragonal pyramid of the third order.”

Fig. 53 shows a combination of a tetragonal prism of the first order with a tetragonal bipyramid of the third order and the basal pinacoid, and represents a crystal of fergusonite. Scheelite (q.v.), scapolite (q.v.), and erythrite (C4H10O4) also crystallize in this class.

Pyramidal Class

(Hemimorphic-tetartohedral).

Here the only element of symmetry is the tetrad axis. The pyramids of the first {hhl}, second {hol} and third {hkl} orders have each only four faces at one or other end of the crystal, and are hemimorphic. All the simple forms are thus open forms.

Examples are wulfenite (PbMoO4) and barium antimonyl dextro-tartrate (Ba(SbO)2(C4H4O6)·H2O).

Ditetragonal Pyramidal Class

(Hemimorphic-hemihedral).

Here there are two pairs of vertical planes of symmetry intersecting in the tetrad axis. The pyramids {hhl} and {hol} and the bipyramid {hkl} are all hemimorphic.

Examples are iodosuccimide (C4H4O2NI), silver fluoride (AgF·H2O), and penta-erythrite (C5H12O4). No examples are known amongst minerals.

Trapezohedral Class

(Trapezohedral-hemihedral).

Here there are the full number of axes of symmetry, but no planes or centre of symmetry. The general form {hkl} is bounded by eight trapezoidal faces and is the tetragonal trapezohedron.

Examples are nickel sulphate (NiSO4·6H2O), guanidine carbonate ((CH5N3)2H2CO3), strychnine sulphate ((C21H22N2O2)2·H2SO4·6H2O).

Bisphenoidal Class

(Bisphenoidal-tetartohedral).

Here there is only a single dyad axis of symmetry, which coincides with the principal axis. All the forms, except the prisms and basal pinacoid, are sphenoids. Crystals possessing this type of symmetry have not yet been observed.

3. ORTHORHOMBIC SYSTEM

(Rhombic; Prismatic; Trimetric).

In this system the three crystallographic axes are all at right angles, but they are of different lengths and not interchangeable. The parameters, or axial ratios, area:b:c, these referring to the axesOX,OYandOZrespectively. The choice of a vertical axis,OZ = c, is arbitrary, and it is customary to place the longer of the two horizontal axes from left to right (OY = b) and take it as unity: this is called the “macro-axis” or “macro-diagonal” (fromμακρός, long), whilst the shorter horizontal axis (OX = a) is called the “brachy-axis” or “brachy-diagonal” (fromβραχύς, short). The axial ratios are constant for crystals of any one substance and are characteristic of it; for example, in barytes (BaSO4),a:b:c= 0.8152 : 1 : 1.3136; in anglesite (PbSO4),a:b:c= 0.7852: 1 : 1.2894; in cerussite (PbCO3),a:b:c= 0.6100 : 1 : 0.7230.

There are three symmetry-classes in this system:—

Holohedral Class

(Holohedral; Bipyramidal).

Here there are three dissimilar dyad axes of symmetry, each coinciding with a crystallographic axis; perpendicular to them are three dissimilar planes of symmetry; there is also a centre of symmetry. There are seven kinds of simple forms:—

Bipyramid (figs. 54 and 55). This is the general form and is bounded by eight scalene triangles; the indices are {111}, {211}, {221}, {112}, {321}, {123}, &c., or in general {hkl}. The crystallographic axes join opposite corners of these pyramids and in the fundamental bipyramid {111} the parametral plane has the interceptsa:b:c. This is the only closed form in this class; the others are open forms and can exist only in combination. Sulphur often crystallizes in simple bipyramids.

Prism. This consists of four faces parallel to the vertical axis and intercepting the horizontal axes in the lengths a and b or in any multiples of these; the indices are therefore {110}, {210}, {120} or {hko}.

Macro-prism. This consists of four faces parallel to the macro-axis, and has the indices {101}, {201} ... or {hol}.

Brachy-prism. This consists of four faces parallel to the brachy-axis, and has the indices {011}, {021} ... {okl}. The macro- and brachy-prisms are often called “domes.”

Basal pinacoid, consisting of a pair of parallel faces perpendicular to the vertical axis; the indices are {001}. The macro-pinacoid {100} and the brachy-pinacoid {010} each consist of a pair of parallel faces respectively parallel to the macro- and the brachy-axis.

Figs. 56-58 show combinations of these six open forms, and fig. 59 a combination of the macro-pinacoid (a), brachy-pinacoid (b), a prism (m), a macro-prism (d), a brachy-prism (k), and a bipyramid (u).

Examples of substances crystallizing in this class are extremely numerous; amongst minerals are sulphur, stibnite, cerussite, chrysoberyl, topaz, olivine, nitre, barytes, columbite and many others; and amongst artificial products iodine, potassium permanganate, potassium sulphate, benzene, barium formate, &c.

Pyramidal Class

(Hemimorphic).

Here there is only one dyad axis in which two planes of symmetry intersect. The crystals are usually so placed that the dyad axis coincides with the vertical crystallographic axis, and the planes of symmetry are also vertical.

The pyramid {hkl} has only four faces at one end or other of the crystal. The macro-prism and the brachy-prism of the last class are here represented by the macro-dome and brachy-dome respectively, so called because of the resemblance of the pair of equally sloped faces to the roof of a house. The form {001} is a single plane at the top of the crystal, and is called a “pedion”; the parallel pedion {001}, if present at the lower end of the crystal, constitutes a different form. The prisms {hko} and the macro- and brachy-pinacoids are geometrically the same in this class as in the last. Crystals of this class are therefore differently developed at the two ends and are said to be “hemimorphic.”

Fig. 60 shows a crystal of the mineral hemimorphite (H2Zn2SiO5) which is a combination of the brachy-pinacoid {010} and a prism, with the pedion (001), two brachy-domes and two macro-domes at the upper end, and a pyramid at the lower end. Examples of other substances belonging to this class are struvite (NH4MgPO4·6H2O), bertrandite (H2Be4Si2O9), resorcin, and picric acid.

Bisphenoidal Class

(Hemihedral).

Here there are three dyad axes, but no planes of symmetry and no centre of symmetry. The general form {hkl} is a bisphenoid (fig. 61) bounded by four scalene triangles. The other simple forms are geometrically the same as in the holosymmetric class.

Examples: epsomite (Epsom salts, MgSO4·7H2O), goslarite (ZnSO4·7H2O), silver nitrate, sodium potassium dextro-tartrate (seignette salt, NaKC4H4O6·4H2O), potassium antimonyl dextro-tartrate (tartar-emetic, K(SbO)C4H4O6), and asparagine (C4H8N2O8·H2O).

4. MONOCLINIC5SYSTEM

(Oblique; Monosymmetric).

In this system two of the angles between the crystallographic axes are right angles, but the third angle is oblique, and the axes are of unequal lengths. The axis which is perpendicular to the other two is taken asOY = b(fig. 62) and is called the ortho-axis or ortho-diagonal. The choice of the other two axes is arbitrary; the vertical axis (OZ = c) is usually taken parallel to the edges of a prominently developed prismatic zone, and the clino-axis or clino-diagonal (OX = a) parallel to the zone-axis of some other prominent zone on the crystal. The acute angle between the axesOXandOZis usually denoted as β, and it is necessary to know its magnitude, in addition to the axial ratiosa:b:c, before the crystal is completely determined. As in other systems, except the cubic, these elements,a:b:cand β, are characteristic of the substance. Thus for gypsuma:b:c= 0.6899 : 1 : 0.4124; β = 80° 42′; for orthoclasea:b:c= 0.6585 : 1 : 0.5554; β = 63° 57′; and for cane-sugara:b:c= 1.2595 : 1 : 0.8782; β = 76° 30′.

Holosymmetric Class

(Holohedral; Prismatic).

Here there is a single plane of symmetry perpendicular to which is a dyad axis; there is also a centre of symmetry. The dyad axis coincides with the ortho-axisOY, and the vertical axisOZand the clino-axisOXlie in the plane of symmetry.

All the forms are open, being either pinacoids or prisms; the former consisting of a pair of parallel faces, and the latter of four faces intersecting in parallel edges and with a rhombic cross-section. The pair of faces parallel to the plane of symmetry is distinguished as the “clino-pinacoid” and has the indices {010}. The other pinacoids are all perpendicular to the plane of symmetry (and parallel to the ortho-axis); the one parallel to the vertical axis is called the “ortho-pinacoid” {100}, whilst that parallel to the clino-axis is the “basal pinacoid” {001}; pinacoids not parallel to the arbitrarily chosen clino- and vertical axes may have the indices {101}, {201}, {102} ... {hol} or {101}, {201}, {102} ... {hol}, according to whether they lie in the obtuse or the acute axial angle. Of the prisms, those with edges (zone-axis) parallel to the clino-axis, and having indices {011}, {021}, {012} ... {okl}, are called “clino-prisms”; those with edges parallel to the vertical axis, and with the indices {110}, {210}, {120} ... {hko}, are called simply “prisms.” Prisms with edges parallel to neither of the axesOXandOYhave the indices {111}, {221}, {211}, {321} ... {hkl} or {111} ... {hkl}, and are usually called “hemi-pyramids” (fig. 62); they are distinguished as negative or positive according to whether they lie in the obtuse or the acute axial angle β.

Fig. 63 represents a crystal of augite bounded by the clino-pinacoid (l), the ortho-pinacoid (r), a prism (M), and a hemi-pyramid (s).

The substances which crystallize in this class are extremely numerous: amongst minerals are gypsum, orthoclase, the amphiboles, pyroxenes and micas, epidote, monazite, realgar, borax, mirabilite (Na2SO4·10H2O), melanterite (FeSO4·7H2O) and many others; amongst artificial products are monoclinic sulphur, barium chloride (BaCl2·2H2O), potassium chlorate, potassium ferrocyanide (K4Fe(CN)6·3H2O), oxalic acid (C2O4H2·2H2O), sodium acetate (NaC2H3O2·3H2O) and naphthalene.

Hemimorphic Class

(Sphenoidal).

In this class the only element of symmetry is a single dyad axis, which is polar in character, being dissimilar at the two ends.

The form {010} perpendicular to the axis of symmetry consists of a single plane or pedion; the parallel face is dissimilar in character and belongs to the pedion {010}. The pinacoids {100}, {001}, {hol} and {hol} parallel to the axis of symmetry are geometrically the same in this class as in the holosymmetric class. The remaining forms consist each of only two planes on the same side of the axial planeXOZand equally inclined to the dyad axis (e.g.in fig. 62 the two planesXYZandXYZ); such a wedge-shaped form is sometimes called a sphenoid.

Fig. 64 shows two crystals of tartaric acid,aa right-handed crystal of dextro-tartaric acid, andba left-handed crystal of laevo-tartaric acid. The two crystals are enantiomorphous,i.e.although they have the same interfacial angles they are not superposable, one being the mirror image of the other. Other examples are potassium dextro-tartrate, cane-sugar, milk-sugar, quercite, lithium sulphate (Li2SO4·H2O); amongst minerals the only example is the hydrocarbon fichtelite (C5H8).

Clinohedral Class

(Hemihedral; Domatic).

Crystals of this class are symmetrical only with respect to a single plane. The only form which is here geometrically the same as in the holosymmetric class is the clino-pinacoid {010}. The forms perpendicular to the plane of symmetry are all pedions, consisting of single planes with the indices {100}, {100}, {001}, {001}, {hol}, &c. The remaining forms, {hko}, {okl} and {hkl}, are domes or “gonioids” (γωνία, an angle, andεἶδος, form), consisting of two planes equally inclined to the plane of symmetry.

Examples are potassium tetrathionate (K2S4O6), hydrogen trisodium hypophosphate (HNa3P2O6·9H2O); and amongst minerals, clinohedrite (H2ZnCaSiO4) and scolectite.

5. ANORTHIC SYSTEM

(Triclinic).

In the anorthic (fromἀν, privative, andὀρθός, right) or triclinic system none of the three crystallographic axes are at right angles, and they are all of unequal lengths. In addition to the parameters a : b : c, it is necessary to know the angles, α, β, and γ, between the axes. In anorthite, for example, these elements area:b:c= 0.6347 : 1 : 0.5501; α = 93° 13′, β = 115° 55′, γ = 91° 12′.

Holosymmetric Class

(Holohedral; Pinacoidal).

Here there is only a centre of symmetry. All the forms are pinacoids, each consisting of only two parallel faces. The indices of the three pinacoids parallel to the axial planes are {100}, {010} and {001}; those of pinacoids parallel to only one axis are {hko}, {hol} and {okl}; and the general form is {hkl}.

Several minerals crystallize in this class; for example, the plagioclastic felspars, microcline, axinite (fig. 65), cyanite, amblygonite, chalcanthite (CuSO4·5H2O), sassolite (H3BO3); among artificial substances are potassium bichromate, racemic acid (C4H6O6·2H2O), dibrom-para-nitrophenol, &c.

Asymmetric Class

(Hemihedral, Pediad).

Crystals of this class are devoid of any elements of symmetry. All the forms are pedions, each consisting of a single plane; they are thus hemihedral with respect to crystals of the last class. Although there is a total absence of symmetry, yet the faces are arranged in zones on the crystals.

Examples are calcium thiosulphate (CaS2O3·6H2O) and hydrogen strontium dextro-tartrate ((C4H4O6H)2Sr·5H2O); there is no example amongst minerals.

6. HEXAGONAL SYSTEM

Crystals of this system are characterized by the presence of a single axis of either triad or hexad symmetry, which is spoken of as the “principal” or “morphological” axis. Those with a triad axis are grouped together in the rhombohedral or trigonal division, and those with a hexad axis in the hexagonal division. By some authors these two divisions are treated as separate systems; or again the rhombohedral forms may be considered as hemihedral developmentsof the hexagonal. On the other hand, hexagonal forms may be considered as a combination of two rhombohedral forms.

Owing to the peculiarities of symmetry associated with a single triad or hexad axis, the crystallographic axes of reference are different in this system from those used in the five other systems of crystals. Two methods of axial representation are in common use; rhombohedral axes being usually used for crystals of the rhombohedral division, and hexagonal axes for those of the hexagonal division; though sometimes either one or the other set is employed in both divisions.

Rhomobohedral axes are taken parallel to the three sets of edges of a rhombohedron (fig. 66). They are inclined to one another at equal oblique angles, and they are all equally inclined to the principal axis; further, they are all of equal length and are interchangeable. With such a set of axes there can be no statement of an axial ratio, but the angle between the axes (or some other angle which may be calculated from this) may be given as a constant of the substance. Thus in calcite the rhombohedral angle (the angle between two faces of the fundamental rhombohedron) is 74° 55′, or the angle between the normal to a face of this rhombohedron and the principal axis is 44° 36½′.

Hexagonal axes are four in number, viz. a vertical axis coinciding with the principal axis of the crystal, and three horizontal axes inclined to one another at 60° in a plane perpendicular to the principal axis. The three horizontal axes, which are taken either parallel or perpendicular to the faces of a hexagonal prism (fig. 71) or the edge of a hexagonal bipyramid (fig. 70), are equal in length (a) but the vertical axis is of a different length (c). The indices of planes referred to such a set of axes are four in number; they are written as {hikl}, the first three (h+i+k= 0) referring to the horizontal axes and the last to the vertical axis. The ratioa:cof the parameters, or the axial ratio, is characteristic of all the crystals of the same substance. Thus for beryl (including emerald)a:c= 1 : 0.4989 (often writtenc= 0.4989); for zincc= 1.3564.

Rhombohedral Division.

In the rhomobohedral or trigonal division of the hexagonal system there are seven symmetry-classes, all of which possess a single triad axis of symmetry.

Holosymmetric Class

(Holohedral; Ditrigonal scalenohedral).

In this class, which presents the commonest type of symmetry of the hexagonal system, the triad axis is associated with three similar planes of symmetry inclined to one another at 60° and intersecting in the triad axis; there are also three similar dyad axes, each perpendicular to a plane of symmetry, and a centre of symmetry. The seven simple forms are:—

Rhombohedron (figs. 66 and 67), consisting of six rhomb-shaped faces with the edges all of equal lengths: the faces are perpendicular to the planes of symmetry. There are two sets of rhombohedra, distinguished respectively as direct and inverse; those of one set (fig. 66) are brought into the orientation of the other set (fig. 67) by a rotation of 60° or 180° about the principal axis. For the fundamental rhombohedron, parallel to the edges of which are the crystallographic axes of reference, the indices are {100}. Other rhombohedra may have the indices {211}, {411}, {110}, {221}, {111}, &c., or in general {hkk}. (Compare fig. 72; for figures of other rhombohedra seeCalcite.)

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), whereh− 2k+l= 0.

Hexagonal prism of the first order (211), 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°.

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° 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 {321}, {532} ... {hkl}, whereh+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 seeCalciteandCorundum. 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, crystallizein 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 {211} or {211}, 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 {hkl} (whereh+k+l= 0), of six faces parallel to the principal axis, and with two sets of angles between them.

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

Fig. 73 represents a crystal of tourmaline with the trigonal prism (211), 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° 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.

In this class the rhombohedra {hkk}, the hexagonal prism {211}, 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}, whereh− 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}, whereh− 2k+l= 0, is also a rhombohedron, being the hemihedral form of the hexagonal bipyramid. The dihexagonal prism {hkl} of the holosymmetric class becomes here a hexagonal prism. The rhombohedra (hkk), hexagonal prisms {211} 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 rhombohedronr{100} and the hexagonal prism of the second orderm{101} combined with the rhombohedrons{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,pqr}, (hkl) for faces above the equatorial plane of symmetry and (pqr) 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 {211} 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;vin fig. 80). The indices are {2131}, &c., or in general {hikl}. This form may be considered as a combination of two scalenohedra, a direct and an inverse.

Hexagonal bipyramid of the first order, bounded by twelve isosceles triangles (fig. 70;panduin fig. 80); indices {1011}, {2021} ... (hohl). 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 (sin figs. 79 and 80); indices {1121}, {1122} ... {h.h.2h.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 (min figs. 71, 78-80).

Hexagonal prism of the second order {1120}, consisting of six faces also parallel to the hexad axis, but perpendicular to the other set of three vertical planes of symmetry (ain fig. 78).

Basal pinacoid {0001}, consisting of a pair of parallel planes perpendicular to the hexad axis (cin 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 {hohl} and (h.h.2h.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 (fromδένδρον, 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. Mügge, “Die regelmässigen Verwachsungen von Mineralien verschiedener Art,”Neues Jahrbuch für 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 faceb. 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° 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° 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° 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. Haüy introduced the term “hemitrope” (from the Gr.ἡμι-, half, andτρόπος, a turn); the word “macle” had been earlier used by Romé 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).

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