--Crystallography is the study of crystal shapes--
                          --in this section we will study crystal symmetry, crystal forms
                            crystallographic axes, unit cells, axial ratios, face intercepts 
                            and Miller Indices

  A. Introduction
       -atoms of elements combine in mineral formation to build specific geometric shapes
         on the smallest scale (we will study this later in a section dealing with the nature of atomic 
         shapes)--these shapes combine to form crystal shapes--since each mineral is restricted 
         by the crystal shape (s) it can have, the study of crystallography can aid in the 
         determination of the name of a mineral--

        -the determination of the symmetry functions present on a crystal is paramount and allows 
         the crystal to be categorized into one of 32 possible crystal classes

        -to become familiar with crystal class determination we will analyze symmetry functions
         on many crystals--wooden crystals precut to show symmetry functions will be used for
         practice before doing the same for real mineral crystals--

  B. Symmetry Functions (Crystal Symmetry)
       1. axis of rotation
           -is an imaginary line through a crystal around which a crystal is rotated through 360
            degrees--the complete occurrence of the original face or set of faces in a number of
            multiple appearances during the full rotation determines the kind (type) or fold of axis of
                   A2 = (2-fold axis of rotation) a reproduction of original face or faces twice
                   A3 = (3-fold axis of rotation)  "                        "                         "        three times
                   A4 = (4-fold axis of rotation)  "                        "                         "        four times
                   A6 = (6-fold axis of rotation)  "                        "                         "        six times
           -a crystal can have more than one type of fold axis and may have a multiple of that
             fold type--an example--1A6, 6A2 indicates the crystal has one six fold axis and six two fold
             axes-in all but the isometric crystal system additional rotation axes types and their multiple,
             if present, occur only in the 90 degree horizontal plane from an axis already found--in the
             isometric system, fold axes may also be present at 45 degrees from another axis-

           -an axis of rotation can represent only one yAx---in order to have more than one kind of
            rotation axis or a multiple of the same rotation axis, you must move your fingers and hold
            the crystal in another place to define these other axes


       2.  mirror plane (symmetry plane)
           -is an imaginary plane that divides a crystal into two equal halves, each a mirror image of
             the other--there may be 0-9 different mirror images on a crystal--a crystal which has
             6 different mirror images as an example is designated as 6m-
           -in all but the isometric system mirror images, if present, occur in the vertical plane of
             rotation axes and one possibly in the horizontal plane of an axis--in the isometric
             system, mirror planes may be present as above and also can occur at 45 degrees
             to the rotation axes-
           -in the determination of rotation axes and mirror planes, it is paramount not to 
             count the same individual axis or mirror plane more than once

       3.  center of symmetry
           -occurs if the same feature is located on the exact opposite sides of a crystal and both
             are equal distance from the center of the crystal--features include points, corners, edges
             or faces--
           -there is a center of symmetry or none--if there is a center present, there is an infinite
             number of places where this analysis will occur and the crystal has an (i) appearing in the
             designation of its symmetry functions-

       4. axis of roto-inversion
           -is present if the original face or faces are reproduced after a crystal is rotated either at 
           180degrees (A2), 120 degrees (A3), 90 degrees (A4) or 60 degrees (A6) and then inverted-
           -if present, a crystal can have only one type of roto-inversion axis and the designation for
            this is similar to that of an axis of rotation except the AX includes a bar above the letter
            or number
                    -each roto-inversion A3 is equivalent to an (A3) + i
                    -each           "            A4             "               A2
                    -each           "            A6             "              (A3 + m)
            -since the roto-inversion axis has equivalents of the other functions of symmetry, we need
              only to use the first 3 functions to categorize a crystal in a class--even so you need to
              be able to identify a roto-inversion axis on crystals--
         If the total symmetry of a crystal is ascertained, the table below can be used to
        determine to which crystal class and system the crystal belongs

            -the table below includes the equivalent non-rotoinversion symmetry where ap-         

Crystal System

Crystal Class

Name of Class

Symmetry Content


















i, 1A2, 1m








1A2, 2m



i, 3A2, 3m











i, 1A4, 1m



1A4, 4A2



1A4, 4m



3A2, 2m



i, 1A4, 4A2, 5m









i, 1A3




1A3, 3A2



1A3, 3m


Trigonal (Hex.)-scalenohedral

i, 1A3, 3A2, 3m








1A3, 1m




i, 1A6, 1m



1A6, 6A2



1A6, 6m



1A3, 3A2, 4m



i, 1A6, 6A2, 7m






3A2, 4A3



i, 3A2, 3m, 4A3



3A4, 4A3, 6A2



4A3, 3A2, 6m

4/m 32/m


i, 3A4, 4A3, 6A2, 9m

         You should note from the table there are certain associated symmetry functions as:
                    1.  a crystal which has an A6 cannot have an A4 and vice versa
                    2.  a crystal with only 1A4 is tetragonal, and if more present, has to have 3A4 and
                         must belong to a specific isometric crystal class  
                    3.  a crystal with an A6 cannot have an A3 and must be in the hexagonal
                         division of the hexagonal system
                    4.  a crystal with only 1A3 must be in the hexagonal system, and if more present
                         there has to be exactly 4A3 and belong in the isometric system
             List other similar symmetry functions associations from the table above
Holohedral refers to the highest symmetry class in a crystal system-list those
                              classes--know them 

  C.  Crystal Forms
-a crystal form is a group of faces, all of which have the same relation to the functions of
          symmetry on the crystal--the symmetry on a crystal has a direct influence on which specific
          forms are present
        -the forms present on crystals may be classified as (1) non-isometric and (2) isometric--
          isometric crystals have essentially a different set of forms than the same in the other
          crystal systems
        -the faces of a crystal form may intersect faces of another form on a crystal resulting in shapes
          being altered not resembling the perfect texbook "full form" shown below, hence 
          the faces comprising a form must be extended visually to resemble
          the picture drawings below--this may be more difficult to do for multiform crystals in some
          systems and especially those in the isometric system--practice will solve this problem 
        -note in the picture drawings, each form is shown related to axes (is) of rotation or roto-
          inversion--see page 127 in text if you are not familiar with symbols for these axes

       1. non-isometric forms         
          -CLICK to see drawings of single full form non-isometric forms taken from the textbook
            and explained below--important to note is the name of a form with a                     
   specific number of faces can indicate the crystal system or
            possibly the crystal class-
-an example is a prism--if comprised of:          
                          3, 6 or 12 faces it must belong in the hexagonal system
                          4                 "     "   "        "       "   "orthorhombic, tetragonal or monoclinic system
                          8                 "      "   "       "       "   "tetragonal system
           -specific names are given to a form depending on how many faces it has--from the
            example above: 3 faces= (trigonal prism); 4 faces= (tetragonal, orthorhombic or monoclinic
            prism--depending on which crystal system it belongs based on symmetry);6 faces=
           (hexagonal prism or ditrigonal prism--based on symmetry);12 faces (dihexagonal prism)

            a. Pedion--a single face comprising a form
            b. Pinacoid--an open form comprised of 2 parallel faces
            c. Dome--an open form comprised of 2 nonparallel faces symmetrical with respect to
                            a mirror plane
            d. Sphenoid--two nonparallel faces symmetrical with respect to a 2-fold axis
            e. Prism--an open form consisting of 3, 4, 6, 8 or 12 faces all parallel to the same axis--
                           except for some prisms in the monoclinic system, the prisms are parallel
                           to the vertical or highest fold axis
            f.  Pyramid--an open form comprised of 3, 4, 6, 8 or 12 nonparallel faces that meet at a
                                point at the top of a crystal
            g. Dipyramid--a closed form comprised of  6, 8,12, 16, or 24 faces--basically it is a
                                   pyramid appearing at the top and bottom of a crystal with a mirror plane
                                   separating each
            h. Trapezohedron--a closed form comprised of 6, 8 or 12 faces with 3, 4, or 6 upper
                                          faces offset from 3, 4, or 6 lower faces--this results in a 3, 4, or 6
                                          fold axis with 3 2-fold perpendicular axes
            iScalenohedron--a closed form with 8 or12 grouped in symmetrical pairs
            j.  Rombohedron--a closed form comprised of 6 faces of which 3 faces on top are offset
                                         with 3 faces on the bottom each by 60 degrees
           k.  Disphenoid--a closed form comprised of 2 upper faces that alternate with 2 lower
                                    faces, offset by 90 degrees

       2. isometric forms
          -CLICK to see drawings of single isometric forms taken from text book and explained
          -many of the forms have their nature based on 3 elementary forms--this triad of forms
           include the cube (hexahedron), octahedron and the tetrahedron--many form names
           include a prefix with the suffix being one of these 3 basic forms--an example is the
           tetrahexahedron having 4 (tetra) faces on each hexahedron face (6) for a total of 24 faces

            a. Cube(hexahedron)--6 equal faces intersecting at 90 degrees 
            b. Octahedron--8 equilateral triangular faces
            c. Dodecahedron--12 faces, each rhomb shaped
            d. Tetrahexahedron--24 isosceles triangular faces
            e. Trapezohedron--24 trapezium shaped faces
            f. Trisoctahedron--24 isosceles triangular faces
            g. Hexoctahedron--48 triangular faces
            h. Tetrahedron--4 equilateral triangular faces
            i. Tristetrahedron--12 triangular faces
            j. Deltoid dodectahedron--12 faces corresponding to 1/2 of the faces of a trisocta-
            k. Hextetrahedron--24 faces-6 triangular faces formed on each side of the tetrahedron
            l.  Diploid--24 faces
            m. Pyritohedron--12 pentagonal faces

       3.  forms present in the 32 crystal classes
          -one or a combination of the forms above can be present on individual crystals based on
           crystal symmetry--it is possible to determine the crystal class based on the com-
           bination of forms present--
          -in some cases it is possible to determine the crystal class if a special form is present on the
           crystal since that form can occur only in that class--some examples would be:

               -the rhombic dipyramid occurs only in the rhombic dipyramidal class;
                the ditrigonal dipyramid occurs only in the ditrigonal dipyramidal class;
                the hextetrahedron occurs only in the hextetrahedral class;
                the tetrahexahedron occurs in the hextetrahedral class;
-crystal class names are based on the most outstanding form possible in that class-                  
          -form tables below list all possible forms that can be present on crystals in each of the 32 classes-
          -in the tables, numbers in parentheses after the form name represent the number of
           faces comprising the form-
          -CLICK HERE  and page down to the desired system or CLICK below on a crystal system           
                              1. triclinic classes
                              2. monoclinic classes
                              3. orthorhombic classes
                              4. tetragonal classes
                              5. hexagonal-rhombohedral division classes
                              6. hexagonal-hexagonal division classes
                              7. isometric classes



       D.  Crystallographic Axes
         -are imaginary reference lines constructed through crystals and in most instances
          coincide with symmetry axes or normals to symmetry planes
--these axes aid in the
          orientation of crystals and help to explain other crystal concepts like unit cells and
          Miller Indices which we will discuss below
         -crystals in all crystal systems have 3 axes associated with them except for 4 axes in the
          Hexagonal System
         - these axes are compared by lengths and angles of intersection with each other
           axes are designated as a, b and c when unequal in lengths in a crystal or by the same letter, 
          "a" whenever equal in length--the "c" axis when present is the principal axis and can be
           larger or smaller than the other axes associated with it
         -the axes have a preferred vertical and horizontal orientation--the c axis when present 
          always occurs in the vertical--when an a and b axis are present, they are located in the
          horizontal plane, the a is in a front-back orientation, and the b in a left to right position
          --if there is no b but a c axis present, the a axes are located horizontally in the front-back,
          and left-right positions if 2 are present, or if 3 are present  the same exists  and the third lies 
          between the other 2--
         -the following explains relative lengths of axes and angles of intersection of axes with each other
          and the association with symmetry functions in each of the crystal systems--CLICK to
          see drawings of the crystallographic axes

                  1. Triclinic
                     -consists of 3 unequal axes (a, b, c) all intersecting at oblique angles--since there
                      are no symmetry axes or symmetry planes the a, b, and c axes are not associated
                      with any symmetry functions

                  2. Monoclinic
                       -consists of 3 unequal axes (a, b, c), in which a and c are inclined to each other
                        at an oblique angle and the third (b axis) perpendicular to the other two--the b
                        axis corresponds to the 2-fold axis and lies in a perpendicular plane to the m if
                        either or both are present--the faces of the monoclinic prism (main prism), if 
                        present, parallel the c axis

                 3.   Orthorhombic
                       -consists of 3 unequal axes (a, b, c) all mutually perpendicular--the c axis
                         corresponds to one of the 2-fold axes and is that on which the pyramid,
                         dipyramid or disphenoid are found if present--the faces of the orthorhombic
                         prism, if present, parallel the c axis 

                 4.   Tetragonal
                     -consists of 3 mutually perpendicular axes (a, a, c), two of which are equal in
                       length (a, a) and the third axis (c) is shorter or longer than the other two--the faces of
                        most forms in this crystal system form on or parallel the c axis

                 5.   Hexagonal
                     -consists of 4 axes, three of which are equal in length, found in the horizontal
                       plane, and intersect at 60 degrees (a, a ,a); the fourth axis (c) is perpendicular to
                       and shorter or longer than the other 3-- faces of most all forms in this crystal system 
                       form on or parallel the c axis

                 6.   Isometric
                     -consists of 3 mutually perpendicular axes of equal length

   E. Unit Cell
       -is a 3-dimensional geometric figure constructed in the array of atoms of a mineral so that
        there are specific (nodes) atoms occurring at the corners of the figure and possibly at the center
        of  face or faces and/or at the very center of the figure--there are additional (nodes) atoms located
        within the cell but not at these specific locations

      -is the smallest unit of a mineral that retains all of the physical, chemical, and crystallographic
        properties of a mineral.  The unit cell comprises the symmetry of each holohedral class
        of the crystal system

      -there are only 14 unique figures possible for the 6 crystal systems and include the following:
               1. primitive
                  -a cell with (nodes) atoms at all the corners and not at the center of faces nor an atom at the
                    very center of the cell-
2. non primitive
                  -a cell with (nodes) atoms at all corners and also an atom on each of 2 opposite faces--if at
                    top and bottom, C-centered; if on both side faces, B-centered; if on
                    front and back faces, A-centered; or on all faces, F-or face-centered--the A, B,
                    and C lattices are symmetrically identical and can be converted to each other by
                    an appropriate change of the crystallographic axes
                  -also, a cell with (nodes) atoms at the corners and one at the very center of the unit cell is a
                    body-centered ( I ) non primitive cell

        CLICK to see the 14 types of unit cells (Bravais Lattices)
                 -although many minerals may have the same type of primitive or non primitive unit
                  cell, each mineral has a sole combination of crystallographic axes lengths which acts
                  like a genetic code for that mineral--hence no two minerals will have the same 3 unit
                  cell axes dimensions--this fact allows X-ray analysis to generate the name of a mineral

        CLICK HERE to see how unit cells on an atomic scale pack together to form crystal forms on
         a much larger seeable scale (micro or macroscopic)

        CLICK HERE to see Bravais Lattices that rotate

      -The volume of the unit cell is important in Specific Gravity calculations--the table below indicates the
        formula to calculate the volume of the unit cell for each crystal system: a, b and c are axes lengths (10-8cm)
        and alpha, beta and gamma are axes angles




 V = a3


 V = a2c


 V = a2csin(60o)


 V = a2csin(60o)


 V = abc


 V = abcsin(beta)


 V= abc(1-cos2alpha-cos2beta-cos2gamma + 2cos alpha cos beta cos gamma)1/2

   F.  Axial Ratio
        -is a ratio of the unit cell lengths measured in Angstrom units () using the length of the b axis as 
         the common denominator, or the a axis as the same if the b axis is not present--thus the unit
         cell lengths of sulfur with the following axial lengths: a = 10.47 ;   b=12.88 ;
          c = 24.49 will have an axial ratio of a:b:c = 0.8135:1:1.9029--as you can see all unit cells
          in the isometric must have an axial ratio of 1:1:1  

   G.  Face Intercepts (Parameters)
        -face(s) of forms (is) are defined by their intercepts or non-intercepts on the crystallographic
         axes--an example is, do faces parallel one or two axes and intersect the other?-- and if so,
         it must be determined at what relative distance the faces intersect the different axes--this
         relative distance is expressed as a multiple or fraction related to the axes lengths or ratios
         of the unit cell as measured from the vortex--for instance, if the aforementioned unit cell of
         sulfur is used and a face representing a form intercepts the a axis at 21.94 , the b axis at
          6.44 , and the c axis at 24.49 , the parameters or face intercepts would be
          2a, 1/2b, c--if the face parallels an axis (axes) an infinity sign (looks like an 8 turned
          90 degrees) is placed before the axis letter--a minus intercept on an axis is then so
          noted with a minus sign above the intercept number

        CLICK HERE to see other examples of face intercepts

   H.  Miller Indices
the indices of a face is a series of whole numbers derived from the intercepts by inverting
         the whole or fraction numbers and clearing of fractions if present--examples of conversion
         from intercepts to indices would be: ( 2a, 1/2b, c) = (1/2, 2/1,1/1) = (1,4,2): ( infinity a, 3b,
         2/3c) = (0, 1/3, 3/2) = (0,2,9)--once again the minus sign intercept should be included
         above the appropriate number    
     Now let us consider the nature of some basic mineral chemistry principles--this
     includes the nature of mineral formulas, element and element oxide weight
     percents, specific gravity determinations and other principles--we will solve problems
    to aid in understanding these concepts