### Crystal morphology and Symmetry

```Stereographic Projection
References:
Dexter Perkins, 2002, Mineralogy, 2nd edition. Prentice Hall, New
Jersey, 483 p.
Bloss, F.D., 1971, Crystallography and Crystal Chemistry: Holt,
Reinhardt, and Winston, New York, 545 p.
Klein, C., and Hurlbut, C.S.Jr., 1993, Manual of Mineralogy (after James
Dana), 21st edition: John Wiley & Sons, New York, 681 p.
Introduction
 Crystals have a set of 3D geometric relationships among their
planar and linear features
 These include the angle between crystal faces, normal
(pole) to these faces, and the line of intersection of these
faces
 Planar features: crystal faces, mirror planes
 Linear features: pole to crystal faces, zone axis, crystal axes
 Question: How can we accurately depict all of these planar
and linear features on a 2D page and still maintain the correct
angular relationships between them?
 Answer: With equal angle stereographic projection!
Stereographic Projection
 Projection of 3D orientation data and symmetry
of a crystal into 2D by preserving all the angular
relationships
 Projection lowers the Euclidian dimension of the
object by 1, i.e.,
 planes become lines
 lines become point
 In mineralogy, it involves projection of faces,
edges, mirror planes, and rotation axes onto a
flat equatorial plane of a sphere, in correct
angular relationships
 In mineralogy, in contrast to structural
geology, stereograms have no geographic
significance, and cannot show shape of
crystal faces!
Two Types of Stereonet
 Wulff net (Equal angle)
 Used in mineralogy & structural geology when
angles are meant to be preserved
 e.g., for crystallography and core analysis
 Projection is done onto both the upper and
lower hemispheres
 Schmidt net (Equal area)


Used in structural geology for orientation
analysis when area is meant to be preserved for
statistical analysis
Uses projection onto the lower hemisphere
-90
The ρ angle, is
between the c
crystal axis and
the pole to the
crystal face,
measured
downward from
the North pole of
the sphere
-135

0

+135
Wulff net for
Minerals 
+90
 = -90
-

20
40
60

       
80


=0
+
 Pole to face
http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm
axis
The ρ angle, is between the
c crystal axis and the pole
to the crystal face,
measured downward from
the North pole of the sphere
axis
a axis
The φ angle is
measured in the
horizontal
equatorial plane.
Wulff Stereonet (equal angle net)
 Shows the projection of great circles and small circles
 Great circle: Line of intersection of a plane, that passes
through the center of the sphere, with the surface of the
sphere (like lines of longitude on Earth)
 NOTE: Angular relationships between points can only
be measured on great circles (not along small circles)!
 Small circle: Loci of all positions of a point on the surface
of the sphere when rotated about an axis such as the
North pole (like lines of latitude on Earth)
Why need projection?
 Projection of all crystal faces of a crystal leads to
many great circles or poles to these great circles
 These great circles and poles allow one to determine
the exact angular relationship, and symmetry
relationships for a crystal
 For example, the angle between crystal faces and
rotation axes, or between axes and mirror planes
 To understand these, we first give an introduction
to stereographic projection!
Stereographic projection of a line
• Each line (e.g., rotation axis, pole to a mirror plane) goes
through the center of the stereonet (i.e., the thumb tack)
• The line intersects the sphere along the ‘spherical projection
of the line’, which is a point
• A ray, originating from this point, to the eyes of a viewer
located vertically above the center of the net (point O),
intersect the ‘primitive’ along one point
• The point is the stereographic projection of the line.
• A vertical line plots at the center of the net
• A horizontal line plots on the primitive
Special cases of lines
 Vertical lines (e.g., rotation axes, edges) plot at the
center of the equatorial plane
 Horizontal lines plot on the primitive
 Inclined lines plot between the primitive and the
center
Projection of Planar Elements
 Crystals have faces and mirror planes which are
planes, so they intersect the surface of the sphere
along lines
 These elements can be represented either as:
 Planes, which become great circle after projection
 Poles (normals) to the planes, which become
points after projection
Stereographic projection of a plane:
•
Each plane (e.g., mirror plane) goes through the
center of the stereonet (i.e., the thumbtack)
•
The plane intersects the sphere along the
‘spherical projection of the plane’, which is a
series of points
•
Rays, originating from these points, to the eyes
of a viewer located vertically above the center of
the net (point O), intersect the ‘primitive’ along a
great circle
•
The great circle is the stereographic projection of
the plane. The great circle for a:
• vertical plane goes through the center
• horizontal plane parallels the primitive
Special cases of planes
 Stereographic projection of a horizontal face or mirror
plane is along the primitive (perimeter) of the equatorial
plane
 Stereographic projection of a vertical face or mirror plane
is along the straight diameters of the equatorial plane
 They pass through the center
 They are straight ‘great circles’
 Inclined faces and mirror planes plot along curved great
circles that do not pass through the center
http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html
3
Face 1: vertical
Face 2: inclined
Face 3: horizontal
2
1
Face 3
 pole to the
crystal face
Preparing to plot
 Mark N of the net as -90, E as 0, S as =+90 )(for ).
 Mount the stereonet on a cardboard. Laminate it. Pass
a thumbtack through the center from behind the board
 Secure the thumbtack with a masking tape from behind
the cardboard
 Place a sheet of tracing paper on the stereonet
 Put a scotch tape at the center, from both sides of the
tracing paper; pierce the paper through the pin
 Tracing paper can now rotate around the thumbtack
without enlarging the hole (because of scotch tape)
-90
-135
00
+135
+90
http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html
Symbols used
 For faces below the equator (when using lower hemisphere), place an
open circle symbol () where the ray connecting the spherical
projection of the pole to the plane intersects the equatorial plane
 This is the stereographic projection of the pole to the face
 For faces above the equator (when using the upper hemisphere),
place a solid circle symbol () the ray connecting the spherical
projection of the pole to the plane intersects the equatorial plane
 Use a bull’s-eye symbol () to show a point above the page that
coincides with one directly below it (when using both hemispheres)
 Reorient the stereogram such that lines of symmetry are north-
south or east-west
Projection of Linear Elements
 We can show all of the symmetry elements of a crystal
and their relative positions stereographically
 Edges, pole to crystal faces, and rotation or roto-inversion
axes are lines
 When extended through the origin of the sphere, lines
intersect the surface of the sphere as points
 Each of these points, when connected to the upper or
lower pole of the sphere (viewer’s eyes), is projected onto
the equatorial plane, and depicted as a polygon symbol
with the same number of sides as the ‘fold’ of the axis
Mirror and Polygon Symbols
 To plot symmetry axes on the stereonet, use the
following symbol conventions:
Mirror plane:
― (solid line great circles)
Crystal axes (lines):
Plotting the rotation axes
 Vertical axes (normal to page) will only have one
polygon symbol
 Horizontal axes (in the plane of page) intersect the
primitive twice, hence they have two polygons
 Inclined axes will have one polygon symbol
 An open circle in the middle of the polygon shows
there is a center of symmetry
Measuring angle between faces
 This is done using the poles to the faces!
 Three cases:
1. On the primitive, the angle is read directly on the
circumference of the net
2. On a straight diameter, the paper is rotated until the
zone is coincident with the vertical diameter (i.e., N-S or
E-W) and the angle measured on the diameter
3. On a great circle (an inclined zone), rotate the paper
until the zone coincides with a great circle on the net;
read the angle along the great circle
Going from one hemisphere to another
 During rotation of the pole to a face by a certain angle,
we may reach the primitive before we are finished with
the amount (angle) of rotation
 In this case we are moving from one hemisphere to
another
 Move the pole back away from the primitive along that
same small circle you followed out to the primitive, until
it has been moved the correct total number of degrees
 Then note its new position with the point symbol for the
new hemisphere
How to find reflection of a point
 Having a symmetry (mirror) plane and a point p, find
the reflection of point p (i.e., p’) across the mirror:
 Align the mirror along a great circle
 Rotate point p along a small circle to the mirror plane
 Count an equal angle beyond the mirror plane, on the
same small circle, to find point p’
 If the primitive is reached before p’, then count
inward along the same great circle
Crystallographic Angles
 Interfacial angle: between two crystal faces is the
angle between poles to the two faces.
 The interfacial angle can be measured
with a contact goniometer
 These angles are plotted
on the stereonet
Making a stereographic projection of a crystal face pole
Use a contact goniometer to measure the interfacial
angles (also measures poles)
Convention
 By convention (Klein and Hurlbut, p.62), we place the
crystal at the center of the sphere such that the:
 c-axis (normal of face 001) is the vertical axis
 b-axis (normal of face 010) is east-west
 a-axis (normal of face 100) is north-south
 See next slide!
http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm
Up and down
axis
Stereonet is
the equator
a axis
N-S
axis
E-W
The ρ and φ angles
 Generally, it is the angles of the spherical
projection, ρ and φ, that are given for each face
of a crystal
 These are measured with goniometer
 If these are known, then the actual angles
between any two faces can easily be obtained
through trigonometry, or by the use of the
stereonet
The ρ angle
 The ρ angle, is between the c axis and the pole to the
crystal face, measured downward from the North pole of
the sphere
 A crystal face has a ρ angle measured in the vertical plane
containing the axis of the sphere and the face pole.
 Note: the (010) face has a ρ angle of 90o
 (010) face is perpendicular to the b-axis
 The φ angle is measured in the horizontal equatorial plane.
 Note: the (010) face has a φ angle of 0o!
Plotting ρ and φ
Suppose you measured ρ = 60o and φ = 30o for a face with goniometer.
Plot the pole to this face on the stereonet.
Procedure: Line up the N of the tracing paper with the N of the net.
From E, count 30 clockwise, put an x (or a tick mark). Bring x to the E,
and then count 60 from the center toward E, along the E-W line. Mark
the point with .
NOTE: The origin for the φ angle is at E (i.e., φ =0).
-φ is counted counterclockwise, horizontally from E to the N on the primitive.
+φ is counted clockwise horizontally from E to S (i.e., clockwise) on the primitive.
http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html
-135
135
http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html
 These angular measurements are
similar to those we use for latitude
and longitude to plot positions of
points on the Earth's surface
 For the Earth, longitude is similar
to the φ angle, except longitude is
measured from the Greenwich
Meridian, defined as φ = 0o
 Latitude is measured in the vertical
plane, up from the equator, shown as
the angle θ. Thus, the ρ angle is like
what is called the colatitude
(90o - latitude).
http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm
Zone plotting
 Zone: Two or more faces whose edges of intersection are parallel to a
specific linear direction in a crystal
 This direction is called the zone axis. A zone is indicated by a symbol similar
to that for the Miller Indices of faces, the generalized expression for a zone
is [uvw], e.g., all faces parallel to the c axis in an orthorhombic crystal are
said to lie in the [001] zone
 All faces in a zone lie on a great circle; i.e., a zone is constructed by
aligning the poles to these faces on a great circle
 The zone axis (pole to the zone) is normal (i.e., 90o) to this great circle
 On the stereogram, the lower hemisphere part of the zone great circle is
dashed, while the upper great circle is solid
 Lower-hemisphere faces are depicted by open circle symbol ()
 Upper hemisphere faces are depicted by filled circle symbol ()
(111) (100) (111)
(011) (100) all
coplanar (= zone)
Thus all poles in a
zone are on the
same great circle
The following rules are applied:
 All crystal faces are plotted as poles (lines perpendicular to the
crystal face. Thus, angles between crystal faces are really
angles between poles to crystal faces
 The b crystallographic axis is taken as the starting point. Such
an axis will be perpendicular to the (010) crystal face in any
crystal system. The [010] axis (note the zone symbol) or (010)
crystal face will therefore plot at φ = 0o and ρ = 90o
 Positive φ angles will be measured clockwise on the stereonet,
and negative φ angles will be measured counter-clockwise on
the stereonet
Rules cont’d
 Crystal faces that are on the top of the crystal (ρ < 90o) will be
plotted with the closed circles () symbol, and crystal faces on
the bottom of the crystal (ρ > 90o) will be plotted with the ""
symbol
 Place a sheet of tracing paper on the stereonet and trace the
outermost great circle. Make a reference mark on the right
side of the circle (East)
 To plot a face, first measure the φ angle along the outermost
great circle, and make a mark on your tracing paper. Next
rotate the tracing paper so that the mark lies at the end of the
E-W axis of the stereonet
Rules cont’d
 Measure the ρ angle out from the center of the stereonet
along the E-W axis of the stereonet
 Note that angles can only be measured along great circles.
These include the primitive circle, and the E-W and N-S
axis of the stereonet
 Any two faces on the same great circle are in the same zone.
Zones can be shown as lines running through the great circle
containing faces in that zone
 The zone axis can be found by setting two faces in the zone on
the same great circle, and counting 90o away from the
intersection of the great circle along the E-W axis.
 That is, the zone axis is the pole to the great circle of the poles to
the faces in a zone
http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm
•
As an example, the ρ and φ
angles for the (111) crystal
face in a crystal model is
shown here
Note again that the ρ angle
is measured in the vertical
plane containing the c axis
and the pole to the face, and
the φ angle is measured in
the horizontal plane,
clockwise from the b axis.
D and E are spherical
projections, i.e., where the
pole to the faces intersect the
inside of the sphere
D' and E' are stereographic
projections, when DS and ES
intersect the equator (when
projected to the south pole)
Distance GD' = f(ρ)
as ρ  90 D’  G
as ρ  0 D’  O
We are looking along the primitive, and
viewer’s eye is at the S pole for the two shown
faces. The upper an lower hemisphere shown!
Fig 6.3
Example for an isometric crystal
a3 axis
See the isometric
crystal axes in
the next slide!
NOTE:
This is a 3D view!
a2 axis
a1 axis
The stereonet is the equatorial plane of the sphere!
Stereographic projection of the isometric crystal in the
previous slide
Upper hemisphere
stereographic
projection of the
poles to the upper
crystal faces are
shown by the ()
symbols
Viewer’s eyes are at
the south pole
Symmetry
elements of an
isometric crystal.
Legend
 Pole to the upper ()
and lower () crystal faces
Stereographic projection of an isometric crystal
NOTE: ρ is measured as the distance (in o) from the center of the
projection to the position where the pole to the crystal face plots
φ is measured around the circumference of the circle, in a
clockwise direction away from the b crystallographic axis (010)
(1-00)

(1-1-0)
 (1 10)
-

(1-01)


(1-1-1)

-
(01-1)
(01 0)

-
(11 1)

(1-11)
(001)
(010)



a3
(011)
(111)


(101)

(110)
(11-0)

(100)
a1 axis

a2 axis
Explanation of Previous Slide
• In the previous slide, only the upper faces of an isometric
crystal are plotted. These faces belong to forms {100}, {110},
and {111}
• Form: set of identical faces related by the rotational
symmetry (shown by poles/dots in stereograms)
• Faces (111) and (110) both have a φ angle of 45o
• The ρ angle for these faces is measured along a line from the
center of the stereonet (where the (001) face plots) toward
the primitive. For the (111) face the ρ angle is 45o, and for the
(110) face the ρ angle is 90o
As an example all of the faces, both
upper and lower, are drawn for a
crystal in the class 4/m 2/m in the
forms {100} (hexahedron - 6 faces),
{110} (dodecahedron, 12 faces), and
{111} (octahedron, 8 faces) in the
stereogram to the right
Rotation axes are indicated by the
symbols as discussed above
Mirror planes are shown as solid lines
and curves, and the primitive circle
represents a mirror plane. Note how
the symmetry of the crystal can easily
be observed in the stereogram
http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm
How rotational axes are shown
 Axes that are parallel to the page are indicated by
straight lines with proper polygons at the end
 Solid line if these are parallel to a mirror plane
 Dashed otherwise
 Oblique axes plot as polygons between center and
primitive
 The distance between the polygon and center is
proportional to the angle between the axis and
pole to the face (ρ angle )
Mirror planes (see Figure 9.20 of Perkins)
 Horizontal mirror planes (in the plane of the page) plot as
solid primitive
 Vertical mirror planes (i.e., normal to the page) plot as
solid straight line through the center
 Inclined mirror planes (inclined to the page) plot as solid
curved great circle
3-D Symmetry Conventions
http://www.kean.edu/~csmart/Mineralogy/Lectures
http://www.kean.edu/~csmart/Mineralogy/Lectures
http://www.kean.edu/~csmart/Mineralogy/Lectures
```