Mineralogy Homework 1 – Winter 2016
Name _________________________________________
Symmetry operations for plane figures, frieze patterns (strips) and tilings. Symmetry operations apply to many
things in nature especially to crystal lattices. Minerals are made of lattices of points (atoms, ions etc.). They
densely fill space and do so via “tilings or tessellations”. This means taking the basic unit cell shape and
repeating it across space. The simplest symmetry operation is translation. It involves a shape or motif (unit cell)
and a distance in a particular direction (in a planar or 3-D grid). For our page this is up and down the page or
across. Other symmetry operations are mirror reflections, rotations and glides. Mirror reflections are a
translation perpendicular to a plane (or a line in a plane) and the same translated distance out the other side.
Rotations involve a centre of rotation and an angle. Glide planes involve a translation and a reflection across a
mirror plane. Because matter is dense and minerals have complete and filled lattices there is often more than 1
symmetry operation that yields the same point.
Translation:
1. What are the 2 elements required by every translational symmetry operation? ________________________________
________________________________________________________________________________________________ (2)
2. Explain why translation is the fundamental symmetry element for all crystal systems and unit cell types. _____________
________________________________________________________________________________________________ (2)
3. The following patterns were created using translation. Circle the initial design element (unit cell) and Draw an
arrow for the translation vector that was used to create each of them. (4)
a.
b.
c.
&
Reflections : reflections involve a mirror plane & a translation perpendicular to that plane (across). In
mineralogy we label these reflection planes as m – for mirror.
4. Note the dashed mirror planes in the first panel of the figure below, then find and label all possible mirror
planes for the examples in the second row. (7)
5. For each of the following bilateral designs, locate the line (plane) of symmetry. (4)
7. Draw a vertical mirror plane through the letter m below. Construct the reflected image of the obtuse scalene
triangle xyz to the other side. Compare the distances and directions to a point on your object from the mirror
plane and the equivalent point on its reflection. Give the distance and direction for each point with respect to the
mirror plane. (2)
m
8. Reflect a shape across 2 parallel mirror planes. What does the final (second) shape resemble? Describe another
symmetry operation that could do tessellation (pattern repetition) this in 1 step. (4)
m
m
9. Will this previous example work if the mirror planes are tilted (not vertical) across the page? Explain. (3)
10. Draw the mirror planes on shapes with (a, b) bilateral symmetry, (c) 3-fold dihedral symmetry, and (d) 4-fold
dihedral symmetry.
a.
name this polygonal shape? _________________________________ (2)
b.
name this polygonal shape? _______________________________ (3)
c.
d.
(7)
11. If you see a design with dihedral symmetry, how do you determine where the mirror lines are? (draw one
and explain) (3)
Rotation:
All rotational symmetry involves a centre of symmetry (a point) and an angle of rotation. All even symmetry in a
plane must repeat a whole integer number of times on 1 complete rotation of 360°. This number for the times
that the symmetry repeats is called the fold of the rotational axis. For example a 4 fold rotational axis has a
rotational angle of 90° or 360°/4 = 90°. While solitary plane figures might have any conceivable rotational
symmetry from 1 to n, in minerals and densely packed networks only: 1, 2, 3, 4 and 6 fold rotation is possible to
fill space without leaving gaps or overlaps in a grid. In the 1st example below the figures are chiral (handedness).
(2)
12. For each of the plane figures below, calculate the angle of rotation: (3)
5-fold = _____________,
3-fold= ______________, 2-fold=_________
13. For the figures below given the rotational symmetry and note whether the figure is chiral or not.
a.
fold - ______________,
Chiral? ____________
(2)
b.
fold - ______________,
Chiral? ____________
(2)
c.
fold - ______________,
Chiral? ____________
(2)
Glide Planes:
14. Glide planes are a combined translation and reflection. Unlike simple translation or simple reflection the next
element in the pattern is both along and across. In the example below the hand is moved laterally & reflected
The hand changes from right to left and the watch from 12:15 to 11:45.
With patterns containing glide planes the motif or unit cell is often chiral. In the example below the motif is a
double comma. Not that for any row or column the motifs all rotates the same way but in the adjacent row or
column the rotation is reversed. This is a clue there might be glide planes present as a symmetry element. A
vertical glide plane is labelled and like mirror planes this is positioned between the points or pattern motifs.
a. Find and label the second horizontal glide plane. (2)
In the next example there are 3 different grid elements so the pattern itself is chiral. Find and label at least 2 other
glide planes in this network.
b.
(2)
15. Putting it together. For the following 2-D patterns assume they fill the entire plane: 1) define a unit cell, 2)
find translation vectors, 3) Label mirror planes, 4) Label rotational symmetry axes, 5) label any glide planes.
a.
name grid type __________________________ (9)
b.
c.
name grid type __________________________ (11)
name grid type __________________________ (11)
16. Label the maximum symmetry elements in these individual motiifs.
a.
b.
(6)