Name __________________
Mineralogy PreLab 1 Homework: Web Assignment (50 points, complete by 1st lab)
The Planar symmetry stuff from Math Forum requires a Java compatible web browser
and runs off of NCTM 9-12 high school math software material. If you do not have Java
installed, go to www:java.com and install it, ~5 minutes.
Part I: For specific exercises on rotation, reflection, translation, and glides, see the
appropriate nctm sites and work through their examples on line. After doing this ~1 hour
series of exercises, answer or perform the comprehensive set of exercises below.
For background on planar symmetry and symmetry operations see Susan Addington’s
Page below.
http://mathforum.org/sum95/suzanne/symsusan.html
For Symmetries involving rotations see nctm site:
http://illuminations.nctm.org/LessonDetail.aspx?ID=U138
Answer the following questions on Rotation:
1. What are the 2 elements required by every rotational symmetry operation? (2)
__________________________________________________________________
2. Rotate the letter shown below about the point by 90°, 180° and 270°. Describe the
maximum order of rotational symmetry present in the figure you have just generated. (2)
Y
.
3. How does a single rotation of 0° or 360° compare with 6 successive rotations of 60°?
Generalize this in a simple statement regarding sequences of rotations and the final
resultant angle. (2)
4. Draw a simple design with a central rotocentre and “Point Symmetry”. (2)
1
For Symmetries involving reflections see nctm site:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L470
5. For each of the following bilateral designs, locate the line (plane) of symmetry. (4)
6. Draw a shape and reflect it across a mirror plane. Compare the distances and directions
to a point on your object and the equivalent point on its reflection. Give the distance and
direction for each point with respect to the mirror plane. (2)
6. Reflect a shape across 2 parallel mirror planes. What does the final shape resemble?
Describe another symmetry operation that could do this in 1 step. (2)
7. Reflect a shape across 2 perpendicular mirror planes. Completely describe another
symmetry operation that could accomplish this in 1 step. Will this work if the mirror
planes are at non-right angles? Explain. (3)
8. Draw shapes with (a) bilateral symmetry, (b) 3-fold dihedral symmetry, and (c) 4-fold
dihedral symmetry. (3)
9. If you see a design with dihedral symmetry, how do you determine where the mirror
lines are? (draw one and explain) (3)
2
For Symmetries involving translations see nctm site:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L474
10. What are the 2 elements required by every translational symmetry operation?
_______________________________________________________________ (2)
11. Explain why translation is the fundamental symmetry element for all crystal systems
and unit cell types.
________________________________________________________________________
____________________________________________________________________ (2)
12. 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.
13. Translate a pattern with bilateral symmetry (internal mirror plane, palindrome). Is the
translated pattern also bilaterally symmetric? (2)
14. Translate a pattern with rotational symmetry (the simplest has point symmetry). Does
the translated pattern also have this rotational symmetry? (2)
15. State the general rule for translations which combines your inference from 8 & 9. (2)
3
For Symmetries involving glides see nctm site:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L475
16. Which of the following band ornaments were made using glide reflections? For those
that were, find the mirror line and the translation vector. (4)
a.
b.
c.
17. The following wallpaper patterns were created using glide reflections. Find two
different glide reflections in each pattern and give the mirror line and the translation
vector for each. (3)
a.
b.
c.
18. What information do you need to give in order to define a glide reflection? Why do
we require that the mirror line of a glide reflection be parallel to the translation vector of
the glide reflection? (4)
4