Quiz 4 Comments

Problem 1.

This problem is about the division of isometries into "even"  (also called "direct" or "orientation-preserving") and "odd" (also called "indirect" or "orientation-reversing").

• Some papers showed a lack of understanding of the whole even-odd idea.  If a paper had line reflection or glide reflection circled, it was in this category.  Anyone who made this mistake needs to review the whole concept and sequence of ideas.
• A great number of papers explained the concept correctly, but did not include all the possible "evens" (e.g., the identity and point reflection). 

Problem 2

This problem is about the concept of symmetry (the definition and the symmetry group concept) and also some practical work with compositions.

• Parts (a) and (b) both are 5.  These numbers will always be equal if the number of line reflections > 0. 
• The most common error in Part (c) was getting the line reflections in the opposite order.  It is impossible to do such problems correctly without sorting out how the order of composition and the direction of rotation are related.
• In Part (d), there were several kinds of incorrect answers.  The answer of "rotation" shows problems with even-odd again.  The answer of "glide reflection" probably comes from an incomplete understanding of the facts about triple line reflections, but it is a seriously incorrect because there is no way a glide reflection can be a symmetry of a bounded figure such as a regular polygon.  (Just try to move a pentagon or a circle or a square by a glide reflection so that the image is the same as the original.). 
• Once on realizes that the product is a line reflection (triple reflections in concurrent lines), one should recognize that it also must be one of the 5 line reflections that are symmetries.  Thus the answer comes easily by tracing the image P''' of any single vertex P and observing the perpendicular bisector of PP'''.

Problem 3

This problem is about half-turns (point reflections). 

• It is easy if you use the results of Lab 9 or Section 2.3. It is very hard to do by composing 6 line reflections.  A few papers did not recognize the product as half-turn.

Problem 4

The bulk of this problem was constructing the center of the product rotation. 

• This uses a method that has been much explained and emphasized.  It is laid out very clearly in Brown 2.4 among other places.
• A very surprising phenomenon was that the majority of papers had the incorrect rotation angle for the product.  Somehow the simple way of computing this seems to have gotten lost in the mirrors.  ANGLES ADD when rotations are composed!

Problem 5

This asks for the definition of isometry, the meaning of angle-preserving, and a proof of the angle-preserving properties of isometries.

• The definition should start with something like "An isometry is a transformation that …"  If you don't say what kind of thing it is, it is probably not going to be correct.  Also, the definition does not mention angles.
• Part (b) should be an operational explanation with ABC not just a rewording of the phrase.
• Part (c) uses SSS congruence for triangles.

Problem 6

 This asks for construction of the defining data of a glide transformation.

• This construction is pretty simple if one uses the relationship between midpoints and invariant line of a glide reflection.  Also, it is not too bad to construct a half-turn and a line reflection.  Using triple l ine reflection is the hardest.

Problem 7

The underlying big idea of this is that if you have a couple of 90-degree rotations in a symmetry group, the pattern of rotations is forced and is always the same (it is the pattern in any p4-type symmetry).  A lot of attention has been paid to these square-based symmetry patterns in lab and homework.

• The building block is the isosceles right triangle with 90s at the 45-degree vertices and 180s at the right angle.
• The overall pattern is made of squares with rotation centers at the corners.
• There are two sets of 90-degree rotation centers and the translations do not intermingle these centers.  The same is true for the 180-degree centers.