Math 123H - Symmetry
Questions on Readings (Hargittai & Hargittai)

Q1: What is the difference between geometric symmetry and symmetry in nature or the real world?

Geometric symmetry can be thought of in our minds as perfectly precise and exact (though it is often realized through our somewhat imperfect drawings of examples).  Symmetry in nature or the real world is often only approximate and we must fill in or fix the missing or imperfect detail in order to create complete symmetry.

Q2: How does the material we have already studied relate to the point group and space group symmetries (p. xvi, xvii) and to symmetry operations and symmetry elements (p. 2, 39)?  More specifically:
Q3: Why do they refer to a mirror plane rather than a mirror line?  Hint: where do the objects that Hargittai and Hargittai consider naturally live?  Where do the objects that we have reflected so far live?

The objects considered in the book by Hargittai & Hargittai are primarily three-dimensional, so to reflect them they use a two-dimensional plane.  The objects we have considered previously are primarily two-dimensional, so to reflect them we used a one-dimensional line.

Q4: Why do they refer to a rotation axis rather than a rotocenter on page 39?  Same type of hint as for Q3.

The objects considered in the book by Hargittai & Hargittai are primarily three-dimensional, so to rotate them they use a one-dimensional line (think of the axis for the wheels on a car or the Earth's axis of rotation).  The objects we have considered previously are primarily two-dimensional, so to rotate them we used a zero-dimensional point, the rotocenter.

Q5: Explain the reason behind the "Fact to Consider" on page 68.  Do you agree with their statement that the "only case where reflection is not accompanied by rotation is when there is bilateral symmetry?

It is true that bilateral symmetry involves one reflectional symmetry and only one other symmetry, the identity, so figures with bilateral symmetry do not appear to have any rotational symmetry.  However, bilateral symmetry is the same as symmetry type D_1.  When discussing dihedral symmetry type D_n, it is most natural to include the identity symmetry as a rotation so that the figure can be considered to have n reflection symmetries and n rotation symmetries (as opposed to n reflection symmetries and n-1 rotation symmetries).  Thus it is reasonable to consider the identity to be a rotational symmetry, in which case one might dispute or at least want to clarify the statement on page 68.

Q6: Is the comment on page 45 about jellyfish consistent with the discussion of symmetry in animals in Bunch, Chapter 1?  Explain.

The comment that the 4-fold rotational symmetry of the jellyfish may be due to a circling motion in capturing food is consistent with Bunch's point that symmetry often occurs in the absence of any differentiating or distinguishing force.  Since the jellyfish is not moving in a straight line direction, but rather in a circle, it is reasonable for the jellyfish to not have a front and a back.  This does not explain however why there is 4-fold rotational symmetry as opposed to some other type C_n symmetry.


Questions on Rosen

Q1 (Foreword, Preface, and Chapter 1):  How many symmetries does he view a square as having?  What are the additional symmetries and why do they arise? 
Rosen identifies 12 symmetries of the square.  He finds more than the eight we have already studied because he is considering the square to be sitting in a 3-dimensional space. 
In addition to the rotations by 0, 90, 180, and 270 degrees about an axis through the center perpendicular to the plane of the square, and reflection in planes perpendicular to the plane of the square and passing through the axes formed by diagonals and lines through midpoints of opposite sides, he also find rotations by 180 degrees about these same lines.  When viewing the square as simply a 2-dimensional object, the effects of these additional 4 rotations and the 4 reflections are the same.

Q2 (Chapter 1):  Rosen takes a much broader view of what constitutes a symmetry transformation than we have so far.  Name 3 types of symmetry transformations that he mentions but we have not considered.

Q3 (Chapter 2):  Suppose the depressions labelled A, B, and C are arranged in an equilateral triangle with A at the top vertex, B at the bottom left vertex and C at the bottom right vertex.  Give the symmetries described on page 10 in terms of Farmer's standard notation (using r and m).  (It may help to draw out the triangle diagrams and show the effect of the maps Rosen describes on the diagram.  Remember that standard notation means the m is written on the left, but as always, the motion on the right is applied first.)

Q4 (Chapter 2):  Write an equation, using the symbols a and b to stand for two general symmetry transformations, that shows the property Rosen describes in the last three paragraphs of Chapter 2 (page 16-17) before the group theory section.  (Let a stand for covering the pot and b stand for filling the pot.  Notice that Rosen is discussing how to find the inverse of a combined transformation.  Write an equation for this using the symbols a and b.  It may help to think of this as translating the sentences on the bottom of page 16 and top of page 17 into symbols, or as translating part of Figure 2.13 into symbols.)
(ab)^{-1}=b^{-1}a^{-1}
The crucial observation is that the inverse is NOT a^{-1}b^{-1} unless a and b commute, i.e., satisfy ab=ba.

Q5:  Write one or more questions you have after reading Chapters 1 and 2.