Math 110-02, Fall 2008
Information On Second First Test

The second test in this course will take place in class on Friday, November 7. The test is not cumulative; it covers only what we have done since the first test. This includes: Material on symmetry from Sections 5.1 through 5.4 of Symmetry, Shape, and Space; readings on the mathematics of voting; and Chapters 0 through 3 of ZERO: The Biography of a Dangerous Idea. The test will include some short answer questions and longer essay questions, as well as exercises on symmetry groups and voting.

You should understand the general idea of symmetry of patterns in the plane, as well as the four basic types of symmetry (rotation, reflection, translation, and glide reflection) and how they can be combined. You should be able to recognize the four types of symmetry groups for patterns in the plane: cyclic groups (Cn), dihedral groups (Dn), frieze groups, and wallpaper groups. You will be asked to find the symmetry groups of several patterns. For the frieze groups and wallpaper groups, you will be given a copy of the "decision trees" that we used for classifying the patterns.

For voting, you should understand some of the criteria that can be used for evaluating various voting systems. You should understand the problems of spoilers, vote splitting, and Condorcet cycles. You should understand the implications of Arrow's Theorem. Given data about voter preferences, you should be able to find the winner of the election using various voting systems (plurality, plurality with runoff, instant runoff voting, Condorcet voting, Borda count, range voting). You should know what is meant by strategic voting. It would be good to be familiar with some of the examples from the reading (such as vote splitting in the Academy Awards or the use of Borda Count in athletics for things like NCAA rankings). You won't be asked about specific examples, but you might want to use them in your essays.

As for ZERO, this book is an attempt to look at a particular mathematical idea (the number zero) that is a lot more subtle than it might at first appear, and to put it into an historical and cultural context. Zero is different from other numbers and is closely related to the idea of infinity. ZERO claims that cultural, philosophical, and religious differences made it more difficult for zero to be invented and accepted in the "East" than in the "West," but that eventually the superiority of the Indo-Arabic system caused it to be accepted in the West as well. You should understand something about the nature and history of zero and infinity as discussed in this book.

Here is a list of some terms and ideas that you should know for the test:

    symmetry operation
    reflection symmetry; line of reflection
    mirrors and reflection symmetry
    what can be done with two or three mirrors
    rosette groups (cyclic groups and dihedral groups)
    how to recognize Cn and Dn symmetry
    how to draw Cn and Dn symmetry
    applying rotations and flips to regular polyhedra
    following one symmetry operation by another (such as: RF)
    "RF" and "FR" can be different
    frieze patterns and frieze groups
    types of symmetry of frieze patterns:
        translation, horizontal reflection, vertical reflection, 
        glide reflection, 180-degree rotation
    using a decision tree to classify a frieze pattern
    drawing frieze patterns with various frieze groups
    wallpaper patterns and wallpaper groups
    types of symmetry of wallpaper patterns:
       translations in multiple directions, reflections, glide reflections,
       rotations of 180 or 90 or 120 or 60 degrees
    using a decision tree to classify a wallpaper pattern
    voting systems
    strategic voting
    vote splitting
    runoff election
    how runoffs help with the problem of spoilers
    the "lizard versus wizard" phenomenon in runoff elections
    ranked ballot
    instant runoff voting (IRV)
    Arrow's Theorem
       criteria:  transitivity, unanimity, non-dictatorship,
                  independence of irrelevant alternatives
    Condorcet voting system
    Condorcet winner, Condorcet cycle
    Borda Count voting system
    range ballots and range voting
    number systems without zero (tallies, Egyptian, Roman numerals)
    zero as placeholder versus zero as actual number
    unusual properties of zero, such as division by zero doesn't make sense
    mathematics in classical Greece
    numbers as geometric shapes
    the Pythagorean idea of the equivalence of numbers and music/harmony
    music or harmony of the spheres
    Aristotle's rejection of infinity
    the nutshell universe
    Zeno's paradox of Achilles and the Tortoise
    how the paradox is resolved by allowing the addition of an infinite number of
        quantities that get smaller and smaller
    the missing "year zero" in the Western calendar
    Indian philosophy and mysticism accepted infinity and nothingness
    invention in India of the base-10 place-value system with zero
    adoption of Indian system by Muslims, giving the "Arabic" number system
    Al-Khowarizmi, algebra, and algorithm
    adoption of Arabic numbers in Europe, especially by bankers

Revised Schedule

We have departed somewhat from the original tentative syllabus for the course. For a revised schedule for the rest of the semester, see the web page for this course at