## 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
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
plurality
spoiler
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
burying
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
Pythagorus
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 http://math.hws.edu/eck/math110.