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 (C_{n}),
dihedral groups (D_{n}), 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 C_{n}and D_{n}symmetry how to draw C_{n}and D_{n}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

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.