## Math 110-02, Fall 2008 Information On the Final Exam

The final exam for this course takes place in our regular classroom at the time scheduled by the registrar: Thursday, December 18, at 8:30 AM. It will be the equivalent of a one hour test (four or five pages). For the most part, the exam is not cumulative, although there will be a few general questions on material and reading from earlier in the term. In particular, you should expect an essay question on the topic, What have you learned in this course about the nature of mathematics? You should prepare to answer this question and to illustrate your answer with specific examples from the course. Also, I might well ask you for the proof that there are only five regular polyhedra.

Most of the exam will be on material covered since the second test. That includes Chapters 4 and 5 of ZERO; Chapters 12 and 13 of The Math Instinct; dimensions, including the Flatland video and Sections 6.1 and 6.2 of Symmetry, Shape, and Space; some basic ideas about fractals and the Mandelbrot set from Assignment 11; material on infinity, summarized in Assignment 13; and Sections 7.1 and 7.2 of Symmetry, Shape, and Space on polyhedra.

As for material from earlier in the course, I would like you be aware of the main themes in the books The Math Instinct and ZERO: The Biography of a Dangerous Idea. The first of these makes the argument that people have some built-in mathematical ability and can learn to tap that ability reliably in everyday life, but their math instinct is generally not very well-served by the way math is commonly taught and used in school. The second, on the other hand, shows how even something as simple and basic to modern mathematics as the idea of zero had to be slowly developed over a period of centuries. It also shows how passionately mathematicians and philosophers have argued about mathematics. And it shows how far modern mathematics has come from "instinct" and common sense, especially with regards to infinity. Aside from these two books, you might want to review the way math has been used in art, especially in the art of M.C. Escher -- although I won't ask you any math problems about tesselations, tilings, and symmetry, these concepts might come up in essay questions.

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

```    life in a two-dimensional world
shapes for two-dimensional worlds:  (plane, sphere, Möbius strip, etc.)
how a two-dimensional being could think about three dimensions
the fourth dimension, and the possibility of higher dimensional worlds
how a three-dimensional being could think about four dimensions
the hypercube (or "tesseract")
how art shows three dimensions in a two-dimensional picture
how a hypercube can be projected into a three-dimension world

fractals and self-similarity
the Sierpinski triangle fractal
the Koch curve fractal
how fractals can be made from "maps" in the ChaosGame program
the Mandelbrot set and what it has to do with infinity

ZERO, Chapter 4:
the introduction of perspective into art
how experiments proved that Nature does not abhor a vacuum
how Aristotle's objections to zero and infinity were finally overcome

infinity
two sets have the same "size" if their elements can be matched up in pairs
infinite sets
Hilbert's Hotel
pairing up elements of various infinite sets to show they are the same size
there are different sizes of infinity
the infinity of all decimal numbers is bigger than the infinity of the integers
the infinity of subsets of integers is bigger than the infinity of the integers
the difference between potential infinity and actual infinity

ZERO, Chapter 5:
calculus works with the infinite and the infinitesimal
finding volumes by adding up an infinite number of infinitely thin slices
finding slopes by dividing infinitesimals
philosophical objection to the original formulation of calculus
how calculus finally became rigorous and tamed infinity

polyhedron
the regular polyhedra (also called "Platonic Solids")
the tetrahedron, octahedron, cube, dodecahedron, and icosahedron
the proof that there are only five regular polyhedra
Euler's characteristic, v - e + f
the Euler characteristic of any "sphere-like" polyhedron is 2
the Euler characteristic of any "torus-like" polyhedron is 0
definition of semi-regular polyhedron
prisms and anti-prisms

The Math Instinct, Chapters 12 and 13:
the trouble with "meaningless math"
the manipulations done by street mathematicians have meaning for them
school math is symbolic and abstract
math as an abstract game of formal [meaningless] symbols
to master math, you have to develop a more abstract type of meaning
natural math + language abilities --> capacity for abstract math
metaphor as a primary means of assigning means to new concepts
```

Math 110 Home