Math 116: Symmetry and Shape
Guide for Exam 1

The first midterm is on Monday, February 32, during the regular class period.

The following is a chapter by chapter guide intended to help you organize the material we have covered as you study for your exam. It is only intended to serve as a guideline, and may not explicitly mention everything that you need to study.

Please start by taking a look at class notes and the material that we discussed in class. Then review all relevant textbook sections and homework problems for the chapters given below, and make sure you can solve all of them.

Ch 1.1. Measurement:

  1. Know how to find the area and perimeter of a figure made up of rectangular, circular or right triangular pieces, such as the ones in Problem 10 and 14.
  2. Know the statement of the Pythagorean Theorem and how to apply it.

Ch 1.2. Polygons:

  1. Know how polygons are defined and named, especially for n-gons where n=3,4,5,6,8,10,12,20.
  2. Identify different types of triangles (ie, equilateral, isosceles, scalene) and quadrilaterals (ie, squares, rectangles, parallelograms, rhombuses, trapezoids).
  3. Know what it means for a polygon to be convex (or not).
  4. Be able to find the sum of the vertex angles of an arbitrary (ie, not necessarily regular) polygon by dividing it up into triangles.
  5. Be able to find each vertex angle, the center and the central angle of a regular polygon.

Ch 3.1. Compass and Straightedge Constructions:

  1. Be able to state Euclid's postulates in your own words.
  2. Know how to construct (and describe the constructions of) the following:
    1. perpendicular bisector
    2. a perpendicular to a line through a given point
    3. an angle bisector
    4. a line parallel to another line
    5. an equilateral triangle
    6. a square
    7. a regular hexagon
    8. a regular octagon
    9. a regular dodecagon
  3. Know the statement of Gauss's Theorem and how to use it to identify which polygons are constructible. (You do not need to know the Fermat primes, which will be given to you as needed.)

Ch 3.2. The Pentagon and the Golden Ratio:

  1. You do not need to know how to construct a pentagon, but you should know how to identify the golden triangle inside.
  2. You should know what it means for two lengths to be related by the golden ratio.
  3. You should be able to use the relationship Φ2 = Φ + 1 to find an exact value for Φ and to simplify higher powers of Φ. (You do not need to know the quadratic formula, which will be given to you as needed.)
  4. You should be able to decompose a golden triangle or golden rectangle to find similar copies of the larger figure inside.

Ch 3.3. Theoretical Origami:

  1. Know how to construct (and describe the constructions of) the following:
    1. perpendicular bisector
    2. a perpendicular to a line through a given point
    3. an angle bisector
    4. a line parallel to another line
    5. an equilateral triangle
    6. a square
    7. a regular hexagon

Ch 4.1. Regular and Semiregular Tilings:

  1. Know the definitions of regular and semiregular tilings, and system used to label them.
  2. You should know all of the possible regular tilings (although you do not need to remember every possible semiregular tiling).
  3. Know how to sketch a given regular or semiregular tiling.
  4. Given a regular or semiregular tiling, you should be able to draw the dual tiling, and identify the vertex angles of the dual tiles.

Ch 4.2. Irregular Tilings:

  1. Know how to identify and sketch irregular tilings.
  2. Be able to describe and identify the following transformations and create examples of tiles formed using them:
    1. parallel translation
    2. glide reflection
    3. midpoint rotation
    4. side rotation

Maintained by ynaqvi and last modified 02/16/15