Math 116: Symmetry and Shape
Guide for the Final Exam

The final exam is on Monday, May 4, from 12:00pm to 3:00pm, in Trumbower 347.

Note that you may bring one page (standard letter size, only one side filled) of handwritten notes to the exam. You will be required to hand this in with your final.

The final exam is cumulative, so you do need to review the material covered in Exam 1 and Exam 2. 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

Ch 2.2. Celtic Knots:

  1. Be able to find the appropriate grid of dots for a knot with specified dimensions, and draw this knot.
  2. Be able to find the number of strands and position of loose ends based on dimensions, and be able to determine dimensions that would give a specified number of strands or position of loose ends.
  3. Know how to draw a celtic knot on a grid with bars.

Ch 7.1. Prisms and Pyramids:

  1. Know the definitions of the terms polyhedron, prism and pyramid.
  2. Be able to classify prisms and pyramids as right or skew, and as regular or irregular.
  3. Be able to sketch a net of a specified polyhedron.
  4. Know how to find the volume of prisms, pyramids, cones, cylinders and spheres.
  5. You should be able to identify the shape of cross-sections of solids.

Ch 7.2. Platonic Solids:

  1. You should know how many Platonic solids there are, and be able to name them all.
  2. You should be able to label these solids in the same way that we labelled regular tilings (for example, 3.3.3.3).
  3. Know how to compute the Euler characteristic of any solid (even non-Platonic), ie, #vertices - #edges + #faces.
  4. Know what it means for a polyhedron to be convex and that the Euler characteristic of any convex polyhedron is 2.
  5. Know what we mean by the dual of a polygon, and know the duals of each of the Platonic solids in particular.

Ch 5.1. Two Dimensional Symmetry:

  1. You should know what we mean by reflection symmetry and be able to identify lines of reflection of a given figure.
  2. Know which angles between two hinged mirrors give you a whole number of images.
  3. You should know what a Coxeter polygon is, and be able to find all of them. You should also know how to sketch the tilings you would get by arranging mirrors into a Coxeter polygon.

Ch 5.2. Rosette Groups:

  1. You should know what we mean by rotational symmetry and be able to identify angles of rotation of a given figure.
  2. You should know the notation for the two types of rosette groups, Cn and Dn, and which elements they contain.
  3. Know how to identify the rosette group of a given figure, and how to produce examples of figures with a given rosette group.
  4. Be able to construct the multiplication table for a given rosette group, and simplify a sequence of transformations to one of standard elements of the rosette group.

Ch 5.3. Frieze Patterns:

  1. Be able to describe and identify the following rigid motions and create examples of frieze patterns formed using them:
    1. rotation
    2. reflection
    3. translation
    4. glide reflection
  2. You should be able to categorize a given frieze pattern as one of the seven possible types based on the types of symmetries that it exhibits, and be able to draw an example of a frieze pattern of a given type. Additionally, you should be able to determine whether two frieze patterns are of the same type or not.

Maintained by ynaqvi and last modified 04/27/15