Topics to study for the final
Multicultural Mathematics
Fall 2004

Remember:  our final exam is Friday, December 17th, at 9 a.m in Science Center 317.  This is not the information published in the exam schedule.  I changed the date so we can all go home sooner!

Topics and typical questions.  You will have some choice of questions.

Sona designs

  1. Explain what a monolinear sona design is, and identify monolinear designs.
  2. Here is a picture of a sona design constructed using ``mirrors.''   Draw the mirrors.  (I'll give you something like the chased-chicken, lion's stomach, etc....)
  3. Which of the following two sonas are variations on the same design?  (I'll give you something like the chased-chicken, flying ducks, lion's stomach, kajana tree etc....)
  4. Here are some variations on the same sona design.  (I'll give you something like variations on the lion's stomach or other design.)  Which of the following is true?
    1. The number of rows must be odd and the number of columns must be even.
    2. The number of rows and the number of columns must be relatively prime.
    3. The number of rows must divide the number of columns.
    4. There may be any number of rows and columns.
Celtic knots
Turn the following design into a Celtic knot.  (I'll give you something like a sona design without the dots.)
Knot theory
  1. Which of the following knots are equivalent to the unknot? The answer may be more than one.

  2. Which pair of the following knots are equivalent to each other?

  3. Why are knots important to modern scientists?
Symmetry
  1. Identify all the symmetries of the following design.  (I'll give you somthing like a titja basket.)  If it has rotational symmetry, draw the rotocenter and indicate the number of degrees of the rotation.  If it has reflections, draw the axes of reflection.
Strip patterns and wallpaper patterns
  1. Classify the following patterns using the pq system.  (Hint:  there is one of each classification.)
  2. Which pair of the following strip patterns have the same symmetries?
  3. Which pair of the following wallpaper patterns have the same symmetries?
    1.  






       
       

  4. Who was M.C. Escher?  How was he inspired by Islamic and Japanese art? (read p. 259-273)
Drumming
  1. What is the total length (in beats) of the following composition?
    1. Pattern A:  dzzdz
      Pattern B:  ddz
      Drum:  ((ABA)2 z4 B)3
  2. Is the following composition ``legal,'' meaning that both voices have the same number of beats?  Show your work.
    1. Patterns A and B as in #1
      Drum 1:  A3 BABB
      Drum 2:  A B8
  3. What is the minimum number of beats necessary to make the polyrhythm 6 against 14?
  4. Write out all the ways to divide 16 beats into an equal number of notes.
  5. Which of the following is an example of AABA form?
    1. dzdd dzdd zdzd dzdd
    2. dzdd zdzd dzdd zdzd
    3. zdzd dzzz zdzd zdzd
  6. Use the following patterns to form a short composition for two instruments.  You must use all the patterns, and each drum must have the same number of beats.  You are also allowed to add more patterns and add rests.
    1. Pattern A:  dzzd
      Pattern B:  dzzddddz
      Pattern C:  dzdzd
Poetry, the Fibonacci numbers, and Pascal's Triangle
  1. If you know the first 16 Fibonacci numbers, how do you find the 17th Fibonacci number?
  2. If you know the first 5 rows of Pascal's Triangle, how do you find the 6th row?
  3. What, precisely, is the relationship between Pascal's triangle and Sanskrit poetry?  Between the Fibonacci numbers and Sanskrit poetry?  Who was the first to discover it, and when?
Essays (should be about 3 paragraphs.  You can do either one or two essays.)
  1. Discuss the Chi Rho page of the Book of Kells from a cultural, mathematical, and artistic standpoint.  (Hint:  mathematical topics you could discuss are symmetry, knots, powers and multiples, and more...)
  2. Discuss the lion's stomach sona design from a cultural, mathematical, and artistic standpoint.
  3. Discuss sipatsi designs from a cultural, mathematical, and artistic standpoint.
  4. Explain precisely how and why the Fibonacci numbers count syllables in Sanskrit poetry.  Discuss the history of this discovery (read p. 274-276 in the course packet).
Required essay (10 points--should be one paragraph)
  1. Discuss your final project.  (Basically, write out what you said when you presented the project.)  What were the goals of the project?  What mathematical concepts does it illustrate?  Did you encounter any mathematical difficulties when constructing the project, and, if so, what were they?