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

Explain what a monolinear sona design is, and identify monolinear designs.

Here is a picture of a sona design constructed using ``mirrors.''
Draw the mirrors. (I'll give you something like the chasedchicken,
lion's stomach, etc....)

Which of the following two sonas are variations on the same design?
(I'll give you something like the chasedchicken, flying ducks, lion's
stomach, kajana tree etc....)

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?

The number of rows must be odd and the number of columns must be even.

The number of rows and the number of columns must be relatively prime.

The number of rows must divide the number of columns.

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

Which of the following knots are equivalent to the unknot? The answer may
be more than one.

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

Why are knots important to modern scientists?
Symmetry

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

Classify the following patterns using the pq system. (Hint:
there is one of each classification.)








Which pair of the following strip patterns have the same symmetries?





Which pair of the following wallpaper patterns have the same symmetries?

Who was M.C. Escher? How was he inspired by Islamic and Japanese
art? (read p. 259273)
Drumming

What is the total length (in beats) of the following composition?
Pattern A: dzzdz
Pattern B: ddz
Drum: ((ABA)2 z4 B)3

Is the following composition ``legal,'' meaning that both voices have the
same number of beats? Show your work.
Patterns A and B as in #1
Drum 1: A3 BABB
Drum 2: A B8

What is the minimum number of beats necessary to make the polyrhythm 6
against 14?

Write out all the ways to divide 16 beats into an equal number of notes.

Which of the following is an example of AABA form?

dzdd dzdd zdzd dzdd

dzdd zdzd dzdd zdzd

zdzd dzzz zdzd zdzd

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.
Pattern A: dzzd
Pattern B: dzzddddz
Pattern C: dzdzd
Poetry, the Fibonacci numbers, and Pascal's Triangle

If you know the first 16 Fibonacci numbers, how do you find the 17th Fibonacci
number?

If you know the first 5 rows of Pascal's Triangle, how do you find the
6th row?

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.)

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...)

Discuss the lion's stomach sona design from a cultural, mathematical, and
artistic standpoint.

Discuss sipatsi designs from a cultural, mathematical, and artistic
standpoint.

Explain precisely how and why the Fibonacci numbers count
syllables in Sanskrit poetry. Discuss the history of this discovery
(read p. 274276 in the course packet).
Required essay (10 pointsshould be one paragraph)

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?