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- Build a true scale model of the solar system
- but be careful because it cannot be
contained within the confines of an exhibit. Illustrate how you would
locate it in your town. Maybe even do so!!
- What is/are Napier's bones and what can you do with it/them?
- Discover how to construct the Koch or ``snowflake'' curve. Use
your computer to draw fractals based on simple equations such as Julia
sets and Mandelbrot sets. References: [Pet], see
[Lau] for example programs.
What is fractal dimension? Investigate it by examining examples
showing what happens to lines, areas, solids, or the Koch curve,
when you double the scale.
- Martin Gardner in [Gard6] defines a paradox to be
``any result that is so contrary to common sense and intuition that it
invokes an immediate emotion of surprise.'' There are different
types of paradoxes. Find examples of all of them and understand how
they differ.
- Knots. What happens when you put a knot in a strip of paper and
flatten it carefully? When is what appears to be a knot really a
knot? Look at methods for drawing knots. References:
[Stei], [F&S]. Also check out the web
sites [KnotPlot], [MathMania] and [MegaMath].
- Another source of knots is the stone-work and ornamentation of
the Celts. Investigate Celtic knotwork and discover how these
elaborate designs can be studied mathematically. References:
[Cro], [M].
- Learn about origamic architecture by making pop-up greeting
cards. Reference: [Cha].
- Is there an algorithm for getting out of 2-dimensional mazes?
What about 3-dimensional? Look at the history of mazes (some are
extraordinary). How would you go about finding someone who is lost in
a maze (2 or 3 dimensional) and wandering randomly? How many people
would you need to find them?
- Explore Penrose tiles and discover why they are of interest.
References: [Pet], most books on tiling the plane.
- Investigate the Steiner problem - one application of which is
concerned with the location of telephone exchanges to minimize costs.
- Use PID (proportional-integral-differential) controllers and
oscilloscopes to demonstrate the integration and differentiation of
different functions.
- Construct a double pendulum and use it to investigate chaos.
- Investigate the mathematics of weaving. References:
[G&S2], [Cla] and [Hos].
- What are Pick's Theorem and Euler's Theorem? Investigate them
individually, or try to discover how they are related. Reference: [D&R]
- Popsicle Stick Weaving: With long flat sticks, which patterns
of ``weaving over and under'' in the plane are stable (as opposed to
flying apart). Find a pattern with four sticks. Is it unique? Does
the stability change when you twist one of the sticks (in the plane)?
Find several patterns with six sticks whose stability depends on the
particular ``geometry'' of where they cross (i.e. the pattern becomes
unstable if you twist one of the sticks in the plane). Can you give a
rule for recognizing the ``good geometric positions''. What kinds of
``forces'' and ``equilibria'' are being balanced here? What general rules
can you give for ``good'' weavings? [Source of some information:
whiteley@mathstat.yorku.ca.]
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CaMEl@camel.math.ca
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