The Synergistic Dynamics of Art, Mathematics, & Science
An Interdisciplinary Project at Stowe Middle School
[©Frederick David Abraham, project
contributors, and sources]
Mathematical Construction of Spirals: Polar Coordinate System
If you look at polar graph paper (example), you see
concentric circles (circles within circles), and lines radiating out from the
center. We can think of this as related to the face of a clock. The hands of a
clock sweep clockwise around the face. The change in the angle of the hands can
be measured in different ways, but the most familiar way is to say that one
revolution around the clock takes 360°, so the angle is measured in degrees
Exercise 1: On an ordinary clock showing 12 hours, how
many degrees does the minute hand go through in one hour? In two hours? In
Exercise 2: On an ordinary clock showing 12 hours,
how many degrees does the hour hand go through in 5 minutes? In 10 minutes? In 15
minutes? For a 24 hr clock, what would the answers be?
The usual convention (way of doing things) is
to call the three o'clock position zero, and go around counter-clockwise. Note
that our graph paper is labelled both ways, for counting either clockwise or
Exercise 3. At what positions are the numbers for
clockwise and counter-clockwise the same? That is, how many degrees are there,
and where are they located on the circle, going clockwise and counter-clockwise
at the same time, to have the degrees travelled be the same? That is similar to
asking the question when two people leave the same spot, at the same time, in
Vermont, one going exactly north, the other going exactly south, how far will
they have gone, and through how many degrees willhave travelled, when they meet
again. Also after they pass each other, when they get back home? Will they be
tired? How much of their trip will be in cross-country skiis, how much in
kayaks, and will it be the same for both of them?
Exercise 4. If the longest radial lines (radiating
out from the center) on the graph paper are spaced every 10°, how many of them
are there in the 360° circle?
Exercise 5. How many degrees does it take to go 1/4
of the way around the circle? 1/2 way? 3/4 way?
Exercise 6. On a globe or polar map of the earth,
what countries does the 0°, 90°, 180°, and 270° longitudinal lines go through?
So far we have been considering how to measure angles. Now lets look at the
distance from the center. The circles are placed evenly. That is, there are 10
circles (on our graph paper), and the distance between them remains constant;
the distance between rings 2 and 3 are the same as between rings 9 and 10. We
don't care what you call these units of distance since they are not exact in
either U. S. Customary Units (actually one unit on our graph paper is about .33
inch) or the metric system (one unit on our graph paper is about 8.5
millimeters-mm or .85 centimeters-cm).
Created: 12/23/96 Updated: 1/11/97;