The Synergistic Dynamics of Art, Mathematics, & Science
An Interdisciplinary Project at Stowe Middle School
[©Frederick David Abraham, project contributors, and sources]

Geometric Construction of Spirals

Let's try putting some points on the polar graph paper that lie on spirals. One kind of spiral is the Archimedean spiral (named after the 3rd Century Greek mathematician, Archimedes, who studied them extensively) where the spiral goes out from the center by adding equal steps of distance to the previous distance for every equal increase in the amount of angle as we go around the circle. The other main kind of spiral is logarithmic, where the amount of distance that is added with each step to the distance from the center of the previous step is not the same at each step. Instead, the amount added increases with increases in angle, as the spiral goes outward.

Now lets go around the circle in steps of 10° instead of 90°, making an Archimedean spiral by going out further by the addition of an equal distance at each step, starting with 0 distance at 0°.

Some people rounded off the distance to be added at each step. Let us see how much small rounding errors make in the spirals we construct. Some people rounded the number down to .25 units of distance to be added every 10°, and some rounded up to .33 units of distance per 10°.

Because we want to run our psychological experiment on the psychophysics of the aesthetics of spirals for spirals with different numbers of turns within the same diameter (space), lets try to make some Archimedean spirals with two and three turns (720° 1080° respectively) about the circle within the 10 units of distance.

That pretty well introduces us to Archimedean spirals. Now we have a difficult exercise just to get you started thinking about logarithmic spirals. Above we mentioned that for these spirals, each step of the geometric construction of the spiral, each advance of an equal number of degrees, we add not a constant amount of distance to the length of the distance from the center of the graph to the position of the spiral at the previous step, but we add an increasing amount. That amount should increase systematically.

Other Places To Go


Ex. 1-12

of Spirals

of Spirals

Defs Rel
to Spirals

Index Page

Index Page

Created: 12/23/96 Updated: 1/11/97