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.

• Exercise 7. Place a point at 0°,0 units of distance (at the center - does the angle make any difference there? Then place one at 2.5 units of distance at 90° another at 5 units of distance at 180° another at 7.5 units of distance at 270° and a fifth point at 10 units of distance (at the outermost circle) at 360°. On what kind of spiral do these points lie?

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°.

• Exercise 8.
How many steps will it take to go around the circle once? What distance do we have to add at each step? Go ahead an put the points on the polar graph paper, and connect the dots to make the spiral.

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°.

• Exercise 9. For each of these rounding errors, what was the distance of the spiral from the center at 360°? How many degrees did it take to reach the outside circle at 10 units of distance from the center?

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.

• Exercise 10. How big is the distance to be added at each increase of 10° for these sprials?

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.

• Exercise 11. Try drawing a logarithmic spiral.