From polygons to circles

If you pin down one end of a string, attach a pen to other end, you can draw a circle as you make a full turn, all the way back to where you started (360 degrees). The string, which we call a “radius,” allows you to make polygons with ends that touch the circle. To get a triangle: make a third of a full turn before putting the pen down to make a dot. Repeat making another third turn, and a final third of a turn before making the third dot. Connect the dots to get a triangle with corners touching the circle. To get a square, make a quarter turn (90 degrees) before putting the pen down to make a dot. Make another quarter turn before making a second dot. Repeat two more times. Connect the dots. With smaller turns, you can get pentagons, hexagons (60 degree turns), and polygons with more and more sides, looking increasingly like a circle. Try this with code for circles and spirals (click here). Click “no radius growth,” click square, then click the green flag (this uses the same code as for the circle, but makes quarter turns.

By moving with tiny turns (small angles) we can approximate a circle, keeping the radius fixed at the same size. To do this in our model, click the circle and “no radius growth” before clicking the flag. With small angles, say five degrees or less, we approximate a circle. In fact, we can consider a perfect circle to be a polygon with an infinite number of sides, each side infinitely small. But this code is a practical approximation that allows us to do other things to like make simple regular polygons (triangles, circles, squares, hexagons and so on, all with the same “radius” from the center to the corners of the polygon.

Try this code to draw polygon with pens and rhythm (click here). Click on the triangle then the flag and hear a drum beat every three angle changes (once every cycle of 360 degrees, three beats per measure).

From circles to spirals

So we can use the same code to create spirals, by having the radius grow. If we increase the radius by a fixed amount every time we turn at a small angle, you get what’s called “Archimedes spiral.”

To get a logarithmic growth spiral, click on “logarithmic growth” before clicking on the flag in this Scratch model. This is the kind of spiral we see in snails, ammonites, and many other life forms. At each turn, the radius grows geometrically (logarithmically). Whereas successive coils of Archimedes’ spiral are equally spaced, the distance between successive coils of the logarithmic spiral increases geometric progression (such as 1, 2, 4, 8,…).

Click here for the musical version of this code. Click any of the curve buttons and when it draws you hear the notes get higher when the pen is higher. Click the spiral growth buttons and hear the notes go wider and wider.