Fermat's Spiral

By the end of this lab you'll be able to:

• Plot Fermat's spiral r = k * sqrt(theta) using math.sqrt
• Explain why square-root growth makes the rings get closer together as they go out
• Use the ± sign to draw two mirrored arms from one equation

Fermat's spiral grows with the square root of the angle, so each new ring is packed a little tighter than the last — unlike the evenly-spaced Archimedean spiral. And because the equation is really r = ±k*sqrt(θ), it has two arms: a plus arm and a minus arm, winding out of the origin in perfect 180° symmetry.

Welcome to Fermat's Spiral!

Pierre de Fermat studied this spiral in 1636 — and nature loved it first! Sunflower seed heads pack their seeds along Fermat spirals. One square root, two graceful arms. Let's draw it!

How It Works

For each angle theta from 0 to 8π (four full turns), compute r = k * math.sqrt(theta), then convert to x = r * cos(theta) and y = r * sin(theta). That draws the plus arm. Then repeat with r = -k * math.sqrt(theta): a negative radius places each point on the exact opposite side of the origin, producing a second arm that is the 180° mirror of the first.

Sample Code

import turtle
import math
monty = turtle.Turtle()
monty.speed(0)
monty.hideturtle()

k = 35
steps = 600
arms = [[1, 'crimson'], [-1, 'royalblue']]

monty.pensize(2)
for arm in arms:
    sign = arm[0]
    monty.pencolor(arm[1])
    monty.penup()
    for i in range(steps + 1):
        theta = 8 * math.pi * i / steps
        r = sign * k * math.sqrt(theta)
        x = r * math.cos(theta)
        y = r * math.sin(theta)
        if i == 0:
            monty.goto(x, y)
            monty.pendown()
        else:
            monty.goto(x, y)

What Do You Think Will Happen?

theta runs from 0 to 8 * math.pi, and one full turn is 2 * math.pi. How many complete times will each arm wind around the center? Make your guess, then click Run!

Try It Now

▶ Run ↻ Reset

4 turns — 8π / 2π = 4 full trips around the center for each arm. Were you right?

How It Works

Look closely at the ring spacing: near the center the gap between one turn and the next is wide, but each ring farther out is a bit closer to the one before it. That is the square root at work. After 1 turn the radius is proportional to sqrt(2π); after 4 turns it is only sqrt(8π) — twice as big, not four times. Growth keeps slowing, so the rings crowd together. The minus arm works because a negative r times cos and sin flips both x and y, rotating every point exactly 180° around the origin — one sign, a perfect mirror.

Explanation Table

Line	What it does
r = sign * k * math.sqrt(theta)	Fermat's rule: radius grows with the square root of the angle
arms = [[1, 'crimson'], [-1, 'royalblue']]	The ±: sign +1 and sign -1 give the two mirrored arms
theta = 8 * math.pi * i / steps	Sweep the angle over four full turns
x = r * math.cos(theta)	Polar to Cartesian; a negative r lands 180° across the origin

Learning Check

Your Turn — Add the Second Arm

The program below only draws the crimson plus arm — the arms list has a single entry. Add one list entry so the royalblue minus arm appears too. Where will the blue arm start relative to the red one?

▶ Run ↻ Reset

Change the list to arms = [[1, 'crimson'], [-1, 'royalblue']] — the blue arm starts at the same origin but winds out on the opposite side, weaving perfectly between the red rings.

Experiments

1. Wind farther out. Change 8 * math.pi to 18 * math.pi and k = 35 to k = 24. You'll know it worked when each arm makes 9 turns and the outer rings look almost evenly packed.

2. Break the square root. Change math.sqrt(theta) to just theta and set k = 7. You'll know it worked when the rings become evenly spaced — you've turned Fermat's spiral into an Archimedean spiral.

3. Grow faster instead. Try r = sign * 1.2 * theta ** 1.5 (and keep 8π). You'll know it worked when the gaps widen outward — the opposite of Fermat.

4. Sunflower dots. Delete the if i == 0: block, keep the pen up the whole time, and after computing x and y call monty.goto(x, y) then monty.dot(5). You'll know it worked when the arms become strings of seeds like a sunflower head.

Spiral Scientist!

Square-root growth, a ± sign, and polar coordinates — you drew a 400-year-old curve that sunflowers have been growing all along. Up next: Astroid — a four-pointed star hiding inside a family of sliding line segments.