Colored Logarithmic Spiral

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

• Use math.cos() and math.sin() to convert polar coordinates to Cartesian (x, y) positions
• Draw a curve by plotting many points with goto() instead of using forward() and right()
• Color each segment by its angular position using angle % 360 to cycle through hues

A smooth, continuously curving spiral — no straight segments or corners. Each revolution of the spiral is colored by cycling through red, orange, yellow, green, blue, and violet, so the arms glow like a rainbow ribbon.

Welcome to the Logarithmic Spiral!

Until now, every spiral was made of straight line segments. This one is different — we use math functions to compute exact positions, giving a truly smooth curve. Let's code it together!

How It Works

A logarithmic spiral in polar coordinates is r = a · e^(b·θ) — the radius grows exponentially as the angle increases. We convert each polar point (r, θ) to Cartesian (x, y) using:

x = r · cos(θ)
y = r · sin(θ)

Then goto(x, y) moves the turtle to that point. With small angle steps and the pen down, this traces a smooth curve.

Sample Code

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

colors = ['red', 'orange', 'yellow', 'green', 'blue', 'violet']
a = 0.5
b = 0.15

monty.penup()
for i in range(600):
    theta = i * 0.05
    r = a * math.exp(b * theta)
    x = r * math.cos(theta) * 30
    y = r * math.sin(theta) * 30
    monty.goto(x, y)
    hue_index = int(math.degrees(theta) % 360 / 60)
    monty.pencolor(colors[hue_index])
    monty.pendown()

What Do You Think Will Happen?

We multiply r by 30 to scale the spiral to fit the canvas. Will the spiral look smooth or jagged? (Each step is only 0.05 radians.) Make your guess — then click Run to find out!

Try It Now

▶ Run ↻ Reset

Smooth — each 0.05-radian step is small enough that the eye can't detect the individual line segments. Were you right?

How It Works

math.exp(b * theta) is the exponential function — the radius grows slowly at first and accelerates outward. Multiplying by 30 scales the result to fill the canvas. math.degrees(theta) % 360 converts radians to degrees and uses modulo to cycle through 0–360 regardless of how many revolutions we've made.

Explanation Table

Line	What it does
theta = i * 0.05	Angle in radians; 0.05 rad per step for smoothness
r = a * math.exp(b * theta)	Radius grows exponentially — logarithmic spiral formula
x = r * math.cos(theta) * 30	Convert polar to Cartesian x, scaled to canvas
y = r * math.sin(theta) * 30	Convert polar to Cartesian y, scaled to canvas
math.degrees(theta) % 360 / 60	Map angle to color index: one color per 60°

Polar and Cartesian Coordinates

Polar coordinates describe a point by (r, θ) — distance from origin and angle. Cartesian coordinates use (x, y) — horizontal and vertical distances. Converting between them is one of the most useful tricks in computational graphics!

Learning Check

Spot the Bug!

The program below draws a straight line instead of a spiral — the radius never changes! Find and fix the bug.

▶ Run ↻ Reset

Change r = a to r = a * math.exp(b * theta) — without the exponential growth, the radius stays constant and the turtle traces a fixed-radius circle instead of a spiral.

Experiments

1. Make it grow faster. Change b = 0.15 to b = 0.25. The spiral widens much faster. You'll know it worked when fewer revolutions fit on the canvas.

2. Make it grow slower. Change b = 0.15 to b = 0.07. More revolutions pack tightly together. You'll know it worked when the arms are much closer together.

3. Remove the color. Replace the pencolor() line with monty.pencolor('black'). You'll know it worked when the spiral is a single clean black curve.

4. Double the resolution. Change i * 0.05 to i * 0.025 and range(600) to range(1200). The curve gets even smoother. You'll know it worked when you can't see any corner at all.

Mathematical!

You used math.cos() and math.sin() — the foundation of all computer graphics — to convert polar coordinates into pixel positions. This is how video games, maps, and simulations place objects in space! Up next: Double Helix Spiral — two interleaved spirals in a single loop.