DNA Double Helix

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

• Plot a sine wave point by point with math.sin()
• Offset a second wave by 180° (math.pi) to make two crossing strands
• Connect matching points with "rungs" to build the DNA ladder

Seen from the side, DNA looks like a twisted ladder: two strands weaving over and under each other, joined by rungs (the base pairs). The secret is pure trigonometry — the two strands are the same sine wave, just shifted half a cycle apart. (This is a different beast from the gallery's Double Helix Spiral, which is an expanding turtle spiral — here we plot points with math.)

Welcome to the Double Helix!

In 1953, the shape of DNA was one of the biggest puzzles in science. You're about to draw it with two sine waves and a phase shift — the molecule of life in under 40 lines of Python!

How It Works

Both strands are plotted as point sequences going across the canvas: for point i, the height is amp * sin(theta + phase) where theta sweeps through 3 full waves. The blue strand uses phase = 0 and the red strand uses phase = math.pi — a 180° offset, so wherever one strand is high the other is low, making them appear to cross like a twisted ladder. The gray rungs are drawn first: every 6th point, a straight line connects the blue strand's height to the red strand's height at the same x.

Sample Code

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

amp = 70
waves = 3
points = 180
offset = math.pi

def strand_y(i, phase):
    theta = 2 * math.pi * waves * i / points + phase
    return amp * math.sin(theta)

def x_at(i):
    return -180 + 360.0 * i / points

monty.pencolor('gray')
monty.pensize(2)
for i in range(0, points + 1, 6):
    x = x_at(i)
    monty.penup()
    monty.goto(x, strand_y(i, 0))
    monty.pendown()
    monty.goto(x, strand_y(i, offset))

monty.pensize(4)
for strand in range(2):
    if strand == 0:
        phase = 0
        monty.pencolor('royalblue')
    else:
        phase = offset
        monty.pencolor('crimson')
    monty.penup()
    for i in range(points + 1):
        monty.goto(x_at(i), strand_y(i, phase))
        if i == 0:
            monty.pendown()

What Do You Think Will Happen?

The red strand is the blue strand shifted by 180°. When the blue strand is at its highest point, where will the red strand be — at the top with it, in the middle, or at its lowest point? Make your guess, then click Run!

Try It Now

▶ Run ↻ Reset

At its lowest — sin(theta + pi) is always the exact opposite of sin(theta). That mirror-image dance is what makes the ladder look twisted. Were you right?

How It Works

This is parametric plotting: instead of steering the turtle with forward() and left(), we compute each point's coordinates from a parameter i and jump there with goto(). The identity sin(theta + pi) = -sin(theta) means the two strands are perfect reflections, so they cross wherever sin(theta) = 0. Notice the rungs are longest halfway between crossings (where the strands are far apart) and shrink to nothing at each crossing — no extra code needed, the geometry does it automatically. Drawing the rungs before the strands keeps the gray lines neatly behind the blue and red.

Explanation Table

Line	What it does
theta = 2 * math.pi * waves * i / points + phase	Sweeps theta through 3 full waves as i counts up
return amp * math.sin(theta)	Turns the angle into a height between -70 and +70
offset = math.pi	180° phase shift — the whole secret of the helix look
for i in range(0, points + 1, 6):	Every 6th point gets a gray rung between the strands

Learning Check

Your Turn — Break the Phase

In the program below, offset has been changed from math.pi to 0, so both strands get the same phase. Predict: what will the rungs look like, and how many separate strands will you see? Run it, then set offset = math.pi back to repair the DNA!

▶ Run ↻ Reset

With offset = 0 every rung connects a point to itself (length zero!) and the red strand draws exactly on top of the blue one — one lonely wave, no helix. The 180° offset isn't decoration; it is the double helix.

Experiments

1. Twist it tighter. Change waves = 3 to waves = 5. You'll know it worked when the strands cross 10 times instead of 6.

2. Add more base pairs. Change the rung step from 6 to 3 in range(0, points + 1, 6). You'll know it worked when the ladder has twice as many gray rungs.

3. Squash the molecule. Change amp = 70 to amp = 35. You'll know it worked when the helix becomes a narrow ribbon through the middle of the canvas.

4. Try a quarter twist. Change offset = math.pi to offset = math.pi / 2. You'll know it worked when the strands still weave but the rungs never quite shrink to zero — a 90° offset never lets the strands meet.

You Cracked the Code!

Two sine waves, one phase shift, a few gray rungs — and the most famous molecule in biology appears on your screen. Up next: Topographic Map — paint an island's elevation with wobbly contour rings, from green shores to a snowy white peak.