Kaleidoscope Slice

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

• Build a whole pattern by repeating one "slice" at different rotation angles
• Mirror alternate copies by flipping the angle inside each slice
• Convert a radius and an angle into (x, y) with cos and sin

A real kaleidoscope uses mirrors to bounce one small wedge of colored glass into a full circle of symmetry. This program does the same thing with math: it draws one 30° slice made of four filled shapes, then repeats it 12 times around the center — mirroring every other copy so the edges line up like a reflection.

Welcome to the Kaleidoscope!

Here's a secret: we only design ONE tiny slice of this pattern. Rotation and reflection do all the rest of the work. That's the magic trick behind every kaleidoscope!

How It Works

Each shape in the slice is stored as a list of (radius, angle) pairs measured from the center. To draw the slice at rotation base, we add base to every angle, then convert to screen coordinates with x = r * cos(a) and y = r * sin(a). For mirrored copies we use base + wedge - off instead of base + off, which flips the slice like a reflection in a mirror.

Sample Code

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

sections = 12
wedge = 360 / sections

shapes = [
    ['gold', [(30, 3), (95, 13), (65, 26)]],
    ['crimson', [(45, 6), (110, 2), (165, 7), (110, 18)]],
    ['royalblue', [(85, 24), (150, 15), (178, 27)]],
    ['darkorchid', [(120, 10), (185, 12), (150, 21)]],
]

def draw_slice(base, mirror):
    for color, points in shapes:
        corners = []
        for r, off in points:
            if mirror == 1:
                ang = base + off
            else:
                ang = base + wedge - off
            a = math.radians(ang)
            corners.append((r * math.cos(a), r * math.sin(a)))
        monty.penup()
        monty.color(color, color)
        monty.goto(corners[0])
        monty.pendown()
        monty.begin_fill()
        for corner in corners:
            monty.goto(corner)
        monty.goto(corners[0])
        monty.end_fill()

for i in range(sections):
    if i % 2 == 0:
        draw_slice(i * wedge, 1)
    else:
        draw_slice(i * wedge, -1)

What Do You Think Will Happen?

The loop runs 12 times, and each slice contains exactly one gold triangle. How many gold triangles will you count in the finished pattern? Make your guess, then click Run!

Try It Now

▶ Run ↻ Reset

12 gold triangles — one per slice, because the loop calls draw_slice twelve times. Were you right?

How It Works

Look closely at where two slices meet: the pattern on one side is a mirror image of the other. That happens because odd-numbered slices use base + wedge - off, which measures angles backwards from the far edge of the wedge. Without that flip you would get plain rotation (like a pinwheel); with it you get true kaleidoscope symmetry, where every boundary between slices acts like a mirror line.

Explanation Table

Line	What it does
wedge = 360 / sections	Splits the circle into 12 equal 30° wedges
ang = base + off	Rotates a point into slice number i
ang = base + wedge - off	Same point, but flipped — a mirror reflection
corners.append((r * math.cos(a), r * math.sin(a)))	Converts radius + angle into an (x, y) position

Learning Check

Find the Bug

This program is supposed to draw all 12 slices, but almost the whole kaleidoscope is missing! Look carefully at the two draw_slice calls inside the loop. Can you spot what never changes — and fix it?

▶ Run ↻ Reset

The bug: both calls pass wedge as the base angle, so every slice lands in the same spot. Change them to draw_slice(i * wedge, 1) and draw_slice(i * wedge, -1) and the full circle of 12 slices returns.

Experiments

1. Try 8 sections. Change sections = 12 to sections = 8. The wedge grows to 45° automatically. You'll know it worked when the pattern has 8 arms with small gaps between them.

2. Recolor the glass. Change 'crimson' to 'limegreen' and 'royalblue' to 'deeppink'. You'll know it worked when the big darts change color in every one of the 12 slices at once.

3. Redesign one shape. In the 'darkorchid' shape, change the radius 185 to 120. You'll know it worked when the purple points pull in closer to the center all around the wheel.

4. Break the mirror. Change draw_slice(i * wedge, -1) to draw_slice(i * wedge, 1) so every copy uses the same direction. You'll know it worked when the pattern stops reflecting and starts spinning — it becomes a pinwheel instead of a kaleidoscope.

One Slice, Twelve Reflections!

You designed one 30° wedge and let rotation + reflection build the rest. That's exactly how symmetry works in snowflakes, mandalas, and real kaleidoscopes. Up next: Pinwheel Tiling — a triangle that tiles the plane by endlessly splitting into five copies of itself.