Christmas decorations, snowflakes and reflection symmetry

An observant reader of Christmas decoration, snowflakes and rotational symmetry (which we recommend you read before this tutorial) might have noticed snowflakes often have more than just rotational symmetry. Snowflakes also have reflection symmetry, meaning that if we look at the mirror image of one of the “arms”, it’s the same as the original image.

Reflective symmetry

Landscape reflected in water (image by Eberhard Grossgasteiger).

Use reflection symmetry to reduce the number of code lines

When drawing an image with reflection symmetry, we can use a loop to cut down on how many lines of code we need (like what we did with rotation symmetry). Below is the code we used to create a snowflake in Christmas decoration, snowflakes and rotational symmetry. Looking closely, we can see that lines 11-14 and 21-24 are very similar, but the angles point in opposite directions.

import turtlethread

needle = turtlethread.Turtle()
with needle.running_stitch(30):

    for arm in range(6):
        # Move forward a little bit
        needle.forward(90)

        # Draw the first "branch" pointing a little downwards
        needle.right(30)
        needle.forward(60)
        needle.backward(60)
        needle.left(30)
    
        # Move forward and then back again
        needle.forward(90)
        needle.backward(90)
    
        # Draw the second "branch" pointing a little upwards
        needle.left(30)
        needle.forward(60)
        needle.backward(60)
        needle.right(30)

        # Move backwards to the starting point
        needle.backward(90)

        # Rotate 60 degrees to the right to draw six branches
        needle.right(60)

needle.visualise()

To take advantage of the reflection symmetry, we first need to notice that rotating 30 degrees to the left is the same as rotating -30 degrees to the right. This means we can replace left with right if we add a negative sign to the angle.

import turtlethread

needle = turtlethread.Turtle()
with needle.running_stitch(30):

    for arm in range(6):
        # Move forward a little bit
        needle.forward(90)

        # Draw the first "branch" pointing a little downwards
        needle.right(30)
        needle.forward(60)
        needle.backward(60)
        needle.right(-30)
    
        # Move forward and then back again
        needle.forward(90)
        needle.backward(90)
    
        # Draw the second "branch" pointing a little upwards
        needle.right(-30)
        needle.forward(60)
        needle.backward(60)
        needle.right(30)

        # Move backwards to the starting point
        needle.backward(90)

        # Rotate 60 degrees to the right to draw six branches
        needle.right(60)

needle.visualise()

Because we can change direction by changing the sign of the angle, we can use a for loop over -1 and 1 to loop over two different directions:

import turtlethread

needle = turtlethread.Turtle()
with needle.running_stitch(30):

    for arm in range(6):
        for direction in [-1, 1]:
            # Move forward a little bit
            needle.forward(90)

            # Draw the "branch", which will point either downards or upwards, depending on the direction
            needle.right(30*direction)
            needle.forward(60)
            needle.backward(60)
            needle.right(-30*direction)
        
            # Move a bit forward and then back to start
            needle.forward(90)
            needle.backward(180)

        # Rotate 60 degrees to the right to draw six branches
        needle.right(60)

needle.visualise()

We can see that this code contains a lot fewer lines!

Finished snowflake

Example snowflakes

Try it yourself:

Use code to embroider your own snowflake with both rotational and reflection symmetry! Below is a gallery of examples you can use as a starting point or for inspiration.

Snowflake 1

Snowflake

Kode for arm

from turtlethread import Turtle

needle = Turtle()
direction = 1

with needle.running_stitch(20):
    needle.forward(30)
    needle.right(120 * direction)
    needle.forward(15)
    needle.backward(15)
    needle.left(120 * direction)

    for branch_length in range(30, 21, -4):
        needle.forward(20)
        needle.right(60 * direction)
        needle.forward(branch_length)
        needle.backward(branch_length)
        needle.left(60 * direction)

    needle.forward(10)
    needle.backward(100)

needle.visualise()

Kode for snowflake

from turtlethread import Turtle

needle = Turtle()

with needle.running_stitch(20):
    for side in range(6):
        for direction in [1, -1]:
            needle.forward(30)
            needle.right(120 * direction)
            needle.forward(15)
            needle.backward(15)
            needle.left(120 * direction)

            for branch_length in range(30, 21, -4):
                needle.forward(20)
                needle.right(60 * direction)
                needle.forward(branch_length)
                needle.backward(branch_length)
                needle.left(60 * direction)

            needle.forward(10)
            needle.backward(100)

        needle.right(60)

needle.visualise()

Snowflake 2

Snowflake

Kode for arm

from turtlethread import Turtle

needle = Turtle()
direction = 1

with needle.running_stitch(20):
    needle.forward(120)
    needle.circle(direction * 50, 220)
    needle.home()

needle.visualise()

Kode for snowflake

from turtlethread import Turtle

needle = Turtle()

with needle.running_stitch(20):
    for arm in range(4):
        for direction in [1, -1]:
            needle.right(arm*90)
            needle.forward(120)
            needle.circle(direction * 50, 220)
            needle.home()

needle.visualise()

Snowflake 3

Snowflake

Kode for arm

from turtlethread import Turtle

needle = Turtle()
direction = 1

with needle.running_stitch(20):
    for radius in range(24, 11, -4):
        needle.forward(30)
        needle.circle(radius*direction)
    needle.backward(120)

needle.visualise()

Kode for snowflake

from turtlethread import Turtle

needle = Turtle()

with needle.running_stitch(20):
    for side in range(6):
        for direction in [1, -1]:
            for radius in range(24, 11, -4):
                needle.forward(30)
                needle.circle(radius*direction)
            needle.backward(120)
        needle.right(60)

needle.visualise()