Coding Flowers in TurtleStitch:
From Curves to a Paper-Cut Design
Snap!Con 2025
Elaine Wolfe
A Gateway to Mathematical Art
TurtleStitch isn't just for embroidery—it's a gateway to mathematical art. By combining precise geometric programming with hands-on making, I’ve developed a process for turning algorithmic flower designs into physical paper creations using the Silhouette Cameo.
Today I'll show you:
TurtleStitch flowers are built on geometry and rotational symmetry. Whether petals are straight, curved, or custom-shaped, each design relies on fitting elements evenly around a circle. Each method reveals how math can lead to surprising and beautiful results.
Designing Petals With The Arc Block
Drawing curves with turtle graphics goes back to the early days of Logo. Cynthia Solomon, Seymour Papert, and others knew that straight lines couldn't capture the beauty of curves in nature, so they added arcs to the turtle's vocabulary.
TurtleStitch continues that legacy with its “arc” block. Students can create flowing, curved petals with just a few simple commands, as you can see on the right, thereby discovering how code and geometry are connected.
The “Petal” Block
Creating a custom 'petal' block allows you to generate complete flowers with just two inputs:
Each petal uses two symmetrical arcs connected by a (180° - degrees) turn which is the key to maintaining perfect symmetry.
This works because 180° represents a straight line, causing the turtle to flip across the centerline and mirror the first arc. The pattern then repeats around the circle with petals spaced using 360° ÷ number of petals.
Another Version of the “Petal” Block
If you replace the 180° turn with a different value, you will get a lopsided, asymmetrical petal. I developed a formula to compensate for the differences:
final turn = 360° ÷ number of petals − ( 2 × arc angle) + inner turn
This formula accounts for three things:
When the turtle's rotation doesn't complete a full circle, adding 360° ensures it's properly oriented for the next petal. This technique maintains perfect symmetry regardless of your petal design. This lets me design with intention and flexibility—pointier petals, rounded petals, and more.
Designing a Variety of Flowers with the Enhanced “Petal” Block
Now let's try a different approach entirely, straight-line petals that rely on interior angle calculations.The turtle draws one petal and then uses this formula:
interior angle = ((n−2) × 180) ÷ n
where n is the number of petals, to align the turtle for the next petal.
In the diagram to the left:
A Straight Line Petal
Building on the straight line equation code, we can create more expressive petals by replacing each straight line with a series of smaller moves and turns. This lets us customize each petal's shape—making them curved, wavy, or zigzagged—while keeping the petals evenly spaced around the flower.
The key is making sure all the turns within each petal add up to exactly 180°. This keeps the flower symmetrical and ensures the turtle automatically faces the right direction for the next petal, without any manual adjustments.
The 'Move it spaces' block handles drawing these detailed petal segments using small moves and turns.
From A Straight Line Petal To Custom Designs
Curving the Equation: Petals With Arcs
To create a curved petal, we insert an arc into the middle of the straight-line blocks. In this example, the arc uses 120°.
Since a complete petal needs a total turn of 180°, we subtract the arc angle: 180° - 120° = 60°
This remaining 60° must be split equally between the two sides of the arc: 60° ÷ 2 = 30°
So the turtle turns 30° before the arc and 30° after the arc. The two "Move it spaces" blocks before and after the arc show this pattern—each contributes 30° of turn, while the arc provides the curved middle section.
This creates a smooth, natural-looking petal because the turtle gradually enters and exits the curved section, ensuring it draws a symmetric petal and ends facing the correct direction for the next petal.
Designing Flowers with Rose Curves
Rose curves take a different approach entirely. One mathematical formula draws the complete flower in a single, flowing motion.
Rose Curve Formula
r = a sin(nθ)
Rose curves
from n = 1 to 7
For the above example, the slider value for amplitude is set to 100 and the n slider value is 6 which creates 2n petals (12 petals).When n is an even integer, 2n petals are produced. An odd integer produces n petals.
Exporting Rose Curves From TurtleStitch To the Silhouette Studio
Once the design is created, I export the rose curve from TurtleStitch as a DXF file (blue line) and open it in Silhouette Studio. I offset the design outward by 0.02 inches (red line), then apply an internal offset of 0.02 inches to cancel out the original offset. This technique creates a flower shape that preserves the center, preventing it from being cut out because the blue lines would otherwise be cut lines.
I use this offset technique with all of my flowers.
From Code to Craft: Creating Paper Vases
To make a lacy flower, I export the design from TurtleStitch as a DXF file. In Silhouette Studio’s Offset menu, I applied an internal offset of 0.075 inches to the entire design. Then, I added an external offset of 0.075 inches to just the outer line of the flower. The interior petals were enlarged by 125%, and the center circular piece was removed.
The blue-lined flower shows the original DXF file from TurtleStitch. The red-lined flower shows the modified version created in Silhouette Studio. The result is a lacy flower that was added to a dodecahedron vase.
An Arithmetic Spiral - A Rolled Paper Flower
Torus - A Different Type of Flower
TurtleStitch combined with the Silhouette Cameo provides both precision and creative freedom. With just a few blocks, I can quickly generate complex, symmetrical designs that would be tedious and time-consuming to edit point-by-point with the Silhouette software. The ability to export these designs directly as cut files allows me to move seamlessly from code to paper. TurtleStitch becomes more than a design tool, it's a time saver! It frees up time for me to focus on creativity, experimentation, and making.
The Versatility of Coding
I hope you enjoyed exploring the mathematical beauty of coding flowers in TurtleStitch.
My TurtleStitch Projects: www.turtlestitch.org/users/Elaine/projects
Please visit my blogs:
🌐 papercraftetc.blogspot.com�🌐 turtlestitch.blogspot.com
There, you will find more fun projects featuring math-based designs.