Torus Blossoms in a Sliceform Vase
The Art of Mathematical Flowers Using a Paper Torus
To enhance the floral appearance, I modified the edges of the slices to resemble petals while preserving the torus’s underlying circular symmetry. Layered petal designs were added to the center of each flower, giving them depth and visual richness. The completed arrangement stands approximately nine inches tall, with the flowers mounted on stems and displayed in a sculptural sliceform vase—also constructed from interlocking flat pieces.
Understanding the Mathematics: Villarceau Circles
At the heart of this project lies the torus—that familiar donut shape with fascinating mathematical properties. The key to creating these floral forms is understanding Villarceau circles: pairs of circles that emerge on a torus’s surface when it is intersected by a plane at a specific oblique angle. Named after French mathematician Yvon Villarceau, who described them in 1848, these circles provide the structural foundation for this model.
When a torus is cut by a plane that passes through its center and touches it at two opposite points, (as shown above from a Wikipedia generated gif), something remarkable appears: a pair of circles known as Villarceau circles. These circles intersect at exactly those two touching points, creating a special geometric relationship.
24-slice paper torus and a 16-slice paper torus
Here's the elegant part: when a torus is sliced along these special circles, the resulting pieces(slices) can slide together to reconstruct the complete three-dimensional form. This remarkable geometric property makes the sliceform technique possible, transforming flat paper into dimensional sculpture. By cutting the torus into circular cross-sections at Villarceau's specific angles, each pair of circles shares the same cross-section, producing perfectly matched components that interlock with mathematical precision. This relationship is what allows paper flowers to bloom directly from pure mathematics.
From Geometry to Paper Sculpture
As my work evolved, I focused on refining my understanding of Villarceau circles, the mathematical foundation that makes torus sliceforms possible. This led me to the paper “Building a Torus with Villarceau Sections” by María García Monera and Juan Monteabout (University of Valencia), published in the Journal for Geometry and Graphics (Volume 15, 2011). Their work clearly explains the mathematics behind constructing a torus using Villarceau sections and provides the formulas needed for precise construction.
Using these formulas, I was able to build an accurate sixteen-slice torus. Prior to this, my sixteen-slice models appeared visually correct, but I sensed that the angles in the Villarceau circles were slightly off. After working through the calculations in the paper, my intuition was confirmed: two of the angles differed by one-tenth of a degree. That small discrepancy had a noticeable impact on the structure, demonstrating how even a minute variation in a complex system can significantly affect the final form.
With this deeper mathematical understanding in place, I turned to Silhouette Studio, the software provided with the electronic paper cutting machine, to refine the geometric forms into something more organic. I point-edited the slice edges of the torus shapes while preserving the circular symmetry of the original torus. Driven by curiosity, I also experimented with adjusting the angles at which the pieces intersect to see what new structures might emerge. To my delight, an unexpected circular form appeared—one that echoed the torus itself.
To support this evolving work, I developed a TurtleStitch program capable of generating Villarceau circles of any size, with adjustable slits designed to slide together precisely. In sliceform construction, slit size is critical: while a circle can be scaled, the slit width must remain constant to account for paper thickness and ensure that each slice maintains the correct orientation. This tool allows me to quickly create toruses of any size while preserving the mathematical precision required for successful assembly.
Once the TurtleStitch program generates the Villarceau circle slices, I refine each slice through point editing in Silhouette Studio, adjusting the geometry while preserving the underlying mathematical structure. Each slice is then precision-cut from cardstock and assembled by sliding the pieces together, mirroring the way a torus is conceptually constructed from its individual slices. The result is a collection of three-dimensional flowers with layered centers and unique edges that reflect patterns found in nature. They’re mounted on stems and displayed in a geometric vase that is itself made from interlocking paper pieces.
Sharing the Beauty of Mathematics
I’ve written a detailed explanation here of how to make this project, and—as always—all instructions and cutting files are available for free. My hope is that by building this model, you’ll not only enjoy the process of making something beautiful, but also gain a deeper appreciation for the mathematical structures behind it.
Mathematics doesn’t just describe the world—it can help us create it.
Examples of the sliceform flowers
Creating Your Own Torus Blossoms
Design Process
110 lb. cardstock was used for the vase and green stems and 65lb. cardstock was used for the flowers.
Cut Files
You can cut out the flowers and vase with scissors by using the PDF file [link to file]
For electronic cutting machine to create this project:
- Silhouette users: Download the .Studio file [link to file]
- Cricut users: Download the SVG file [link to file]
Note: The SVG file extends beyond the initial viewable area. Simply zoom out to see the complete design
What You'll Find in the Files
The included .Studio and SVG files contain:
- Sliceform Vase: Circles and sides of the vase. A total of 10 circles are needed so cut that page twice and cut the vase sides page six times, for a total of 24 slices.
- Three-slit flowers: I recommend starting with these, as they're the simplest to assemble
- Four-slit flowers: Slightly more complex but offering different aesthetic possibilities
Important note: The flower sizes cannot be altered because the slices must maintain specific dimensions for proper assembly. The slits need to be exact so pieces can slide into one another correctly, and paper thickness must remain consistent to hold each slice at the proper orientation.
Make the Vase
Glue the corresponding vase circles together to make them two-ply.
Glue the vase side pieces together to make them four-ply. Do not apply glue to the bud area until later.
Bend the tabs of the buds at a right angle. Apply glue as needed to the bud area.
Repeat to make six side slices.
Slide the vase side slices onto the circle slices. There are five circles. The order of the circles, starting from the top, is small, large, large, small and small.
Completed vase
Make the Flowers
There are two types of torus flowers in the file, a three-slit version and a four-slit version. Begin with a three-slit flower for your first attempt.
Each flower has two corresponding pieces, an upward slit and a downward slit. The pieces are alternated as they are slid into one another and stacked together. There are eight of each type in this three slit design. (There is one flower that has twelve of each type.)
I recommend starting with a center slit as shown above.
Continue adding the slices to create the above pattern.
Once completed, there will be a stack of slices.
There is a hole in each slice. Thread a needle and thread it through these holes.
Once the thread is sewn through all of the slices, flex the flower into a circle and slide the remaining two slices into one another. Pull the thread tight and knot it. The flower is now ready to be added to your vase by sliding it over the bud area.
Please note, the four slit version is the same process as the three slit version. Also, the modified torus version is the same process but with more slices that are slid together.
Make the Flower Centers
Cut out the flower centers and curl the petals upward.
Assemble the flower by gluing one flower piece on top of another as shown above.
Assemble the Flowers
Place the flower on the stem.
Apply glue to the circular bud area and adhere the center of the flower.
Move the slice form flower to the center of the flower. I added a glue dot to make sure that the flower wouldn't move.
Completed flower with center.
Continue adding flowers until the arrangement is complete.
Completed Flower Arrangement
Beauty in Mathematics
This project reveals how abstract mathematical ideas can reshape our perception of geometry, transforming the theoretical into something tangible and beautiful. The flowers mirror patterns found in nature while remaining true to their mathematical origins—a perfect marriage of art and science.
I hope you'll experience the deep satisfaction that comes from creating these mathematical models. There's something profound about holding in your hands a three-dimensional manifestation of centuries-old mathematical concepts, shaped into something that delights the eye and mind alike.
