A STEM Project: A Square Cube Vase: Where Math Becomes Art

A Square Cube Vase: Where Math Becomes Art

Have you ever wondered what happens when geometry, code, and creativity come together in a single project? This is the story of a piece that shows how math can become art when you blend digital tools with the love of making things by hand.

Designing the Cube Vase 

Using Silhouette Studio and TurtleStitch, I created a cube-shaped vase lined with vellum and decorated it with delicate, lacy floral cutouts.The petal design on the sides of the vase were created in TurtleStitch and exported as a DXF file. The design was offset to create the lacy effect. I arranged twelve layered paper flowers on stems, each designed in code and cut with precision with the Silhouette Cameo. The result is a vase that proves papercrafting and coding belong together. 

Cut Files 

You need an electronic cutting machine for cutting the files.  The .Studio file is for the Silhouette machine and the SVG file for the Cricut machine.

Here is the .Studio file. 

https://drive.google.com/file/d/17vfuYHk_q6J65TuIA0UM4ZSHgSPvVLxL/view?usp=sharing

Here is the SVG. The file extends beyond the viewable area.  Zoom out to see the entire file.

https://drive.google.com/file/d/1Klufu3oJgE0vyh-GbrUg0tUoOdpocfSj/view?usp=sharing

Creating Three-Dimensional Flowers 

I coded two Turtlestitch programs to make the flowers. The Petal Block Flower and Petal Arc Block Flower. 

The  Petal Block Flower program uses a custom 'petal' block that allows you to generate complete flowers with just two inputs:

• radius - controls the arc size

• degrees - sets the arc angle

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.  If you replaced 180° with a different value (for example, 120°), you would get a lopsided, asymmetrical petal. 

The Petal Arc Block Flower program takes into account this anomaly. The formula to compensate for the difference is:

final turn = 360° ÷ number of petals − ( 2 × arc angle) + inner turn

This formula accounts for three things:

• The total angle the turtle turns while drawing both arcs.

• The sharp turn between the two arcs.

• The final adjustment needed to evenly space the petals around the circle.

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.

Assembling The Vase

To make the vase, glue the vellum to the inside of the box sides. Glue the side tabs of the vase together to form a square. Bend the tabs of the bottom of the vase at a right angle.  Apply glue to the tabs and slide the bottom into the vase body.

To make the stems,  fold and glue one half of each side together at a 90-degree angle to form an X.  The two ply stems offer stability for the flowers.

The right angled stems are inserted into the base and the tabs were glued as shown.  The base of the stems is glued to the inside of the cube.  The inserted tabs provide a lip around the cube.

The tops of the stems are flared so that the flowers can be glued on to a secure base.
Attach each flower to a paper stem.

Assembling Three-Dimensional Flowers 

I made twelve three-dimensional flowers to accompany the vase. Each flower was cut in graduated sizes so the largest layers formed the base and the smallest topped the bloom. This technique creates depth and realistic form. Each petal was curved by rubbing it against my fingernail.

To assemble: Stack the flower layers from largest to smallest, securing them in the center. 
Combine the two TurtleStitch flower designs to create visual variety.

Arrange the flowers around the vase in a natural, pleasing composition. Have fun mixing shapes, heights, and colors so the bouquet looks interesting from every angle.

The Math Behind the Flowers

This project was powered by math at every stage, using both TurtleStitch and Silhouette software: 

• Rotation and Repetition: Petals repeat evenly around a central point, each rotated by a precise angle (e.g., 60° for six petals). 
• Arc Geometry: Arcs are defined by radius and angle to create smooth curves. 
• Symmetry: Dividing the circle evenly produces perfect balance. 

The result is more than just a container for flowers. It’s a unique vase, part sculpture, part floral arrangement, and completely inspired by the possibilities of math and digital papercrafting. This project shows that math can become art in your hands and can be displayed as a testament to the beauty of mathematics.

A STEM Project: Torus Flowers in a Dodecahedron Vase

Torus Flowers in a Dodecahedron Vase

This project brings together geometry, code, and craft in a celebration of mathematical beauty. I’ve designed intricate lacy flowers based on torus sliceforms, placed them inside a dodecahedron paper vase with windows on each face, and used a combination of TurtleStitch and Silhouette Studio to bring the design to life.

Cut Files 

The .Studio file is for the Silhouette machine and the SVG file for the Cricut machine.

Here is the PDF to see how this project was created.

https://drive.google.com/file/d/1GJ8Gqv_6ek2306GzOMhfxGkt8mUM-PnG/view?usp=sharing

Here is the .Studio file.

https://drive.google.com/file/d/1tkYjbTPnQNhU01hv-3UhY9jgNUkv4AdQ/view?usp=sharing 

Here is the SVG. The file extends beyond the viewable area.  Zoom out to see the entire file.

https://drive.google.com/file/d/1JaqbNKE29jCAbbSXIswA59iD5SbmkC-T/view?usp=sharing

 Designing the Flowers in TurtleStitch

I began with a flower created using a custom Petal Block in TurtleStitch. https://www.turtlestitch.org/users/Elaine/projects/Petal%20Flower The design combines arcs and rotations to form a symmetrical flower pattern. Once complete, I exported the design as a DXF file.

 Modifying in Silhouette Studio - Please check out the  PDF file included in this posting to see the photos of this process and the flower that was produced.

In Silhouette Studio, I refined the flower shape using the Offset menu:

First, I applied an internal offset of 0.075 inches to the entire flower. This created a delicate inner contour.

Then, I added an external offset of 0.075 inches to the outer line only, preserving the boundary shape while adding definition.

To create variation and depth, I enlarged the inner petals by 125% and removed the central circular piece, giving the flower a layered, airy appearance.

The result is a lacy, stylized flower perfect for papercrafting—and just the right size to fit in the openings of a dodecahedron vase. A dodecahedron is a Platonic solid made of 12 regular pentagons. In Silhouette Studio, I created a net, an unfolded layout that could be cut, folded, and assembled into the vase. On ten sides of the dodecahedron, I added windows of lacy flowers using Silhouette software, which were then backed with vellum. 

After cutting, I folded along each edge and carefully glued the structure into place. 

 The Torus as Floral Inspiration

The true inspiration for this flower design comes from the torus, a fascinating shape formed from a stack of Villarceau circles—thin, half-moon slices that curve and weave together to form a donut-like shape.

There are two types of Villarceau circles used in the construction, each with opposite slits that allow them to slide into one another. This assembly process creates not just a torus, but a dynamic, flexing form that can be transformed into petal-like structures when the slices are altered.

I’ve explored this concept in previous work by modifying the edges and angles of the Villarceau circles. From this exploration, four unique torus-inspired flowers emerged. Each flower preserves the essential tension and curvature of the torus but expresses it in a new, floral form.

The fifth flower in this collection is a reimagined torus, approximately half the size of the original. It features a modified Villarceau circle, and must be strung together at the bottom point to maintain alignment. Please note—this smaller torus requires significant patience to assemble. The last few slices must be carefully bent and stretched into place without tearing or creasing.

Want to try it yourself? Cut 16 slices to form one flower.  There will be 8 slices of each type since the slices slide into one another. Here's a basic tutorial on weaving Villarceau circles, originally used for a honeycomb pumpkin. The principle is the same—just with a toroidal twist!

The red flower torus (it looks like a half of a heart) is made with the technique in this blog posting. A thread is used to hold the torus together. https://papercraftetc.blogspot.com/2021/07/a-stem-project-amazing-slice-form.html

Assembly in the Dodecahedron Vase

The vase is made by gluing the vellum to the the sides of the dodecahedron. The dodecahedron is glued together to form the vase. 

The stems are two ply. The buds are splayed outward and are not glued together.

The stems are folded in half and are glued to the base.

Bottom view of base.  Two pentagons are glued to this bottom for support and then glued to the top of the dodecahedron vase.

The twelve flowers are glued to the tops of each stem. The vellum windows on each face allow the light to shine through and highlight the layered offset curves of the TurtleStitch created flowers. Each window becomes a frame for the toroidal geometry inside.

The combination of curved forms and sharp dodecahedral edges creates a striking contrast—one that’s both organic and mathematical.