A STEM Project: A Magical Dining Scene: Bringing Belle and Ariel Together with MicroBlocks and the MakerPort

From Walmart Toy Sets to an Interactive Masterpiece

Watch the video to see the scene come alive!

There are five touch points that play different songs.

After attending a delightful production of Beauty and the Beast at my granddaughter's school, I found myself inspired during a routine shopping trip to Walmart. There in the toy section sat a Beauty and the Beast playset that sparked an idea—what if I could bring this scene to life with lights, music, and movement?

I couldn't resist adding a Little Mermaid playset to my cart as well. Little did I know that these two playsets would become the foundation for one of my most ambitious projects yet, combining MicroBlocks programming, MakerPort technology, and traditional crafting techniques.

Building the Foundation: Elegant Seating

Upholstered chairs adorn the scene

Every magical dining experience needs proper seating. I crafted custom chairs for both Belle and Ariel following these instructions. To give them a more elegant appearance befitting our princesses, I upholstered each chair, transforming simple paper furniture into sophisticated dining seats.

The Table: Where Coding Meets Craftsmanship

For the centerpiece table, I turned to TurtleStitch to code a presentation box design. One of the things I love most about working with code is the flexibility it provides—thanks to variables in the programming, I can adjust the size of this table design to fit any future project needs.  Check out the TurtleStitch code.

The presentation box serves a dual purpose: it provides a sturdy, elegant surface for dining while also concealing the technical components that bring the scene to life.

Illuminating the Magic

The lighting in this project works on multiple levels:

The Centerpiece Neopixel: Embedded in the center of the table, a Neopixel light creates a warm, colorful glow that draws the eye to the table's center.

The Rotating Tea Service: A servo motor continuously rotates a platter featuring Mrs. Potts and Chip, adding movement and whimsy to the scene.

Luminaire's Pixie Lights: To complete the ambiance, I added pixie lights to Luminaire (the candelabra). These affordable lights from the Dollar Store were modified—I cut them from their original battery casing and connected them to the MakerPort using alligator clips. This technique is similar to what I used in my clown project, and it works beautifully.

The Soundtrack: A Musical Compromise

My original plan included five touch sensors programmed to play music from Beauty and the Beast. However, my granddaughter had other ideas. "That's not fair!" she declared. "Ariel wants her music too!"

She was absolutely right. The final version includes beloved songs from both Beauty and the Beast and The Little Mermaid, activated by touch sensors positioned around the scene. Now both princesses can enjoy their signature melodies.

Finishing Touches: Form Meets Function

The tablecloth design balances aesthetics with practicality. The top portion is glued directly to the presentation box, creating a seamless, elegant appearance. However, the skirt features an elastic top that can be easily removed whenever I need to access the MakerPort and electronics housed inside the box.

All of the electronics and interactive elements were programmed using MicroBlocks, which gave me precise control over the lights, sounds, and movements.

The Grand Reveal

A table fit for princesses!

When I finally presented the completed project to my granddaughters, their reaction made every hour of work worthwhile. Their faces lit up brighter than any Neopixel as they discovered they could make the lights change colors, the tea service spin, and their favorite Disney songs play with just a touch.

Watching them play with Belle and Ariel in this interactive, illuminated dining scene reminded me why I love combining traditional crafts with modern technology—it creates experiences that engage children's imaginations in ways that neither approach could achieve alone.

What's Next?

This project has opened up so many possibilities. The modular nature of the TurtleStitch-coded presentation box means I can create different scenes and settings for future adventures. And thanks to the removable tablecloth skirt, I can easily reprogram the MicroBlocks to add new features or change the music selections.

Who knows? Perhaps a third princess will join Belle and Ariel for dinner soon. After all, there's always room at the table for more magic.

Update: I may have already made another trip to Walmart... and a Frozen playset with Elsa may have found its way into my cart. Something tells me my granddaughters are going to insist that Elsa needs a seat at the table—and her own music, of course! The dining party is about to get even more crowded, and I couldn't be happier about it.

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Have you combined traditional crafts with coding and electronics? I'd love to hear about your projects in the comments below!

Exploring N-Gon Prisms: From TurtleStitch Code To Paper Net Templates

N-Gon Prisms

Starting at the bottom row and circling counterclockwise are eight prisms, 

ranging from ten-sided to three-sided.

In this blog post, I continue my quest to create prisms using paper net templates. Please see my previous post to learn how my journey began: Exploring the Five-Color Torus: A Mathematical Approach.

Inspired by the elegant modular structure of that model, I became curious about designing paper versions of n-gon prisms. What began as a simple experiment quickly evolved into a fascinating mathematical puzzle.

Through trial and error (I experimented with over thirty different nets), I discovered something remarkable: odd-sided prisms behave fundamentally differently from even-sided prisms.

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The Odd-Sided Pattern

Three, five, seven, nine-sided prisms

Note: the sides are equilateral triangles

For odd-sided prisms (triangles, pentagons, heptagons, etc.) with an edge length of a:

• Each face consists of two equilateral triangles connected by a rectangle.
• The total number of equilateral triangles needed is 2n (where n is the number of sides).
• All triangles have sides equal to the chosen edge length a.
• The height of the prism equals the height of the equilateral triangle.

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The Even-Sided Pattern

Four, six, eight, ten-sided prisms

Note: the sides are two isosceles right triangles that form a square

For even-sided prisms (squares, hexagons, octagons, etc.) with an edge length of a:

• Each face is made of two isosceles right triangles connected by a rectangle.
• These triangular pieces form squares when paired together on each side.
• The total number of isosceles right triangles needed is 2n.
• The height of the prism equals the side length a.

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Shared Properties of Even and Odd Prisms

• Each paper unit consists of a rectangle flanked by two triangles—one at the top and one at the bottom.
• The rectangle is folded along its diagonal to form the three-dimensional structure of the prism.
• The prism exists in two chiral forms, depending on whether the rectangle’s diagonal slants clockwise or counterclockwise.
• The rectangle’s diagonal represents the longest internal diagonal of the prism.
• The two sides of the rectangle form another diagonal, connecting adjacent sides of the prism (traversing from the top of one face to the bottom of the next, depending on the prism’s chirality).

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Formulas Used

For a regular n-gon prism with side length a and height h:

Height Calculation
Odd n-gon: h = a × √3 ⁄ 2 (height of an equilateral triangle)
Even n-gon: h = a (two isosceles right triangles form a square face)

Circumradius of the Base Polygon
R = a ⁄ (2 × sin(180 ⁄ n))

Maximum Base Chord
D_max = 2R × sin((n ⁄ 2) × 180 ⁄ n)

Longest Internal Diagonal of the Prism
D_space = √(D_max² + h²)

Rectangle Dimensions
Side 1: a (polygon edge)
Side 2: √(D_space² − a²)

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Reference Table: Paper Template Dimensions

Here is a PDF containing calculated dimensions for n-gon prisms with edge length a = 2 units:
Download PDF

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Bringing It to Life: TurtleStitch and Silhouette

Using the dimensions from the table, I coded the prism nets in TurtleStitch, a block-based programming environment ideal for geometric design. TurtleStitch made it possible to automate the creation of these intricate folding patterns.

I exported the designs as DXF files and opened them in Silhouette Studio software.
(Note: TurtleStitch’s DXF export doesn’t preserve scale, so resizing was necessary.)

Using the Silhouette cutting machine, I:

• Resized the nets to the desired dimensions.
• Added decorative windows to each panel.
• Cut the required number of nets for each n-sided prism.

The final results were stunning—geometric forms that twist and turn.

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Taking It One Step Further with TurtleStitch

Since I now understood the necessary calculations, I generalized my TurtleStitch code to eliminate the need to consult the measurement table manually and to add the decorative window to each panel.

Here is that TurtleStitch project:
Net for N-Gon Prism

With this program, creating a prism net is simple: enter the number of sides (n) and the side length (a), and the program automatically generates a net ready for cutting with Silhouette software.

After exporting as a DXF file, I opened it in Silhouette Studio, used the one-inch block for reference, and resized the entire design accordingly. Then, I used the Silhouette cutting machine to cut the required number of nets for each n-sided prism.

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Assembly Instructions

Step 1: Connect the Nets

Three identical net pieces are needed for a triangular prism.

Fold all rectangle diagonals with a valley fold—this step is critical! 

Make sure they are all facing the same directions and their orientation is the same. 

For example, notice the sides and diagonal form the letter 'Z'...look for it when folding the piece.

Apply glue to the side of the rectangles as shown above

1. Take two pieces and align them so their rectangle sides meet.
2. Glue these sides together, carefully matching the corners.
3. Continue adding and gluing each new piece in the same manner until all n faces are connected.

Step 2: Close the Tube

Join the first and last pieces by gluing their remaining rectangle sides together.

Step 3: Attach the Triangles

1. Valley-fold the triangles toward the rectangle’s diagonal.
2. Tape the adjacent triangle sides together.
3. Repeat until all triangles are joined, forming the complete prism

(Note: While these instructions show a triangular prism being made, the procedure is exactly the same for all n-gon's.)

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The Structure’s Underlying Symmetry

A ten-sided prism

It’s amazing to see how all the diagonals of the prism converge at a single point in the center, creating an opening through the middle of the shape. Both ends of the prism curve inward—this concave form appears because of the way the longest internal diagonals connect across the structure.

I made each net a different color so that you can appreciate the structures that are created. 

A multi-colored circle is formed with the intersection of the diagonal symmetry.  If you look carefully at the photo, you can see the longest internal diagonal.  A light pink diagonal goes from the top center of the photo to the bottom right below the light blue diagonal at the top of the prism. Try to see if you can find all ten of the internal diagonals.

The finished prisms are truly captivating—geometric forms that twist and turn in space. Each one feels like a small architectural sculpture, bridging the connection between art, craft, and mathematics.

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Many thanks to: 

• Alba Málaga Sabogal, Alix Kremer, Djatil Krichenane, Samuel Lelièvre, Richard Schwartz and Ulrich Breh for inspiring this exploration with their beautiful torus structures 
• Saul Stahl for his foundational work on map coloring 
• ICERM (Institute for Computational and Experimental Research in Mathematics) for hosting the Illustrating Mathematics program 
• Claude (Anthropic) for patient assistance with the mathematical calculations and HTML development.  I am still working on the HTML development.  I will try to include it in another post.
• The TurtleStitch community, especially Cynthia Solomon, who encouraged me to continue my quest of prisms and using TurtleStitch as a tool for mathematical making

Exploring the Five-Color Torus: A Mathematical Journey

Exploring the Five-Color Torus: A Mathematical Journey

This summer I attended the ICERM Illustrating Mathematics Reunion/Expansion and had the pleasure of meeting Alba Málaga Sabogal from the Université de Lorraine. Alba gave a fascinating presentation on a structure called the five coloring of a torus. You can watch her talk in the ICERM video archive; at about 4:02 she begins discussing the piece.

Because I’ve long been intrigued by the torus, I was immediately drawn to this project. The model was created by undergraduate students Alix Kremer and Djatil Krichenane, under the mentorship of Alba and Samuel Lelièvre. Their work combines geometry, design, and craftsmanship into a tangible structure: five folded modules, each a different color, joined together by hinges. Alba included their work in her ICERM presentation on “Tangible Mathematics Research Internships.” The project itself grew out of two prompts: Samuel Lelièvre’s 7-colored diplotorus and Saul Stahl’s paper “The Other Map Coloring Theorem.”

The result is a pentagonal torus — a surface of genus 1 (topologically equivalent to a doughnut) with pentagonal symmetry. Instead of the familiar smooth torus generated by rotating a circle, this version is polygonal, assembled from five wedge-like modules. What makes it so elegant is that each colored region touches all the others, perfectly embodying the idea of a five-coloring in three dimensions.

At the conference, Alba wanted to build a version to show how the coloring works. She was prepared to cut it by hand, but since I had brought my Silhouette Cameo cutting machine, I offered to help produce a cleaner, more precise version.

When I got home, I couldn’t stop thinking about the model. I decided to build one myself, this time experimenting with hinged paper tabs instead of Scotch tape. It was my first attempt at this joining method, and I quickly discovered the net could be assembled in two ways: clockwise or counterclockwise, depending on how the pieces were arranged.

That experiment sparked a new challenge for me: given that the four-color theorem guarantees any map can be colored with just four colors, I wondered if I could create a four-color torus using only four folded modules that are skewed like Alba's. The theorem shows that you need at most four colors to color any flat map so that adjacent regions—those sharing a border, not just touching at a point—have different colors. Since Francis Guthrie first proposed this concept in 1852 (earning it the alternate name "Guthrie's problem"), I was curious whether this mathematical principle could translate into a physical origami construction.

This turned out to be much trickier than expected. The geometry demanded precise control of the diagonal angles and the heights of the outer triangles. Even the slightest mismatch caused the paper to warp or wrinkle as shown in the above photos.After many trials, I settled on a two-piece approach: one piece for the central diagonal, and another for the two outer triangles. I also scaled the outer triangles to 101% to account for cardstock thickness. At exactly 100%, they were just a bit too small to close smoothly into a ring.

In the end, the process gave me a new appreciation for Alix and Djatil’s original five-color model. What began as a photograph shared at ICERM grew into an exploration of geometry, color, and structure — and left me eager to keep experimenting with new variations.

I have since tried a six color version.  Again, there is warping.  I tried to ask ChatGPT to design a program to determine the correct calculations.  This program does need further modification but it is interesting to see what AI can do with this mathematical color journey. Here is the html to run the program, Dynamic Twisting Polygon https://drive.google.com/file/d/1hJY8uP0mBRd8wXsxd0-RfgXi6y9Klyex/view?usp=sharing Click on the file and then download it.  Once downloaded, open the file to run the html code, toggle on the twisted edges and the diagonals to see the internal structure of the torus. After further investigation, the measurements are wrong!  Interesting to see that AI is not always right.  I will correct its mistake and write about this experience when completed.

Here are the files for the colored torus:

The PDF for a five colored torus. https://drive.google.com/file/d/13C4T0GLoQzU01qpDz258_CiddF5CMQ_J/view?usp=sharing

The .Studio file for a four and five colored torus.

https://drive.google.com/file/d/1CLtkFFHwI5PBigCwHL-FDGvHBsG3eMEd/view?usp=sharing

The SVG file for a four and five colored torus.

https://drive.google.com/file/d/16c9YwpBY3xGA8N7J2SPesNTnNudHOmB3/view?usp=sharing

A Paper Doll Chair for the American Girl Little Bitty Baby

A Paper Doll Chair for the American Girl Little Bitty Baby

The paper doll chair is pictured on the left.  The upholstered version in the middle and right of the photo.

I am so excited to share this fun project with you today.  Your American Girl Little Bitty Baby doll is going to love having her very own perfectly-sized seat! I created the chair with TurtleStitch, a block-based programming language.

 For my papercrafting friends, no coding is necessary.  All the files for a five inch chair are included in the file resources section.

What I absolutely love about this project is how precise and versatile it is! Since our Little Bitty Baby is 5 inches tall, I designed the TurtleStitch code at a 1:1 scale, which means everything translates perfectly to real-world tiny dimensions.

I created this TurtleStitch code with a built-in scale feature that's like magic! Want to make a chair for your other dolls too? Just change that scale number and voilà! You can create perfectly sized chairs for:

• Your full-size 18-inch American Girl dolls
• Barbie and her friends (11.5 inches)
• Any baby dolls you have around the house
• Really, any doll in your collection!

Isn't that amazing? You're not just learning one project - you're getting a whole chair-making system for your doll family!

File Resources

I know not everyone wants to dive into coding right away, so I've prepared all the files you need for the Little Bitty Baby chair. Just pick what works best for you:

TurtleStitch Project: View and modify the original code - For the adventurous souls who want to make different sized chairs.

Ready-to-Print PDF: Download PDF file - For hand cutting

Silhouette Studio File: Download .Studio file - For your Silhouette machine.

SVG File: Download SVG - Works with any cutting machine

Components of the Chair

This little chair has just three simple parts.

• Chair frame: The back and sides 
• Seat: The bottom
• Legs: You'll cut four of these sturdy little supports

Assembly 

The legs are designed with a really smart folding trick. When you fold the sides inward, you create these diagonal creases that turn flat cardstock into strong, three-dimensional supports. It's like origami meets furniture making!

Putting It All Together (It's Easier Than You Think!)

Components of the doll chair

I was making two...so that is why there are duplicates of the frame and seat.

1. Cut everything out from 65 lb. cardstock: One frame, one seat, four legs.  I used chipboard for the upholstery version for the chair frame and I cut an additional chair frame to sandwich the glued sides together.
2. Fold the legs along the diagonal lines to make them stand strong
3. Glue the legs to the four corners of your seat
4. Fold the frame at right angles where the creases show you
5. Pop that seat right inside the frame and glue it in place

And just like that, you've got yourself a chair!

Want to Get Fancy? Let's Talk Upholstery!

This is where you can really let your creativity shine! If you want to give your chair the full designer treatment, add some fabric before you start gluing things together.

Fabric Tips

• Go lightweight: Heavy fabrics are tricky at this tiny scale
• Think thin and flexible: These will cooperate much better with your glue
• Watch those patterns: Big prints might overwhelm such a little chair
• Test first: Some fabrics are just stubborn about sticking to cardstock

Cover your chair frame and seat with fabric by cutting the fabric about a 1/4 inch larger than the pattern piece.  Glue the fabric to the cardstock. Two chair frames are glued together to make a sturdy chair frame. 

Why I Absolutely Love This Project

This little chair project has everything I adore about crafting: it's thoughtfully designed, uses clever techniques, includes helpful calibration tools, and works for any size doll you can imagine. Plus, the engineering principles work beautifully no matter what scale you're working in.

The scalability feature means you're not just making one cute chair - you're mastering a system that can furnish dolls of all sizes! 

Happy crafting, friends! 💕

For those of you who want to resize the TurtleStitch design using the Silhouette Software

Change the scale factor for the new doll chair size

The formula for calculating the scale factor for the new chair is:

Scale Factor = NewDimension / Original Dimension

To scale a 5-inch doll pattern to fit an 18-inch doll, the scale factor would be 18/5 = 3.6. 

Make a new chair pattern

1. Run the program with the new scale factor.
2. Export the TurtleStitch design as a DXF file
3. Import into Silhouette software
4. Release the compound path to access individual components
5. Look at the reference one inch square for its measurement 
6. Calculate scaling factor: (1.00 ÷ reference square measurement) × 100
7. Apply the calculated percentage in the Transform Panel

𝑆𝑐𝑎𝑙𝑒𝐹𝑎𝑐𝑡𝑜𝑟=18𝑖𝑛𝑐ℎ𝑒𝑠(𝑛𝑒𝑤𝑑𝑜𝑙𝑙)5𝑖𝑛𝑐ℎ𝑒𝑠(𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙𝑑𝑜𝑙𝑙)=

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. 

Flowers in a Hexagonal Vase

Flowers in a Hexagonal Vase

Bottom View of the Vase

This vase is a culmination of a variety of mathematically coded flowers.  The flowers were coded with rose curves, petal blocks and arithmetic spirals in TurtleStitch.  I am not going to go into the details of their coding in this blog post as I have mentioned how I created these flowers in a previous blog posting. https://papercraftetc.blogspot.com/2025/05/a-stem-project-turtlestitch-coded.html

I used 110 lb. cardstock for the green stems and 65lb. cardstock for the vase and flowers. I used glitter cardstock for the center of each flower.

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/1EvH8ugDOw_6DfYnGmmiM5K0JCtHJWWkr/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/1bD-63VYGGFief7X2iTMxiZ9JnPVpImW9/view?usp=sharing

Make the Vase

Fold the six sides in half.  Glue three of the sides together.  Four of these sides will be slid into the round base as shown above in the top left of the photo above.

Glue the other three sides together, apply glue as shown to the last two sides, slide this half onto the other half.  Make sure to slide the sides into the correct slots to complete the vase.

Bend the top circle into the center of the vase and slide this piece into each of the six slits on the circle.

Glue the vellum to the inside of hexagonal sides

Bend the tops of the stems at a right angle. Glue two sides of the stem together. Do not glue the top of the stem since this is where the flower will be glued.

The stem is glued to the inside of each spoke. Repeat for all of the stems. 

The result will be a two-ply stem for each flower. 

The vase is now ready for embellishment with its flowers.

Make the Flowers

When making this flower arrangement, I suggest that you have fun with the design. Mix and match the different  flower pieces to create unique flowers. Assemble the flowers in a pleasing pattern. There are a total of 18 flowers in this arrangement. Glue each flower to the top of each stem.

This vase is filled with paper flowers created using three different techniques. Each method brings its own unique style and dimension to the arrangement:

1. Multi-Petaled Flowers
These flowers are made by layering several flower shapes on top of each other. Each layer can feature different petal styles, adding depth and variety. After stacking the layers, gently curl the petals upward to give the flower a more natural, dimensional look. A dab of glue between each layer holds everything securely in place.

2. Rolled Flowers
Rolled flowers are formed by curling a spiral-cut piece of paper inward, starting from the outer edge and working toward the center. Once you reach the center, apply a generous amount of glue to the round base circle and press the rolled spiral onto it. This creates a tight, elegant bloom. You can find detailed instructions for making these rolled flowers here: A STEM Project - Slice Form Flower Vase.

3. Sliceform Flowers
Sliceform flowers are created by interlocking precisely cut paper pieces to form a three-dimensional structure without any glue. This method results in striking geometric flowers that stand out beautifully. Complete instructions for creating sliceform flowers are available here: A STEM Project - Amazing Slice Form.

4. Torus Flowers

Torus flowers are made with two types of Villarceau circles, 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. Check out this posting for instructions on how to make the flowers: https://papercraftetc.blogspot.com/2021/07/a-stem-project-amazing-slice-form.html

With these four techniques, you can mix and match styles to create a stunning arrangement of paper flowers that showcase both creativity and craftsmanship.