A STEM Project: A Frozen Kingdom Comes to Life with MicroBlocks and the MakerPort

From Walmart Toy Sets to an Interactive Masterpiece

Watch the video to see the scene come alive!

There are four touch points that play different songs and sayings

My granddaughter was on Spring Break and she came to visit me for the day. The two of us sat down together to bring a brand-new diorama to life. This time inspired by the magical world of Frozen. What started with a trip down the toy aisle turned into one of my most enchanting projects yet, with Bruni sitting on a plate of spinning snowflakes, glowing blue light, and all of our favorite Arendelle characters gathered around the table.

Building the Foundation: Elegant Seating

Anna and Elsa sit at the enchanted table on iridescent satin chairs 

Every magical dining experience needs proper seating. I crafted custom chairs for both Anna and Elsa following these instructions. To give them a more elegant appearance befitting our princesses, I upholstered each chair with iridescent satin fabric, transforming simple paper furniture into sophisticated little thrones.

My granddaughter took charge of the table, arranging the hot cocoa and all the charming accessories just so. Ever the practical one, she secured Glue Dots to the bottom of each cup and pot, ensuring nothing would topple if the table got a bump or a nudge.

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. The table skirt keeps the look polished. The top is glued cleanly to the presentation box, while the skirt itself features an elastic top so it can be lifted away whenever I need to access the components inside.

The Centerpiece: Bruni in His Element

Every magical scene needs a showstopper, and ours is Bruni, the adorable fire salamander from Frozen 2, delightfully rotating on a platter of snowflakes with a snowflake hanging out of his mouth. A servo motor keeps his platter in constant gentle motion, which my granddaughter declared was "her favorite thing." It's hard to argue with that logic.

The rotating platter uses the same technique as Mrs. Potts and Chip in my Beauty and the Beast diorama but snowflakes suit Bruni just as beautifully. https://papercraftetc.blogspot.com/2025/10/a-stem-project-magical-dining-scene.html

The Royal Guests

Anna and Elsa are seated in handcrafted upholstered chairs because even royalty deserves a comfortable seat at a hot chocolate party.

Baby Sven is positioned just so, eyes fixed on Bruni's gentle spin. And then there's Olaf, he is happily watching from the side of the table, maintaining a careful distance. As he reminded us himself, hot chocolate and snowmen don't always mix.

The Soundtrack: Songs, Sayings, and One Showstopping Exchange

Four touch sensors are positioned around the scene, each triggering a different moment of Frozen magic. Three of them play a variety of beloved songs and character sayings from across the films. But the fourth? That one belongs entirely to Olaf and it plays out the full exchange from the scene where poor Olaf finds himself separated from his lower half.

Olaf: "I can't feel my legs! I can't feel my legs!"
Kristoff: "Those are my legs."
Olaf: "Ooh, do me a favor and grab my butt!"

My granddaughter laughed and laughed...and honestly, so did I. She must have pressed that sensor two dozen times. It is now, officially, her favorite feature of any diorama I have ever made. The buildup of the whole exchange is what makes it so special.  Kristoff's deadpan reply is every bit as funny as Olaf's punchline.

Illuminating the Magic: Blue and White Glow

The lighting for this scene called for something cool and ethereal...perfectly icy, perfectly Frozen. An LED ring light embedded in the table casts a soft, shifting glow in hues of blue and white that bathe the entire scene in a wintery atmosphere.

• The LED Ring Light: Nestled beneath the table surface, it cycles through gentle blue and white tones, evoking Elsa's ice magic and the Northern Lights of Arendelle.
• The Rotating Platter: A servo motor keeps Bruni and his snowflakes in a slow, continuous spin which is  endlessly mesmerizing.
• Four Touch Sensors: Songs, character sayings, and one showstopping Olaf line that will have every child (and grandparent) in stitches.

Programming the Scene with MicroBlocks

All of the interactive elements, the spinning platter, the LED ring light, and the touch-activated music and sayings, were programmed using MicroBlocks. It gave me precise control over timing and behavior, and the MakerPort kept everything neatly connected inside the presentation box.

The Grand Reveal

When my granddaughter and I placed the last character at the table and switched everything on, the room filled with the soft blue glow of the LED ring and the opening notes of her favorite song. Bruni spun. Baby Sven watched. Olaf held his ground at a safe distance from the hot chocolate.

And then she found touch sensor number four. The laughter that followed, all three lines of it, was worth every hour of work. 

All electronics and interactive elements were programmed using MicroBlocks, with the MakerPort housed inside the TurtleStitch-coded presentation box.

What Happened Next

This is the part that made the day visit so unforgettable. Once the diorama was finished and the laughter had settled, something wonderful happened. My five year old granddaughter decided she wanted to make something of her own when she looked at Roger Wagner's Animatronics set.

She grabbed one of my old presentation boxes for a body, figured out an axle, added two wheels, and topped it off with a head complete with two eyes and a happy smile. Then she drew squares on the front of the robot and proudly announced, “This is how you turn on the robot, this is how you make it move and these are the lights that turn on and off! Oh and the squiggly lines, those are the wires that connect everything together.” Just like that, she had built her very first robot. 

She carried it home with her at the end of the day, proudly and she had every right to be proud. That little robot was entirely hers.

That is exactly what making things is all about. Not the lights or the code or the servo motors. Though I do love all of those, but the moment a child realizes that her hands and her imagination are all she needs to build something real. Something that didn't exist before she decided it should.

❄️ Just like the song says, “Let It Go…and the cold never bothered me anyway.”  and in that moment, she discovered who she really was. ❄️ 

A STEM Project: Fabergé-Inspired Easter Eggs

Fabergé-Inspired Easter Eggs

Easter is a time for beauty, renewal, and wonder. And nothing captures that spirit quite like the legendary Fabergé eggs. They are breathtaking jeweled masterpieces that have dazzled the world for well over a century. When I set out to make my own Easter eggs this year, I wanted to channel that same sense of artistry and magic, but with my favorite medium…paper. 

The Legacy of Fabergé Eggs 

Few objects in history carry the mystique of a Fabergé egg. Created by the House of Fabergé beginning in 1885, these extraordinary works of art were originally commissioned by Tsar Alexander III as Easter gifts for his wife, Empress Maria Feodorovna. Each egg was a miniature masterpiece. They were crafted from gold, silver, precious gemstones, and enamel. When the eggs were opened, they concealed a delightful surprise within. 

What made the Fabergé eggs so special was not just their materials, but the intention behind them, a desire to create something so beautiful and so full of care that it would stop a person in their tracks. Each egg told a story, celebrated a moment, or honored a relationship. They were gifts of love elevated to the level of art. 

Over 50 Imperial Fabergé eggs were made, and today they reside in museums and private collections around the world, cherished as some of the greatest treasures of decorative art. 

My Version of Fabergé Eggs 

I may not have access to gold or rubies, but I do have paper, vellum, glue, and a deep love of making things by hand. My paper Easter eggs are my personal tribute to the Fabergé tradition. The design is elegant in its geometry.  They are hexagonal shaped with six bunny scenes backed with vellum panels. 
One of the things I love most about the hexagonal design is that it gives you six panels to work with. Each of my eggs tells its story across six scenes, like turning the pages of a tiny illustrated book. 

Two of the panels feature delicate five-petal spring flowers in full bloom. Another two panels show a curious little bunny gazing up at a butterfly, nose twitching, utterly enchanted. A fifth panel captures a similar moment of wonder - a bunny peering at a small bird who has stopped to visit. And on the sixth panel, the most endearing scene of all, is a bunny sitting contentedly, holding a daisy as though it were the most precious thing in the world. 

Together, the six scenes paint a picture of a spring morning full of curiosity and quiet delight. Much like the surprise hidden inside a Fabergé egg, the scenes reward a slow and careful look. Each of the panels is a tiny world worth lingering in. 

What You'll Need

Materials:

• Neenah 65 lb white gold cardstock from Office Depot (for the six panels)
• Decorative cardstock for the hexagonal egg structure
• Vellum
• Craft Glue

The white gold cardstock is essential to this design—its elegant shimmer catches and reflects light beautifully, creating that special festive sparkle.

Equipment:

• Electronic cutting machine (Silhouette or Cricut)

Cut Files

Choose the file that matches your machine:

• Silhouette users: Download the .Studio file [link to file]
• Cricut users: Download the SVG file [link to file]

Quick tip for SVG users: The design extends beyond the initial viewable area, so just zoom out to see the complete pattern.

Make the Egg Structure

Fold three sides of the egg and glue this half together together as shown on the right..  Repeat for the other side.

Insert the top and bottom base into the corresponding slits of the egg structure.  Apply glue to the side of the egg structure and insert the remaining half onto the top and bottom.

View of the top of the egg structure.  Make sure that all of the pieces are aligned correctly.

Make the Bunny Panels

Glue the vellum to the back of each bunny panel.

Completed Bunny Panels

Insert the top and bottom tabs into the slits on the egg structure.

A Small Treasure to Share 

These paper Easter eggs won't be locked away in a museum vault or sold at auction for millions of dollars. But they will sit on a windowsill and catch the afternoon light, and they were given to my granddaughters as tokens of care and creativity. In that spirit, I think they are very much in the Fabergé tradition. 

Because in the end, what made Fabergé's eggs extraordinary wasn't the expense, it was the intention. The idea that beauty is worth the effort. That the people in our lives deserve something made with our own hands and our whole hearts. 

Happy Easter!

May your season be full of beauty, warmth, and the quiet joy of making something wonderful.

Under the Acacia Tree: A Paper Safari

Under the Acacia Tree: A Paper Safari

Here is a video of my paper safari.

Uganda, the Pearl of Africa, is home to some of the continent's most spectacular wildlife: elephants, giraffes, rhinoceroses, and the magnificent grey-crowned crane, the country's national bird. All of them roam beneath one of nature's most iconic silhouettes: the Umbrella Thorn Acacia (Vachellia tortilis), its vast flat canopy spreading like a sheltering parasol over the golden savannah. Whether you dream of visiting Uganda's great parks like Murchison Falls, Queen Elizabeth, or the remote wilds of Kidepo Valley or simply want to bring a piece of that landscape home, this paper model is a celebration of the scene.

I used Neenah brand 65 lb Champagne Pearl metallic cardstock from Office Depot. I recommend using a new blade and overcut to cut out this design.

Here is the .Studio file.

https://drive.google.com/file/d/1dfbVTY9BVxoyTr0cRQkXl3ft6C25V6qB/view?usp=sharing

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

https://drive.google.com/file/d/1aEvxN1vvZPdMDmwQTZlnLlOCKCIOSbC4/view?usp=sharing

Making the Safari Scene

Each of the ten scenes are creased with a valley fold in the center.  Glue is applied to one side of the outer edges of its frame and to the acacia tree. This panel is then adhered to the next scene.  Take care to align the central crease precisely with the midpoint of the frame before the glue sets. This process is repeated until all of the scenes have been glued together. The sun has additional layers which can be glued to create a 3-D sun.

The panels are then fanned out from the spine to create the paper safari in a circular format. The last two scenes can be attached with a Glue Dot so that it can fold flat again later if desired.

Grey-Crowned Crane Diorama

Uganda Diorama - A Grey-Crowned Crane

This diorama depicts a grey-crowned crane, Uganda's national bird. It is a strikingly beautiful bird with a crown of feathers and elegant plumage. The scene shows a graceful bird striding across the open savannah of East Africa, with mountains in the background and a beautiful sunset in the distance. The grey crowned crane is truly one of Africa's most majestic birds. 

Eight double thickness tabs keep the diorama scenes together.

Pretty paper is glued to the front of the first scene.

The tabs slide into the sides of the scenes.

I used Neenah brand 65 lb Champagne Pearl metallic cardstock from Office Depot. I recommend using a new blade and overcut to cut out this design as the intricate pieces might not cut correctly.

Here is the .Studio file.

https://drive.google.com/file/d/1pr4ORSmHbvHD-UgsRWSx7B2jC1DePMmu/view?usp=sharing

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

https://drive.google.com/file/d/1Q13eHelzdiKYJqSuSk7E_lV_TYzmjtfz/view?usp=sharing

Pursuing Kepler's Inscribed Solid Model

Kepler's Inscribed Solid Model

In my last blog posting, I made a nested inscribed Platonic model. I was fascinated by its structure and wanted to pursue it further.

Kepler's Cosmological Vision

In 1596, Johannes Kepler published Mysterium Cosmographicum (The Cosmographic Mystery), proposing an elegant geometric explanation for the solar system. He theorized that the orbital distances of the six known planets were determined by nesting the five Platonic solids between spherical shells. His sequence, from outermost to innermost, was:

Sphere of Saturn → Cube → Sphere of Jupiter → Tetrahedron → Sphere of Mars → Dodecahedron → Sphere of Earth → Icosahedron → Sphere of Venus → Octahedron → Sphere of Mercury

Kepler believed this revealed God's geometric design of the universe. The fact that there were exactly six known planets and exactly five Platonic solids couldn't be coincidence — each solid inscribed in one planetary sphere and circumscribed by the next would explain both why there were six planets and what determined their spacing.

Though beautiful, the theory proved incorrect. The predicted orbital ratios didn't quite match observations, and later discoveries of additional planets definitively disproved the model. Yet this "failed" theory was far from wasted — it led Kepler to develop his three laws of planetary motion, which correctly describe how planets move and became foundational to modern physics. Sometimes the most productive wrong idea is one that is wrong in exactly the right way.

Exploring the Possibilities

There are five Platonic solids, and mathematically, 120 distinct ways to arrange them as nested inscriptions (5! = 5×4×3×2×1 = 120 permutations). Kepler's original cosmological model used one specific ordering, but I was curious: which sequences would work best for a paper model? Which would create the most balanced proportions? Which would maximize the size of the innermost solid?

To explore these questions, I asked Claude to create an interactive table showing all 120 combinations, with the outermost solid inscribed in a sphere of four inches diameter — a practical constraint ensuring the nets would fit on 8½ × 11 inch paper. This mirrors Kepler's approach, where each Platonic solid is inscribed in a sphere. The table draws on the Croft table of maximal inscribing ratios and calculations from Firsching's research to determine the side length of each solid in every sequence, and allows users to adjust the outer sphere diameter and sort by different criteria.

What emerges from exploring all 120 sequences is genuinely surprising. Mathematically, the ordering matters enormously — the ratio of the innermost solid to the outermost sphere can vary by more than a factor of ten depending on which solid sits at each level. The dodecahedron and icosahedron, as duals of each other, tend to preserve size better when paired consecutively, while placing the cube or tetrahedron early in the sequence causes a steep drop in scale. Kepler's chosen order was driven by his belief that it reflected the orbital spacing of the planets, not by any geometric optimality, and it performs only modestly by most mathematical measures. For a paper model, the practical sweet spot lies in sequences that balance two competing goals: keeping the innermost solid large enough to be worth building, while ensuring no single transition dominates so much that the nesting feels anticlimactic. The most satisfying models tend to be those where the size steps between levels feel roughly even — a kind of geometric rhythm — rather than sequences that plunge suddenly to a tiny core. The table makes it possible to hunt for that rhythm deliberately, turning what was once Kepler's act of cosmological faith into something you can sort, optimize, and make your own.

My Model

After examining all 120 possibilities, I chose to follow Kepler's original sequence: Cube, Tetrahedron, Dodecahedron, Icosahedron, Octahedron, working inward. Not because it is geometrically optimal — the table makes clear it isn't — but because I wanted my model to carry the same spirit as Kepler's: the conviction that the universe is built on geometric principles, that beauty and mathematics are the same thing seen from different angles. By following his sequence, the model becomes a small act of homage to one of history's great wrong ideas, and to the mind bold enough to pursue it.

The colors I chose reflect each solid's planetary association in Kepler's scheme: the cube (Jupiter) in white, the tetrahedron (Mars) in red, the dodecahedron (Earth) in dark blue, the icosahedron (Venus) in light blue, and the octahedron (Mercury) in grey metallic. Saturn, the outermost sphere in Kepler's model, has no solid assigned to it — appropriately enough, it exists only as the invisible boundary that contains everything else.

Make Your Own

If this has sparked your curiosity, why not try building one yourself? It's more approachable than it might seem. Start by visiting Turtlestitch.org and searching for "Platonic" — I created five programs there that generate the nets for each of the five solids. Use the Claude table to find the measurements for whichever sequence appeals to you, then input those dimensions into the program and export the file as a DXF.

Open the DXF in Silhouette software, where you'll find a one-inch reference square included in the file — use it to scale the entire design to the correct size before cutting. Work through all five solids, then assemble them starting from the innermost and working outward. I glued each structure together, and for the solids that needed a little extra rigidity, I reinforced one or two interior faces with acetate sheets. Glue Dots made it easy to align and nest each solid cleanly inside the next.

The finished model is genuinely stunning to look at — a physical object that connects your hands to Kepler's imagination, across four centuries of mathematics. Give it a try and see where the sequence takes you.

Nested Platonic Solids: From Kepler to the 2026 Joint Mathematics Meeting

A nested model of all five Platonic solids with windows cut into the outer layers to reveal the inner solids. At the center is a metallic silver tetrahedron, surrounded by a pale blue octahedron, green icosahedron, royal blue dodecahedron, and white cube.

At the Joint Mathematics Meeting in Washington, D.C. this January, I attended a fascinating session titled "Revisiting Kepler's nested Platonic solids" presented by Andrew Simoson. He showed me a beautiful physical model, "nesting them outwards in the order tetrahedron, cube, octahedron, dodecahedron, and icosahedron, rather than Kepler's order; (they're held together by a bamboo stick capped with little spheres)." His model reignited my interest in these remarkable geometric relationships.

This wasn't my first encounter with nested Platonic solids. After making Da Vinci's divine proportion polyhedra models (see blog posts [link 1] and [link 2]), I had decided to inscribe the polyhedra models into one another, inspired by Pacioli's work. Pacioli wrote in his Divine Proportion book about these inscriptions, explaining that the five Platonic solids—the tetrahedron, cube, octahedron, icosahedron and dodecahedron—can be derived from a single one, the dodecahedron, which, according to Pacioli, "sustains the existence of all the others and governs the manifold harmonies and interrelations among all five." Since the dodecahedron is the basis for all others, Pacioli claimed that it would be only mathematically possible with a specific proportion which he named the "Divine Proportion."

Pacioli wrote about inscriptions of the five Platonic solids using the sphere method of calculating the side lengths of the interior polyhedron. I inscribed the Platonic solids as I thought Da Vinci would have done if he had the Silhouette software and cutting technology.

I found out while making the models that as the inscribed polyhedron's size approached a sphere, the size calculations became more complex and difficult to calculate. With some investigation on the internet, I discovered an article about maximizing the side lengths of the interior polyhedron. The computations for six of the polyhedra were just calculated in 2018 by the article's author, Moritz Firsching. These maximal size calculations were in the annals of unsolved geometric problems for many centuries. Using Firsching's calculations, I was able to complete the 20 inscribed Platonic solid models. 

A New Model Inspired by Simoson's Presentation

Simoson's session at JMM inspired me to create a new nested model, this time exploring a different sequence: tetrahedron, octahedron, icosahedron, dodecahedron, and cube (T-O-I-D-C). Using Firsching's maximal inscribing ratios, I calculated the optimal dimensions for each solid.*

The following calculations were cited in this paper:

Maximum side lengths of polyhedron inscribed in another polyhedron.

Using the above calculations, I created my models. I used Glue Dots to affix the inscribed polyhedra. I also used clear acetate to make a base for some of the models so that I could adhere the inscribed polyhedra model to something since the point of contact was sometimes in mid-space in the outer hollow polyhedron.

 

I used 65lb. cardstock to make the models.

Here is the .Studio file.

https://drive.google.com/file/d/1XN7apVgNBxNUBt-Y-_LRuafkyFM3y_iO/view?usp=sharing

Here is the SVG.

https://drive.google.com/file/d/1u2nMRXpyjuMEbLuLBPz42EsuFMs3WG8N/view?usp=sharing

Note: The SVG file extends beyond the initial viewable area. Simply zoom out to see the complete design

The final model features:

• Tetrahedron (center): 2.362" in metallic silver cardstock
• Octahedron: 2.362" in pale blue cardstock
• Icosahedron: 2.0" in green cardstock
• Dodecahedron: 1.528" in royal blue cardstock
• Cube (outer): 3.878" in white cardstock

I constructed each polyhedron from cardstock using my Silhouette cutting machine, with windows cut into the outer four solids to reveal the layers within. The metallic silver tetrahedron at the core catches and reflects light through the colored layers, creating a luminous effect that changes as you move around the model.

What fascinates me about this project is how it bridges historical mathematical inquiry—from Kepler's 1596 cosmological model to Pacioli's Divine Proportion—with contemporary mathematical research. Firsching's 2018 paper solved problems that had remained open for centuries, and now those solutions enable artists and educators to create precise physical models that would have been impossible to calculate just a few years ago.

The intersection of mathematics, history, and art continues to inspire new explorations of these timeless geometric forms.

*"Computing maximal copies of polytopes contained in a polytope," by Moritz Firsching, Institut für Mathematik FU Berlin Arnimallee 2 14195 Berlin Germany, July 16, 2018 https://arxiv.org/pdf/1407.0683.pdf

Torus Blossoms in a Sliceform Vase: Where Mathematics Becomes Art

Torus Blossoms in a Sliceform Vase

I’m thrilled to announce that my model Torus Blossoms in a Sliceform Vase has been accepted into a juried exhibit at the Joint Mathematics Meetings, taking place January 4–7, 2026, in Washington, D.C. This event is the largest annual gathering of mathematicians in the world, and this year’s theme—“We Champion Mathematics: Highlighting Beauty and Innovation in the Mathematical Sciences”—perfectly reflects the spirit of this project.

Download the exhibit booklet (PDF).

The Art of Mathematical Flowers Using a Paper Torus

These paper-cut flowers emerge from an exploration of torus geometry, revealing how abstract mathematical concepts can transform into something unexpectedly organic and beautiful. Each flower begins as a torus, sliced into carefully designed sections and cut from cardstock using an electronic paper cutting machine. The slices are then slid together using the sliceform technique, allowing a three-dimensional structure to emerge from flat pieces.

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

My exploration of toruses began in 2013, when I first started working with sliceforms. I was immediately fascinated by their structure and curious about what might emerge if I began manipulating their edges. Using the same underlying torus geometry, I created a wide range of forms—including a snowman, a pumpkin, and many different types of flowers—each revealing new possibilities within the shape.

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 used point editing on the slice edges 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. The PDF linked here documents my method for making the torus flowers.

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

This project reveals how abstract mathematical ideas can reshape the classical world into something both unexpected and uniquely beautiful. As a former math teacher, I want others to be able to recreate this beauty and explore the mathematical shapes that can be discovered by making this model.

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 interlocking 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. (If you are having difficulty aligning the final slices, I recommend putting a drop of glue at the endpoints where the two slices meet.) 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.

12 Days of Christmas Decorations: Day 12 - Poinsettia Centerpiece

A Poinsettia Centerpiece

Welcome back to my 12 Days of Christmas Decorations series! For twelve wonderful days, I've shared festive projects as my holiday gift to you, and today we reach our joyful conclusion. My hope is that these decorations have filled—and will continue to fill—your home with warmth, beauty, and Christmas joy.

A Heartfelt Thank You

As we arrive at Day 12, I want to take a moment to express my gratitude. Creating and sharing these projects has been truly a labor of love. Each design represents countless hours of planning, testing, refining, and crafting—but more than that, each one reflects my passion for creating beauty and bringing joy into the world through paper art.

I hope you've felt the love and care I've poured into every tutorial, every template, and every word of encouragement. Whether you've been following along making each project or simply enjoying the inspiration, thank you for being part of this creative journey with me. Your presence here has made these twelve days even more special.

The Perfect Finale: A Poinsettia Centerpiece

For our final day, I'm thrilled to share an elegant poinsettia centerpiece that beautifully captures the spirit of the season! And yes, this project was inspired by my adorable (but mischievous) new kitten. Since live poinsettias can be toxic to curious cats, I decided to create a paper version that's just as stunning—and completely pet-safe. The result is a gorgeous centerpiece that will grace your holiday table without any worry!

A Design That Lasts Beyond Christmas

Here's something special about this centerpiece: it's designed to carry you through the entire winter season! The hexagonal vase lantern features six delightful panels displaying cheerful snowmen and intricate snowflakes (two designs, each repeated). Because these winter motifs aren't exclusively Christmas imagery, you can proudly display this beautiful piece through January and February.

The poinsettia blooms and delicate pine needle foliage with bright red berries complement the winter scenes perfectly, creating a cohesive seasonal display that transitions seamlessly from holiday celebration to winter wonderland.

The Meaningful History of Poinsettias

The poinsettia's connection to Christmas comes from a beautiful Mexican legend that perfectly embodies the spirit of giving from the heart.

The story tells of a young girl named Pepita who wanted to give a gift to baby Jesus at her church's Christmas Eve service. Being very poor, she had nothing of value to offer and was heartbroken. Her cousin Pedro tried to comfort her, saying that any gift given with love would be precious, no matter how humble.

Inspired by his words, Pepita gathered a simple bouquet of roadside weeds on her way to church. As she approached the nativity scene and knelt to place her modest offering before the manger, a miracle occurred—the green weeds suddenly burst into brilliant red blooms, transforming into the stunning flowers we now know as poinsettias!

The congregation witnessed this marvel and declared it a Christmas miracle. The vibrant red petals came to symbolize the blood of Christ and the Star of Bethlehem, while the story itself represents the beauty of giving from a pure heart, the miracle of faith, and the truth that no gift is too small when offered with genuine love.

Today, poinsettias remain one of the most beloved Christmas decorations, gracing homes, churches, and celebrations worldwide—a living reminder that the most meaningful gifts come from the heart.

The Design

This stunning centerpiece builds upon the hexagonal vase design I've featured throughout this series (find the original tutorial  here). The six-sided structure creates the perfect showcase for the alternating snowman and snowflake panels, which glow beautifully when lit from within.

Add a multicolored LED tea light from your local dollar store at the center, and watch your creation come to life! The warm, dancing light illuminates the winter scenes while casting a gentle glow on the poinsettia blooms and pine foliage above. It makes an absolutely enchanting centerpiece for your holiday table or a cozy accent that brightens any room in your home.

What You'll Need

Materials:

• 65 lb white cardstock for the snowman and snowflake panels
• 65 lb red, green, and yellow cardstock for the poinsettia petals and leaves
• Vellum for backing the panels (allows the LED light to shine through beautifully)
• Battery-operated multicolored LED tea light

The combination of crisp white panels, vibrant poinsettia colors, and soft vellum backing creates a professional, polished look that will impress your holiday guests!

Cut Files

You'll need an 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.

Assembly Instructions

Follow these steps to create your winter wonderland centerpiece:

1. Build the lantern base: Follow the assembly directions from the  hexagonal vase project to construct your six-sided structure with the snowman and snowflake panels
2. Back the panels with vellum to create that beautiful glowing effect when the LED is lit
3. Create the floral elements: Assemble the pine needle foliage pieces with red berries at their centers, then construct the poinsettia flowers and attach them to their stems
4. Add realistic dimension: Curl the poinsettia petals and pine needle foliage around a pencil or dowel to give them natural shape and movement
5. Arrange your bouquet: Position the poinsettias and pine sprigs in the top of the hexagonal vase, arranging them until you achieve a balanced, full display
6. Light it up: Place your battery-operated LED tea light inside the vase and enjoy the magical glow!

Safety reminder: Always use battery-operated LED lights rather than real candles for paper crafts—especially important with pets in the home!

Display tip: The multicolored LED creates a gentle, ever-changing ambiance. If you prefer a more traditional look, use a single warm white LED instead.

A Fond Farewell and Warm Wishes

And so we've reached the end of our 12 Days of Christmas Decorations journey! This poinsettia centerpiece feels like the perfect way to conclude—bringing together the light, color, beauty, and symbolism that make this season so magical.

Thank you from the bottom of my heart for following along on this crafting adventure. Whether you made all twelve projects, chose a few favorites, or simply enjoyed reading and dreaming about future creations, your support and enthusiasm have meant the world to me.

I hope these projects have inspired you, brought you joy, and perhaps sparked your own creative spirit. Remember, every decoration you make with your own hands carries a special kind of magic—the magic of time, care, and love made visible.

May your home be filled with beauty, your heart with peace, and your holiday season with countless moments of joy and wonder. Thank you for letting me share my passion with you.

Wishing you and yours a very Merry Christmas and a happy, healthy, and creative New Year!

Happy crafting, my friends! 🎄✨🌟

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P.S. - Don't let the end of this series stop your creative momentum! I'll continue sharing new projects on the blog. Be sure to subscribe so you won't miss future inspiration. And if you create any of these twelve projects, I'd absolutely love to see them—tag me or share in the comments!