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