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.
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