Platonic Nest

Select any of the 120 sequences by number, from the special cases below, or from the full table. Contact points mark where each inner solid's vertices meet the next solid, by type. Drag to rotate; scroll or pinch to zoom.

Choose any solids, order them from the inside out, and set the size of each. All share one center and the same axes, and any solid can pass through another. Contact markers appear wherever a vertex lands on another solid's surface; crossing markers show where an edge passes through a face, and intersection lines join those points along the seams where two solids' faces cut each other.

The fit buttons resize every layer from the innermost outward: touching (each solid the smallest that encloses the one inside it), Kepler's sphere rule, a shared circumsphere, or a shared midsphere. A shared circumsphere or midsphere produces the classical compounds, in which the solids interpenetrate.

Most sequences differ from one another only in degree. A small number differ in kind: they are extremal under one of the measures computed here, or they are the only sequence with a particular contact structure.

Every sequence is a composition of four pairwise fits, read from the inside out. Contact type is fixed by which symmetry axis an inner vertex occupies relative to the outer solid. Octahedron vertices lie on the 2-fold axes, which pass through edge midpoints of the tetrahedron, icosahedron and dodecahedron, so these are vertex–edge contacts, together with the icosahedron on the octahedron's edges at golden-section points. Vertex–vertex contact occurs only where one vertex set is a subset of another: tetrahedron within cube, tetrahedron within dodecahedron, cube within dodecahedron. The remaining thirteen pairs make vertex–face contact.

Partial contact arises where the common symmetry group divides the inner vertex set into orbits at different distances from the outer faces. The dodecahedron's twenty vertices split into the eight of its inscribed cube and twelve others, so only one orbit reaches the tetrahedron (4 of 20), octahedron (8 of 20) or cube (12 of 20). The cube within the tetrahedron touches at 4 of 8.

Each solid's share of the outermost volume is the product of the pair ratios outward from it, and the outermost shell's share depends only on the final pair. That shell is the largest layer in 110 of the 120 nests. The ratio of outermost to innermost volume runs from 9 to 1322.87, a spread of more than two orders of magnitude determined entirely by sequence.

Kepler's sequence ranks in the middle of this distribution, with a volume ratio of 100.94 and partial contact at the dodecahedron–tetrahedron interface. It is not extremal under any measure computed here; the sequence was fixed by the spacing of the planetary orbits.

Inserting a solid between two others normally reduces the size available to the inner one. A three-solid chain is lossless when the composed volume ratio equals the direct pair ratio, so the intermediate solid occupies the gap at no cost. Nineteen of the sixty three-solid chains are lossless. Only two complete sequences are lossless at every step, No. 96 and No. 111, and both traverse the same cycle: tetrahedron, cube, dodecahedron, icosahedron, octahedron.

In No. 111 (cube, dodecahedron, icosahedron, octahedron, tetrahedron) every solid is the maximal coaxial fit inside the outer tetrahedron directly. The cube's eight vertices are eight of the dodecahedron's twenty; the dodecahedron sits in the icosahedron in the dual position, vertices at face centroids; the icosahedron's twelve vertices divide the octahedron's edges in golden section; the octahedron's six vertices lie at the tetrahedron's edge midpoints. With tetrahedron edge E the chain gives octahedron E/2, icosahedron E/(√2·φ²), dodecahedron E/(3√2·φ) and cube E/(3√2). The golden-ratio factors introduced at the edge and face contacts cancel, leaving a cube-to-tetrahedron volume ratio of exactly 1/9.

The cube and dodecahedron share a circumsphere equal to the tetrahedron's insphere, and the octahedron's circumsphere is the tetrahedron's midsphere. No. 96 enters the same cycle at the tetrahedron and has the minimum radius ratio of the set, √3, because two of its four interfaces add no radius.

The tetrahedral symmetry of the tetrahedron and the pyritohedral symmetry the icosahedron and dodecahedron retain on these axes intersect only in the rotation group T. Any nest combining the tetrahedron with the icosahedron or dodecahedron is therefore chiral and has a non-superposable mirror form, obtained by exchanging the two golden orientations. In No. 111 the handedness is visible directly in which end of each octahedron edge the icosahedron vertex approaches.

All five solids are placed on the four 3-fold axes through opposite corners of a common cube. In this orientation each contains the chiral tetrahedral rotation group T, so any pair can be nested symmetrically without relative rotation. The icosahedron, dodecahedron and tetrahedron each have two orientations compatible with these axes, related by a 90° rotation about a coordinate axis. For each ordered pair the inner solid is scaled to the largest size that remains inside the outer solid, with the orientation chosen to maximize that size.

One exception is applied. The icosahedron inside the dodecahedron takes the dual position, vertices at face centroids, which matches Kepler's sphere rule at that interface. A rotated position with vertices on twelve of the thirty dodecahedron edges admits an icosahedron about 12% larger by edge.

The results describe coaxial nests, the configuration that a physical model with shared axes realizes. They are not optimal packings. The unconstrained maximal fits determined by Croft and Firsching exceed the coaxial fits for five pairs:

The individual inscriptions are classical: the cube in the dodecahedron (Euclid XIII.17), the octahedron at the tetrahedron's edge midpoints, the icosahedron on the octahedron's edges in golden section, and the dual icosahedron–dodecahedron pair. Kepler described several in Harmonices Mundi (1619). Sequence No. 28 appears in educational material as a vertex-and-edge inscription chain. A search of the literature did not locate the full coaxial enumeration, the contact classification across all 120 sequences, or the lossless-chain result, though their absence from the sources consulted is not evidence that they are unrecorded.

Every count assumes the five solids share one center and one set of symmetry axes, with each inner solid touching the next. Each level adds one more kind of detail that distinguishes two sets.

Five solids admit 5 × 4 × 3 × 2 × 1 = 120 orders. The tetrahedron, icosahedron and dodecahedron each have two orientations compatible with the shared axes, giving four geometrically distinct nests per order, or two when mirror images are identified.

With every face labeled, each solid takes as many positions as it has rotations: tetrahedron 12, cube 24, octahedron 24, dodecahedron 60, icosahedron 60, equal in each case to the number of faces times the sides per face. The product, divided by the nest's own 12 rotations, gives 2,073,600 labeled sets per nest. Labeling edges or vertices adds nothing: a rigid solid's position is already determined by its faces, and counting by edges (edges × 2) or vertices (vertices × degree) returns the same 12, 24 and 60. Mirror-image labeling doubles each solid's positions and multiplies the total by 2⁵ = 32.

Using two, three or four solids, each at most once, gives 320 nests including the 120 complete ones. Repeating a solid, or tilting any solid off the shared axes, makes the count unlimited: repetition has no end, and tilt angles vary continuously.

This project was a conversation between an artist and an AI model. The solids and their classical inscriptions belong to a long history of geometry.