Nested Countries · Interactive Explorer

Data

• Source: Natural Earth Admin 0 Countries (1:10m) — the public-domain Natural Earth 1:10m Admin-0 countries layer.
• Geometry mode: mainland — only each country's largest contiguous landmass is kept (islands/exclaves dropped), which makes nesting strictly harder but unambiguous.
• Each country is re-projected into its own local Lambert azimuthal equal-area CRS (centred on the country, units of km) so that areas and shapes are faithful where it matters. A single global projection would distort comparisons badly.
• “Sovereign states only” uses Natural Earth’s feature TYPE to hide dependencies and territories.

Containment method

For each ordered pair we ask: can the inner country be rotated and translated to sit strictly inside the outer one? The solver runs a staged search:

• Safe rejection — if the inner’s area or diameter exceeds the outer’s, it is provably impossible (the only certain “no”).
• Rotation sweep — orientations in steps of 1.0°, visited in a spread order so a capped search still samples the whole circle.
• Translation search — candidate positions on a 35.0 km interior grid (auto-refined for small countries), then local refinement to maximise clearance.

A pair is an edge only if a concrete placement clearing 1.0 km is found. A missing edge is search-dependent, never a proof of impossibility.

Computation

• All 33,153 ordered country pairs were evaluated (only inner-smaller-than-outer pairs are tested).
• For speed the search ran on size-adaptive simplified geometry (~25.0 km tolerance), capped at 300 placement tests per pair; rotation enabled.
• Every discovered placement is then re-validated on the full-resolution geometry, so each of the 24,682 edges shown genuinely respects the strict buffer.
• The valid edges form a directed acyclic graph (area always decreases). The longest chain is the longest path, found exactly by dynamic programming.

How to read the explorer

• Countries are drawn nested inside the first, using the exact rotation/translation the solver found; the outermost is faint, inner ones brighter.
• Reorder with ▲▼, remove with ✕, and click a country to hand-place it: rotate by 1.0° and nudge by 35.0 km. Green = inside, red = overlaps.
• Build step-by-step guides you one country at a time: it lists every country that fits inside the current last one (with its clearance). Click to extend, click a breadcrumb step to jump back, or use Back / Clear. The size ladder and nesting map update as you go.