Spherical grid prototypes

Two experiments in building a Civilization-style 4X world on a true sphere, and in keeping keyboard navigation comprehensible to a human player while doing it. Both share the same rule: the sphere is authoritative, and the renderers are diagnostic views over one topology graph. Neither is allowed to fudge adjacency to make the picture easier.

A sphere cannot be tiled by a regular grid. Both worlds therefore pay the same debt — 720° of total angular defect — but they pay it in different currency.

Hex sphere

Subdivided icosahedron. At frequency F there are 10F²+2 tiles, of which exactly 12 are pentagons.

The defect lands on tiles: twelve of them have five neighbours instead of six, and standing on one breaks the pointy-top invariant outright.

12 defects × 60° · movement on numpad 7/9/4/6/1/3

Square sphere

Cube-to-sphere after Red Blob Games. Six faces of N×N squares give 6N² tiles, and every tile has four neighbours.

The defect hides on vertices instead: the eight cube corners are places where three squares meet rather than four. No tile is ever irregular.

Includes the draggable flat map from the article: inside a face the grid is exactly regular, and all the distortion is pushed to the seams.

8 defects × 90° · movement on numpad 8/4/6/2

What to compare

Where the defect shows. On the hex world you can stand on a pentagon, and the tile under you is visibly wrong. On the square world you can only ever stand next to a cube corner — the tile you occupy is always an ordinary quad. That is the square grid's real prize: no game rule ever needs a "what if I'm on a pentagon" branch.
The price: square tiles are sheared. A quad-sphere tile is only a true rectangle near a face centre. Elsewhere it is a rhombus, and no camera roll can fix it. Sweeping every possible roll and taking the best achievable alignment gives a median irreducible error of 3.9° per tile, up to 13.4°. The hexasphere's ordinary tiles stay near-regular essentially everywhere, at a measured median of 0.4°. Both demos report this as Axis error: X° (floor Y°) — Y is the tile's shape, and only the gap is the camera's fault.
The corner tiles have a hard floor of 15°. Three squares share a cube corner, so that vertex opens to 120° instead of 90°, leaving exactly (120−90)/2 of unavoidable skew on all 24 tiles around it. It is the quad sphere's version of "a pentagon has no opposite edge pair".
How big each defect is. Fewer, larger defects (8 × 90°) versus more, smaller ones (12 × 60°). The square world's corners are rarer but more violent to cross.
Camera roll. A hexagon's six-fold symmetry means the worst correction needed to align an edge pair vertically is 30°; a square's four-fold symmetry pushes that to 45°.
Orientation can be solved outright. The obvious cameras both disappoint: parallel transport drifts 57.8° around a closed loop, so the same tile faces differently depending on your route, and north-up leaves the tile under you a median 18.5° off pointy-top. But a hex camera only needs a direction mod 60°, and that weaker object has a global, path-independent solution — read straight off the icosahedral lattice, since a triangular lattice's three axes are 60° apart and adjacent faces share an axis along their common edge. The hex page's default grid-locked mode uses it: zero drift, and a median vertical-edge error of 0.47° against a shape-imposed floor of 0.31°. The sphere's 720° debt reappears as an index-⅙ singularity on each of the 12 pentagons — 12 × ⅙ = 2 = the Euler characteristic, so the pentagons and the orientation singularities are the same twelve points.
A flat map is possible — but only five faces at a time. The square world can be drawn as an exact regular grid, Google-Maps style, by laying the current cube face flat with its four edge neighbours rotated into place. It is the most playable view either world has. The catch is measurable: that layout draws every adjacency correctly except the four seams between the surrounding faces, which is why the four diagonal quadrants are black — three faces meet at a cube corner, so there is no fourth face to put there. The hex world has no equivalent view at all; its flat-projection tab is a labelled failed experiment.
Tile size uniformity. The hex world offers Lloyd relaxation; the square world offers the tangent adjustment, which drops the spread in tile size from 2.06× (plain gnomonic) to 1.16×. Toggle it on the square page and watch the face edges.
Diagonals. The quad grid has four neighbours, not eight — a real gameplay difference from the hex world's six, and the reason its numeric keypad only lights up in the cardinal directions.
The cube net. Each cube face is internally a perfect, regular N×N grid — something the hexasphere can never offer. The square world's third tab shows all six, and the seams between them are exactly where face-local axes disagree.

Design notes, measurements and the reasoning behind the orientation compromises live in AGENTS.md in the repository.