Hexagonal Earth

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About this

In the four polyhedral methods, the sphere is partitioned into four regions and each region is flattened into a regular hexagon. The chosen four-region net has matching joins. It is extended into a repeating honeycomb; all tiles remain fully opaque and mismatched edges are shown in red. Matching letter pairs identify the intended spherical joins.

All 81 layouts are checked for overlaps and compatible contacts. The infinite display repeats a finite pattern built around the selected net. Neighbor assignments are chosen to improve edge agreement, but an entirely seamless honeycomb is not generally possible with these boundary identifications. At some spherical vertices, two hexagon corners meet; a planar honeycomb requires three. Those unavoidable discontinuities are shown explicitly in red.

One and two hexagons

Inspired by Jacob Rus’s minimal hexagonal maps, these alternatives start with Lambert azimuthal equal area. An area-preserving disk-to-hexagon transformation adjusts both radius and angle. One hexagon contains the full sphere, with its antipode spread around the perimeter; it does not reproduce Rus’s triangular dihedron. Two hexagons contain opposite hemispheres, with all six equatorial edges paired. Cutting unwarped disks into hexagons would omit parts of the globe. These maps preserve area, but not angles.

Lambert azimuthal equal area

The four polyhedral constructions

1. Tetrahedron: each face becomes one hexagon. Its three vertices and three edge midpoints alternate around the boundary; six triangles connect them to the face center.
2. Octahedron: four alternating faces become central triangles. Each remaining face is divided at its centroid into three congruent pieces and distributed to its neighbors.
3. Rhombic dodecahedron: its 12 rhombi are grouped around four alternating cube vertices. Each rhombus is mapped to two equilateral triangles, producing three diamonds per hexagon. This is the 12-face solid referred to in the request.
4. Tetrakis hexahedron: six square pyramids on a cube give 24 triangular faces. Groups of six become four hexagons. Pyramid height controls the surface used for central projection; at a tip distance of 2, adjacent triangles merge into the rhombic dodecahedron.

Projection controls

Central projection interpolates points on the actual planar polyhedral faces and normalizes onto the sphere. Normalized vertex blend first places the vertices on the unit sphere. Shape bias raises triangle barycentric weights to an adjustable power before normalization. Both controls preserve paired borders, but neither provides a general equal-area or conformal guarantee. Longitude, latitude and roll rotate the entire texture together, preserving all joins. Zoom and pan navigate the flat map.

Distortion overlays

Area color shows relative map-area inflation: blue is smaller, white is average scale, and amber is larger. Purple shows maximum angular deformation derived from the ratio of local principal stretches (Tissot axes). The combined view blends purple into the area color. Estimates use local derivatives of the inverse spherical projection, normalized by the total area of the projection’s hexagons. They describe the projection, so rotating the globe moves continents through a fixed distortion field. Cut discontinuities are not included.

PROJ · areal scale, angular distortion and Tissot scale factors

Map layers & ecological classes

Blue Marble uses the supplied bluemarble-high.jpg image, with brighter oceans and visible seafloor detail. Country outlines and color groups come from Natural Earth Admin 0, 1:50 million, with its default de facto boundaries. The continent-cut search continues to use the supplied silhouette mask for every map source.

Land classes aggregate the Leemans / Holdridge dataset distributed by UNEP-WCMC (1992, based on 1931–1960 climate; approximately 0.5° resolution). This represents potential climatic vegetation, not present-day land cover. Missing land cells borrow the nearest available land climate on the source raster. These filled cells are estimates, not observations. Antarctic land south of 60°S is explicitly assigned to the Polar group. The 3/6/10/15 presets form triangles with fewer moisture distinctions in colder climates; each legend swatch lists its source classes.

Marine classes use the sampled frequency of significant wave height above 2 metres. Copernicus WAVERYS, 2015–2024, sampled every 7 days 3 hours at 0.8°. Classification boundaries are 10%, 25%, 50% and 75%. Wave-exposure distinctions merge toward colder water as a design choice. Hue indicates temperature; within a temperature row, darker colours indicate greater wave exposure. Shared climate labels describe illustrative sea-surface-temperature bands, not marine Holdridge zones. Missing marine temperature and wave values borrow the nearest available ocean value on the source raster. This is not a navigation safety rating.

Wave exposure provenance · Dataset provenance and processing metadata · NASA Blue Marble · Holdridge documentation · Bathymetry · NOAA OISST

Site & privacy

Created by Alex Van de Sande. Source code · CC BY 4.0

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Literature

Alex Van de Sande · Gosper World (Bridges, 2024)
Rhombic dodecahedron, four hexagons, and the discontinuities of repeated layouts.

Carlos A. Furuti · World Map on a Rhombic Dodecahedron (2014)
Gnomonic rhombic dodecahedron map cited and used as the starting point for the globe projection in Gosper World.

Jacob Rus · Flowsnake Earth (Bridges, 2017)
Hexagonal networks folded onto an octahedron and tetrakis hexahedron.

B. J. S. Cahill · Octahedral cartography
Historical background for triangular octant maps; this app implements the described regrouping, not Cahill’s conformal formula.

The supplied continent image is used as the texture. Missing Lifezones data is filled from neighboring source values while preserving the land/sea mask.