Lay out the faces of a grid's solid in the plane

Builds a flat map of the sphere from the faces of the solid an ISEA grid is built on: the face projection of the grid (Snyder's equal-area projection, or Fuller's) maps the sphere onto the faces, and the layout cuts the faces into pieces and places each piece in the plane by a rotation and a translation (the rhombic layout also shears). Draw a grid on it with plot(grid, surface = "net", layout = ), and place points on it with net_project.

Usage

net_layout(
  x,
  layout = c("plane", "gosper", "gosper_flower", "gosper_land", "rhombic", "land"),
  centre = NULL,
  mirror = FALSE,
  land = TRUE
)

Arguments

x: A HexGridInfo object from hex_grid, built on the icosahedron or the octahedron.
layout: Name of the layout; see Details.
centre: For the Gosper layouts, the point c(lon, lat) whose tile lies in the middle, at the origin. The map is turned so the hexagons have horizontal top and bottom sides, with north at that point within 30 degrees of up the page. NULL uses the north pole.
mirror: For the Gosper layouts, TRUE builds the hexagons on the other four faces of the octahedron, the mirror image of the default tiling.
land: For layout = "land" and "gosper_land", the land the net keeps joined: TRUE for the built-in hexify_world, or an sf/sfc object of polygons.

Value

A hexify_net object: a list with the layout's name, the grid's solid in polyhedron, the icosa argument of the grid, and pieces, one list per placed piece with its face (from 0), its region (three corners in the face's triangle coordinates), their labels, the affine map A and b placing a point t at A t + b, and the tile it belongs to.

Details

The layouts are:

"plane": The faces in the PLANE layout of DGGRID, the layout hexify_cell_to_plane reads: five strips of four triangles on the icosahedron, four diamonds side by side on the octahedron.
"gosper": Van de Sande's Gosper World on the octahedron: four regular hexagons, each one face of the octahedron and a third of each of its three neighbours, the three neighbouring faces cut along the lines from their centres to their corners. The tile holding centre lies in the middle and the other three are attached on alternate sides.
"gosper_flower": The tile holding centre surrounded by the six tiles that border it, so each of the other three tiles appears twice (Van de Sande 2024, Fig. 5). The map is continuous across the sides of the middle tile; the three seams run out from its corners that are vertices of the octahedron.
"gosper_land": The four Gosper tiles joined along the three hexagon sides, out of the twelve, that cross the most land of land, so the continents stay whole where they can, as in Van de Sande's Fig. 1 (2024).
"rhombic": DGGAL's 5 x 6 rhombic space on the icosahedron (Jacovella-St-Louis et al. 2025): each face sheared onto half of a unit square, the ten diamonds a staircase of squares. Areas keep one constant. DGGAL's y axis points down the page, so a point at DGGAL's (x, y) is placed at (x, -y).
"land": A net cut along the solid's edges that cross the least land of land, so the faces stay joined across land where they can, in the spirit of Fuller's Dymaxion map, which also splits two faces. With hex_grid(orientation = "dymaxion", projection = "fuller") it is a Dymaxion-style map of whole faces.

On the octahedron with Snyder's projection the Gosper World is equal-area, where Van de Sande built it from a gnomonic projection onto a rhombic dodecahedron. His poles lie at midpoints of octahedron edges, which is hex_grid(polyhedron = "octahedron", orientation = "gosper"); Rus's four-hexagon map puts them at octahedron vertices, which is the standard orientation of the octahedron.

References

Van de Sande, A. (2024). Gosper World: A Hexagonal Map Using Gosper Fractals. Bridges 2024 Conference Proceedings, 507-510.

Rus, J. (2017). Flowsnake Earth. Bridges 2017 Conference Proceedings, 237-244.

Jacovella-St-Louis, J., Padilla Ruiz, M., Moreira de Sousa, L., Peterson, P., Stefanakis, E. (2025). Interoperable global and local indexing of discrete global grid systems based on the ISEA projection for efficient storage, processing and transmission. Abstracts of the International Cartographic Association 10, 126. doi:10.5194/ica-abs-10-126-2025

Examples

grid <- hex_grid(resolution = 3, aperture = 4, polyhedron = "octahedron",
                 orientation = "gosper")
world <- net_layout(grid, "gosper")
plot(grid, surface = "net", layout = world, seams = TRUE)
plot(grid, surface = "net", layout = "gosper_flower", land = TRUE)