Icosahedral Voronoi geometric context

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What kind of projection this is

This projection replaces one global plane, cylinder, or cone with twenty small tangent planes. A regular icosahedron divides the spherical Earth into twenty triangular regions. The center of each triangle is a Voronoi site; a geographic point uses whichever site is nearest on the sphere. A gnomonic projection maps that one region into a planar triangle, and a fixed tree says how the twenty triangles remain hinged when the surface is unfolded.

flowchart LR
  G["Geographic point<br/>latitude and longitude"]
  R["Rotate longitude<br/>by 108 degrees"]
  V["Nearest of 20<br/>spherical sites"]
  P["Face-centered<br/>gnomonic plane"]
  T["19-edge unfolding tree<br/>11 cut adjacencies"]
  D["D3 registration<br/>scale, center, translation"]
  F["48:25 frame<br/>screen coordinates"]

  G --> R --> V --> P --> T --> D --> F

The stages are evaluated directly from formulas and fixed arrays. The C++ implementation does not build a physical solid or consult a polygon mesh at each call.

This map should not be confused with the repository’s Myriahedral projection. Both start from an icosahedron and unfold a tree, but their working scales and local maps differ:

Twelve vertices and twenty faces

An icosahedron has twelve vertices, thirty edges, and twenty triangular faces. This orientation puts one vertex at each geographic pole and the other ten on two staggered latitude rings:

                         v0  North Pole, +90 degrees
                         /|\

 northern ring, +theta: v11   v3   v5   v7   v9
                         \   zigzag face belt   /
 southern ring, -theta:  v10   v2   v4   v6   v8

                         \|/
                         v1  South Pole, -90 degrees

 theta = atan(0.5) = 26.56505117707799... degrees

The sketch shows the layers, not a perspective drawing or the exact cyclic longitude order. The coordinates are:

Five faces meet at each pole. Ten more triangles alternate around the band between the two rings. The chosen latitude makes every chord edge the same length, so all twenty spherical triangles are congruent under icosahedral symmetry.

Face IDs 0–4 touch the north pole, 5–14 form the equatorial belt, and 15–19 touch the south pole. These IDs are stable implementation topology; they are not geographic quadrants or drawing layers.

Why the faces are Voronoi cells

For each triangular face, normalize the sum of its three vertex vectors. The result is a unit vector s[i] pointing through the spherical face center. These twenty centers are the sites of a spherical Voronoi diagram.

For a geographic unit vector g, the nearest site’s angular distance is:

distance(g,s[i]) = acos(g dot s[i])

The selected cell can therefore be found by the largest dot product. Its boundaries occur where two site dot products are equal:

g dot s[i] = g dot s[j]

Each equality traces a great-circle bisector. Three bisectors bound one spherical triangular cell, and five cells meet at every original icosahedron vertex.

There are two useful dual views of the same structure:

• the original icosahedron has 20 triangular faces whose centers are the sites;
• connecting neighboring sites produces the dodecahedral face-adjacency graph, with 20 nodes, 30 edges, and degree 3 at every node.

“Voronoi” here refers to this nearest-site face dispatch. It does not mean the output contains arbitrary planar Voronoi polygons or that callers can supply their own sites.

One face and its gnomonic plane

The face site s is also the outward normal of a plane tangent to the unit sphere. A ray from the sphere’s center through geographic point g meets that plane at the gnomonic image:

                         tangent plane at s
                         +------------------+
                        /        p         /
                       /        *         /
                      +--------/---------+
                              /
                         g   *       unit sphere
                            /
                           /
                          O  sphere center

                    ray O -> g -> p

With tangent basis vectors east and north, the scale along the ray is the reciprocal of g dot s:

p = ((g dot east) / (g dot s),
    -(g dot north) / (g dot s))

This central projection has an important cartographic property: every great-circle arc maps to a straight line on a gnomonic plane. Since every cell is much smaller than a hemisphere, all of its points have a positive denominator and a finite local image.

The benefit is straight triangular edges and a clean hinge construction. The cost is distortion that increases with angular distance from the face site. Gnomonic projection is neither equal-area nor conformal, and joining twenty planes does not change that local fact.

From face adjacency to a planar net

The full face-adjacency graph has thirty edges. Keeping all of them while flattening would impose cycles that cannot close in a plane. The selected spanning tree retains nineteen edges as hinges and cuts the other eleven.

The exact parent tree is:

flowchart TD
  F0["0 root"] --> F5["5"]
  F5 --> F6["6"]
  F6 --> F7["7"]
  F6 --> F15["15"]
  F7 --> F1["1"]
  F7 --> F8["8"]
  F8 --> F9["9"]
  F8 --> F16["16"]
  F9 --> F2["2"]
  F9 --> F10["10"]
  F10 --> F11["11"]
  F10 --> F17["17"]
  F11 --> F3["3"]
  F11 --> F12["12"]
  F12 --> F13["13"]
  F12 --> F18["18"]
  F13 --> F4["4"]
  F13 --> F14["14"]
  F14 --> F19["19"]

For each arrow, the child and parent project their two common icosahedron vertices into their own tangent planes. A similarity transform rotates, uniformly scales, and translates the child until those two edge images coincide. Composing transforms back to face 0 places every triangle in one root plane.

retained hinge                       cut adjacency

       /\  /\                          /\       /\
      /__\/__\                        /__\     /__\
       one shared                       two separated
       planar edge                      edge copies

Across a hinge, the two faces agree exactly on position along the shared edge. Across a cut, nearby spherical points can be far apart in the map. That discontinuity is necessary to open a closed surface into a plane.

Geographic rotation and map registration

The icosahedron above is defined around the ordinary north/south axis, but the public map yaws geography by 108° before choosing a face. This moves the net cuts relative to continents without changing latitude or the icosahedron’s shape.

After unfolding, the default D3 settings apply:

geographic center = [162 degrees, 0 degrees]
scale             = 131.777
translation       = [480, 250]
planar angle      = 0 degrees

Rotation and center work together. Public longitude 54° E becomes 162° E after the 108° yaw, so public (0°,54°) maps to (480,250). Public (0°,0°) maps to approximately (349.228,250).

The C++ implementation keeps this complete coordinate registration, then scales it proportionally to the requested frame.

Geographic quadrants

Latitude and longitude define the familiar four input quadrants before the registered rotation:

                         north, +latitude
                                ^
                                |
       northwest (NW)           |           northeast (NE)
       -longitude,+latitude     |           +longitude,+latitude
                                |
 west, -longitude <-------------+-------------> east, +longitude
                                |
       southwest (SW)           |           southeast (SE)
       -longitude,-latitude     |           +longitude,-latitude
                                |
                                v
                         south, -latitude

Reference anchors at the 960 x 500 registration illustrate argument order, signs, and screen orientation:

These are validation groups, not the projection’s working regions. One geographic quadrant intersects several of the twenty cells and may cross several net cuts. Conversely, faces containing different longitude signs can be adjacent after the 108° rotation. There is no rule that maps NE, NW, SE, and SW into four rectangular quarters of the output.

Points on latitude 0° or longitude 0° lie on quadrant axes rather than in one open quadrant. Their face choice still follows the same nearest-site rule.

Boundaries, ties, and the antimeridian

A point on a Voronoi boundary is equally close to two sites; a point at an icosahedron vertex is tied among five. The C++ scan keeps the lowest face index in such a tie. At the geographic poles, explicit handling chooses the same stable faces without allowing floating-point longitude residue to break the tie.

If the chosen boundary is a retained hinge, either face gives the same planar edge position. If it is a cut, both separated edge copies are geometrically valid and the tie rule selects one. A forward point API can return only one of those images.

Public longitudes -180° and +180° name the same meridian. Both rotate to -72°, and tests require exactly equivalent projected coordinates from pole to pole. This input equivalence does not remove the net’s other interrupted edges; the map has eleven polyhedral cuts rather than one cylindrical date-line seam.

Screen axes and the 48:25 canvas

The public coordinate system follows SVG and raster screen convention:

  (0,0) +------------------------------> +x
        |
        |
        |
        v
       +y

D3’s default translation [480,250] is the center of a 960 x 500 area:

960 / 500 = 48 / 25 = 1.92

The C++ frame contract preserves that entire area, including intentional whitespace around and between net branches. Valid examples include 48 x 25, 960 x 500, and 1920 x 1000.

This ratio comes from the selected registration canvas, not polyhedral geometry. A different parent tree, rotation, scale, translation, or crop could require another ratio even while using the same regular icosahedron.

What is and is not preserved

• Great-circle portions are straight within each gnomonic face.
• Positions agree across the nineteen retained hinges.
• The eleven cut adjacencies are intentional global discontinuities.
• Scale, area, angle, and distance are not globally preserved.
• The derivative can change at a face boundary even when position is continuous there.
• Distance between points on separated net branches has no direct geographic meaning.
• A renderer must split a path when it crosses a cut; connecting projected endpoints blindly can draw a false line across the map.

The design trades one continuous global formula for twenty simple local projections and an inspectable cut topology. Its useful invariant is faithful compatibility with D3’s fixed icosahedral view, not a claim of equal-area or conformal behavior.

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