Myriahedral geometric context

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One method, many maps

“Myriahedral” describes a family of polyhedral world maps made from a very large number of faces. The name combines the idea of “myriad” faces with a polyhedron. Unlike equirectangular projection, there is no single equation and rectangle that uniquely identifies every Myriahedral map.

A configuration answers two separate questions:

1. How is the globe divided into small faces, and how is a point mapped within one face?
2. Which shared edges remain hinges, and which edges are cut so the surface can unfold?

Changing the cut tree can move an entire connected branch without changing the local geometry inside any triangle. This implementation fixes both the mesh and tree so every call has a deterministic answer.

From globe to planar net

flowchart LR
  G[Unit sphere] --> I[20-face icosahedron]
  I --> M[Four subdivision rounds<br/>5120 spherical triangles]
  M --> D[Dual graph<br/>one node per face]
  D --> T[Land-aware minimum<br/>spanning tree]
  T --> U[Unfold retained hinges<br/>cut all other edges]
  U --> C[Center in a 16:9 canvas]
  C --> P[Scale to frame.frame_area]

The spherical triangle mesh is the primal graph. A second, conceptual dual graph puts a node at every triangle and connects nodes whose faces share an edge. The fixed minimum spanning tree has all 5120 face nodes but only 5119 connections. Tree connections are hinges; every omitted dual edge corresponds to a cut in the primal mesh.

Because a tree has no cycles, opening its hinges cannot encounter a conflicting closure constraint. The result can branch freely across the plane.

The depth-5 icosahedral mesh

A regular icosahedron starts with 20 nearly equilateral spherical regions. Each face is recursively divided into four by placing normalized points at its edge midpoints:

                 p1
                 /\
                /  \
               b----c
              / \  / \
             /   \/   \
            p0---a----p2

The four children are (p0,b,a), (b,p1,c), (a,b,c), and (a,c,p2). Four rounds turn each original face into 256 small faces, for 5120 total.

The triangles are planar chords whose vertices lie on the unit sphere. The projection sends a ray from the sphere center through each geographic point to the chord plane associated with its spherical region. A finer mesh reduces the visible distortion of this piecewise gnomonic transform.

Why land influences cuts

If cuts were chosen without geographic information, they could split large land masses into many disconnected pieces. The historical myriaworld configuration estimates how much land occupies each face, smooths that signal across neighbors, and weights the dual graph. Prim’s algorithm then favors a tree whose retained connections keep land together while allowing cuts to travel mainly through ocean.

This is a preference, not a hard topological guarantee. Islands, narrow isthmuses, the date line, and the finite Natural Earth source geometry all affect the result. The 5120-face tree embedded here is the completed decision; land geometry is not consulted during an ordinary point projection.

Local coordinates within a face

Every spherical face and its unfolded planar copy have the same ordered three vertices:

Spherical chord face              Planar face

       p1                              q1
      /  \                            /  \
     / g  \      central ray         / q  \
    /______\          -->            /______\
   p0      p2                       q0      q2

The ray through the 3D direction g intersects the face’s chord plane. Triple products provide barycentric weights (w0,w1,w2) for that intersection, and the same weights place q in (q0,q1,q2). On a shared great-circle edge the opposite weight is zero, so neighboring faces agree on a retained hinge. Along a cut, each face owns a separate planar copy of the shared edge.

This deliberately differs from the earlier implementation, which projected g orthogonally into a chord-plane basis. That approximation introduced a small false hinge discontinuity and could place a point outside its selected planar triangle. The implementation notes preserve the measurements and the reason for choosing the cleaner gnomonic construction.

Geographic quadrants

Latitude and longitude still describe the usual four geographic quadrants:

Those quadrants are useful for checking input orientation, but they are not four rectangular regions of the output. The cut tree, not signs of latitude and longitude, decides which planar branch contains a point. Two nearby coordinates on opposite sides of a cut can be far apart in (x,y), while points from different geographic quadrants may lie on adjacent branches.

The API test therefore exercises more than one point per quadrant. It includes the center, near-poles, both sides of the antimeridian, southern-ocean seam probes, and these city anchors:

New York     Los Angeles   Paris       Durban
Delhi        Tokyo         Sydney      Moloka'i
São Paulo    Reykjavik     Suva        Villa Las Estrellas

Together these locations cover every sign combination, extreme latitudes, the Pacific date-line neighborhood, and several narrow branches of the net.

Cuts, hinges, and boundary choices

Consider two adjacent geographic faces:

Retained hinge                    Cut edge

   /\  /\                         /\       /\
  /__\/__\       unfolds         /__\     /__\
  shared edge                    separate planar copies

Across a retained hinge, the projected coordinate is continuous. Across a cut, the map is intentionally discontinuous. A point exactly on a cut belongs to both spherical faces and therefore has two valid planar images. The C++ search uses stable face order to choose one. This is analogous to selecting one side of the antimeridian in a conventional cylindrical projection.

Longitude -180 and +180 are canonicalized to the same input before this choice, so spelling the same geographic meridian two ways cannot change the selected branch.

Axes and screen coordinates

Unfolding happens in an ordinary mathematical plane:

             +qy
              ^
              |
              +----> +qx

Rendering uses an upper-left origin:

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

After unfolding, the implementation applies the fixed 335-degree rotation, fits the whole net uniformly inside a normalized canvas, and reverses the vertical coordinate. Scaling by the frame width and height produces the public (x,y) result.

The 16:9 canvas

The checked-in source raster is 4480 pixels wide by 2520 pixels high. Its canvas ratio is exactly:

4480 / 2520 = 16 / 9

The net is fitted with one scale factor and centered; unused space is retained instead of stretching the geometry. Any other supported frame must preserve that ratio—for example 1600 x 900, 1920 x 1080, or a frame::area {16*h/9, h}.

This is the required ratio of this raster-registered configuration, not a law of all Myriahedral projections. Another cut tree can have different raw bounds, and another rendering can choose a different crop.

What is and is not preserved

The small-face construction aims to keep local distortion low and to move interruptions away from important land connections. It should not be described as an exact global equal-area or conformal transform:

• every face uses a finite central gnomonic approximation;
• cuts introduce intentional global discontinuities;
• distance between different branches has no direct geographic meaning;
• orientation changes abruptly only at cuts, but derivative behavior changes at every face boundary;
• using more, smaller faces generally reduces within-face distortion.

The method’s visual strength is different: it turns the single long seam of a conventional projection into a configurable branching set of mostly oceanic cuts while preserving recognizable land groupings.