Dymaxion geometric context

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Which Dymaxion map this implements

“Dymaxion map” names more than one historical object. Fuller’s 1943 map used a cuboctahedron. The familiar later map, developed with Shoji Sadao and published in 1954 as the Airocean World Map, uses a specially oriented icosahedron. This implementation is the later icosahedral form.

Its geometric pipeline is:

flowchart LR
  G["Spherical Earth<br/>latitude and longitude"]
  I["Fuller-oriented icosahedron<br/>20 spherical facets"]
  E["Exact Fuller transform<br/>one equilateral triangle per facet"]
  C["Ocean-aware cuts<br/>Australia and Japan splits"]
  N["Horizontal Airocean net<br/>23 drawable triangles"]
  F["Variable a60::carto::frame<br/>upper-left screen axes"]

  G --> I --> E --> C --> N --> F

The polyhedron is a mathematical organizer. The implementation does not construct a solid, render it in 3D, or project through a camera. It directly selects a spherical facet and evaluates the corresponding planar equations.

Why an icosahedron

A regular icosahedron has 20 congruent equilateral faces, 30 edges, and 12 vertices. Projecting its edges radially to the sphere divides the globe into 20 congruent spherical triangles. Each triangle is small enough that its flattening error is distributed among many facets instead of accumulating at one global seam or pole.

regular icosahedron

vertices:       12
edges:          30
faces:          20
edges per face:  3
faces at vertex: 5

The central angle of each spherical edge is:

2 asin(sqrt(5-sqrt(5)) / sqrt(10))
  = 63.43494882292201 degrees

Fuller’s orientation is as important as the choice of solid. It places most facet cuts through ocean, allowing the continents to read as one connected world island in one world ocean. The result avoids treating north as an obligatory top or any one meridian as an obligatory center.

The unfolded net

The gray triangles are the native drawable pieces. Eighteen correspond one-to-one with complete icosahedron faces. Two parent faces are cut again in the plane so that land is not forced across an exterior map cut:

• one parent face becomes two Australia pieces; and
• one parent face becomes three Japan pieces.

Thus the flat drawing has 23 triangles while the spherical polyhedron still has 20 faces. A subface does not introduce a new spherical projection model; it uses its parent’s local Fuller transform and selects another placement in the net.

The topological distinction is visible in the outline:

• a hinge is an edge where neighboring planar triangles remain attached;
• a cut is an icosahedron or subface edge whose two sides appear in different places on the page; and
• a split copy is a portion of one parent face moved to preserve a land connection elsewhere.

Adjacent geographic points must not be joined blindly. Two points can be very close on the globe and far apart on the flat map when their shared spherical edge is a cut.

The local face picture

For one complete spherical face with vertices q0, q1, and q2, the implementation builds a local orthonormal frame:

                         z
                         ^  face-center direction
                         |
                         o----> y  toward q0
                        /
                       x       y cross z orientation

Every point in the selected face is expressed as (u,v,w) in that basis. The threefold symmetry of an equilateral triangle then supplies three equivalent arc-distance coordinates, one associated with each side.

flowchart TD
  P["Geographic unit vector p"]
  B["Local face components<br/>u, v, w"]
  D["Three symmetric distances<br/>a1, a2, a3"]
  T["Canonical equilateral point<br/>X, Y"]
  R["Face rotation, common scale,<br/>and net translation"]

  P --> B --> D --> T --> R

The canonical transform is the same for every complete face. Only the local basis and final face placement change.

Why the transform is not gnomonic

A gnomonic face projection draws a ray from the center of the sphere to the flat icosahedron face. It is simple, and it maps great circles to straight lines, but its scale along an icosahedron edge is not uniform: equal spherical arc intervals crowd differently near the edge midpoint and vertices.

The Fuller/Gray transform instead computes spherical arc distances and uses those distances in the planar equilateral triangle. Its defining boundary property is:

fraction of spherical facet edge
    = fraction of corresponding planar facet edge

The two methods share the same icosahedron orientation and the same vertex positions, so coast-only images can look remarkably similar. Their graticules and numeric coordinates are nevertheless different. This implementation uses the exact Fuller equations; PROJ’s current Airocean projection is used for its permissively licensed face/net registration, not as the face-transform formula.

Reading the Earth result

White page regions are outside the unfolded polyhedron. Blue regions are pieces of one connected spherical ocean, separated only because the icosahedron has been opened and laid flat. Beige land uses the same cuts, but the selected aspect avoids the most disruptive continental separations.

The phrase “one island in one ocean” is a way to read the relationships in this aspect, not a topological claim that the paper outline is itself one uncut polygon. Some ocean cuts are necessarily prominent because flattening a closed sphere requires an interruption tree.

There is no universally privileged up direction. The checked-in aspect uses the official net’s horizontal orientation because it fits print and video carriers efficiently. Rotating the entire page would not change the projection; selecting a different hinge/cut arrangement would define a different aspect of the same icosahedral projection.

Graticules, poles, and facet edges

The graticule makes several properties visible:

• meridians and parallels are complex curves assembled from face-local pieces;
• lines break at exterior facet cuts but remain continuous across hinges;
• neither pole lies at a net vertex; both are ordinary points inside facets;
• no meridian or parallel is generally identical to a facet edge; and
• curve direction can change at a retained facet boundary because the two planar triangles have different orientations.

The implementation detects each face transition in geographic space, bisects it, and compares the two limiting planar positions. This uses topology rather than an arbitrary visual jump threshold.

Facets versus quadrants

Dymaxion’s mathematical units are facets, not geographic quadrants. A north/south-east/west partition would cut across the Fuller-oriented icosahedron and has no role in the forward formula.

The geometry artifact nevertheless contains four blue screen quadrants. They are the generator’s common diagnostic carrier partition, shared by all projections. For the standard 44-inch Dymaxion frame they are:

These rectangles are useful for print inspection and possible carrier slicing, but they cut through native triangles and have no stable continent, hemisphere, or face identity. A seam-safe Dymaxion slice should be specified in terms of complete face/subface masks or explicit clipping geometry, not by assuming those four bands are geographic quadrants.

There are also ordinary signs of the local (u,v) coordinates inside each face, but calling those four sign combinations “quadrants” is not helpful: the exact formula is threefold symmetric and is more naturally described by its three edge distances or six 60-degree local sectors.

Aspect ratio from the triangular lattice

Let e be the native planar edge length. The selected horizontal net is 11/2 edges wide and three equilateral altitudes high:

width  = (11/2)e
height = 3(sqrt(3)/2)e

width / height = 11 / (3sqrt(3))
               = 2.116950987028628...

Uniformly enlarging this lattice is harmless. Forcing it into 2:1, 16:9, or another convenient rectangle would stretch every face in one direction, adding anisotropic distortion unrelated to the Fuller projection. That is why the public constructor validates the aspect ratio rather than silently resizing axes independently.

Distortion and intended use

The Fuller projection is a compromise projection:

• it is not conformal;
• it is not equal-area;
• most distances and directions are not preserved; and
• its facet edges do have correct uniform scale.

Distortion is limited spatially because every location is within one small icosahedron face. Esri describes distortion as increasing away from facet edges. This differs from many familiar world projections, where error is organized around one pole, equator, or central meridian.

The heavily interrupted net is intended for the entire globe and for seeing relationships among all land masses. A regional street, navigation, or continuous ocean route map usually benefits from a projection with a single local seam and a directly useful inverse.

Naming and trademarks

These documents use “Dymaxion” for the familiar 1954 icosahedral Fuller/Sadao map and “Airocean net” when referring specifically to its unfolded aspect. The Buckminster Fuller Institute identifies Dymaxion, Spaceship Earth, and Fuller Projection Map as BFI trademarks and invites licensing inquiries. Repository filenames are technical identifiers; they do not imply ownership, certification, sponsorship, or endorsement by BFI.

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