Waterman butterfly projection
Encyclopedia
The Waterman "Butterfly" World Map is a map arrangement created by Steve Waterman
. Waterman first published a map in this arrangement in 1996. The arrangement is an unfolding of a globe
treated as an octahedron
, evoking the butterfly map principle first developed by Bernard J.S. Cahill
(1866–1944) in 1909. Cahill and Waterman map
s can each be shown in various profiles, typically linked at the north Pacific
or north Atlantic
oceans.
As Cahill was an architect
, his approach tended toward forms that could be demonstrated physically (i.e. his flattenable rubber-ball map). Waterman, on the other hand, derived his design from his work on close-packing of spheres. This involves the interpretation of a spherical extraction from cubic closest-packed spheres into a corresponding convex hull, as demonstrated in the following pictures.
Then, for its projection
, straight lines are used to define each 5 × 5 section onto this convex hull. According to annotations on modern versions of the map, the projection divides the equator
equally amongst the meridians. Latitude
s are drawn in three straight-line sections in each octant
: from pole
to fold-line; from fold-line to largest line parallel to equator; and then from there to the equator. The largest line parallel to the equator also has equal-length delineations. Waterman chose a specific Waterman polyhedron
and central meridian to avoid interrupting land masses.
Like Buckminster Fuller
's 1943 Dymaxion Projection, an octahedral butterfly map can show all the continents uninterrupted if its octants are divided at a suitable meridian
(in this case 20° W) and are joined, for example, at the North Atlantic, as in the 1996 version.
Steve Waterman
Steve Waterman is a producer for both movies and television.He was co-producer for Casper and executive producer for Stuart Little and Stuart Little 2.He is currently producing the upcoming live-action/CGI animated Alvin and the Chipmunks movie....
. Waterman first published a map in this arrangement in 1996. The arrangement is an unfolding of a globe
Globe
A globe is a three-dimensional scale model of Earth or other spheroid celestial body such as a planet, star, or moon...
treated as an octahedron
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
, evoking the butterfly map principle first developed by Bernard J.S. Cahill
Bernard J.S. Cahill
B.J.S. Cahill , cartographer and architect, was the inventor of the octahedral "Butterfly Map" ; an early proponent of the San Francisco Civic Center ; and designer of the Columbarium of San Francisco.His Butterfly World Map, like Buckminster Fuller's later Dymaxion Map of 1943 and 1954, enabled all...
(1866–1944) in 1909. Cahill and Waterman map
Map
A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....
s can each be shown in various profiles, typically linked at the north Pacific
Pacific Ocean
The Pacific Ocean is the largest of the Earth's oceanic divisions. It extends from the Arctic in the north to the Southern Ocean in the south, bounded by Asia and Australia in the west, and the Americas in the east.At 165.2 million square kilometres in area, this largest division of the World...
or north Atlantic
Atlantic Ocean
The Atlantic Ocean is the second-largest of the world's oceanic divisions. With a total area of about , it covers approximately 20% of the Earth's surface and about 26% of its water surface area...
oceans.
As Cahill was an architect
Architect
An architect is a person trained in the planning, design and oversight of the construction of buildings. To practice architecture means to offer or render services in connection with the design and construction of a building, or group of buildings and the space within the site surrounding the...
, his approach tended toward forms that could be demonstrated physically (i.e. his flattenable rubber-ball map). Waterman, on the other hand, derived his design from his work on close-packing of spheres. This involves the interpretation of a spherical extraction from cubic closest-packed spheres into a corresponding convex hull, as demonstrated in the following pictures.
Then, for its projection
Map projection
A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...
, straight lines are used to define each 5 × 5 section onto this convex hull. According to annotations on modern versions of the map, the projection divides the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....
equally amongst the meridians. Latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
s are drawn in three straight-line sections in each octant
Octant
An octant is one of eight divisions.-Octant in the plane :Traditionally wind direction is given as one of the 8 octants because that is more accurate than merely giving one of the 4 quadrants, and the wind vane typically does not have enough accuracy to bother with more precise indication.-Octant...
: from pole
Pole
-General:*Poles, people originating from inbitating or inhabiting the country of Poland*Pole -Fictional:*Jill Pole, a fictional character from C. S...
to fold-line; from fold-line to largest line parallel to equator; and then from there to the equator. The largest line parallel to the equator also has equal-length delineations. Waterman chose a specific Waterman polyhedron
Waterman polyhedron
Waterman polyhedra were invented around 1990 by the mathematician Steve Waterman.Waterman polyhedra are created by packing spheres according to the cubic close packing , then sweeping away the spheres that are farther from the center than a defined radius...
and central meridian to avoid interrupting land masses.
Like Buckminster Fuller
Buckminster Fuller
Richard Buckminster “Bucky” Fuller was an American systems theorist, author, designer, inventor, futurist and second president of Mensa International, the high IQ society....
's 1943 Dymaxion Projection, an octahedral butterfly map can show all the continents uninterrupted if its octants are divided at a suitable meridian
Meridian
-Earth science:* Meridian , imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon* Meridian , an imaginary arc on the Earth's surface from the North Pole to the South Pole* Meridian arc...
(in this case 20° W) and are joined, for example, at the North Atlantic, as in the 1996 version.
External links
- Carlos A. Furuti, Map Projections: Polyhedral Maps
- Carlos A. Furuti, All major map projections
- Rotating Waterman as globe Izidor Hafner
- Flat Worlds, Earth Literacy, and Map Alphabetism (Write-up comparing other world maps.)
- Positive News Youth Views site (Waterman selected as their signet.)