View source for Dymaxion map

{{Short description|Polyhedral compromise map projection}}
{{Stack|
[[Image:Dymaxion projection.png|thumb|300px|The world on a Dymaxion projection, with 15° graticule]]
[[File:Dymaxion with Tissot's Indicatrices of Distortion.svg|300px|thumb|Dymaxion projection with [[Tissot's indicatrix]] of deformation]]}}
The '''Dymaxion map projection''', also called the '''Fuller projection''', is a kind of [[polyhedral map projection]] of the Earth's surface onto the unfolded [[net (polyhedron)|net]] of an [[icosahedron]]. The resulting map is heavily [[interrupted projection|interrupted]] in order to reduce shape and size distortion compared to other [[world map]]s, but the interruptions are chosen to lie in the ocean.

The projection was invented by [[Buckminster Fuller]]. In 1943, Fuller proposed a projection onto a [[cuboctahedron]], which he called the ''Dymaxion World'', using the name ''[[Dymaxion]]'' which he also applied to several of his other inventions. In 1954, Fuller and cartographer [[Shoji Sadao]] produced an updated Dymaxion map, the '''Airocean World Map''', based on an icosahedron with a few of the triangular faces cut to avoid breaks in landmasses.

The Dymaxion projection is intended for representations of the entire Earth.

== History ==

The March 1, 1943, edition of [[Life (magazine)|''Life'' magazine]] included a photographic essay titled "Life Presents R. Buckminster Fuller's Dymaxion World", illustrating a projection onto a [[cuboctahedron]], including several examples of possible arrangements of the square and triangular pieces, and a pull-out section of one-sided magazine pages with the map faces printed on them, intended to be cut out and glued to card stock to make a three-dimensional cuboctahedron or its two-dimensional net.<ref name=life1943>{{cite magazine|title=Life Presents R. Buckminster Fuller's Dymaxion World|magazine=[[Life (magazine)|LIFE]]|date=1 March 1943|pages=41–55|url=https://books.google.com/books?id=WlEEAAAAMBAJ&q=March%201%2C%201943%20Life&pg=PA41}}</ref> Fuller applied for a patent in the United States in February 1944 for the cuboctahedron projection, which was issued in January 1946.<ref>{{US Patent|2393676}}</ref>

In 1954, Fuller and cartographer [[Shoji Sadao]] produced a new map onto an icosahedron instead of the cuboctahedron. It depicts Earth's continents as "one island", or nearly [[Geographic contiguity|contiguous]] land masses. References today to the Fuller projection or Dymaxion usually indicate this version.

== Projection of each triangle ==

Unlike other [[polyhedral map projection]]s, the Dymaxion map does not use a [[gnomonic projection]] (perspective projection through the Earth's center onto the polyhedral surface), which causes length distortion away from the center of each face. Instead each triangle's three edges on the Dymaxion map match the scale along the corresponding arcs of [[great circle]]s on the Earth (modeled as a sphere), and then the scale diminishes toward the middle of the triangle.<ref>Fuller, ''Ideas and Integrities'' (1969 ed., p. 139).</ref> The transformation process was formally mathematically defined in 1978.<ref>{{Cite journal|last=Gray|first=Robert W.|date=1994-01-01|title=Fuller's Dymaxion Map|url=https://www.tandfonline.com/doi/abs/10.1559/152304094782540628|journal=Cartography and Geographic Information Systems|volume=21|issue=4|pages=243–246|doi=10.1559/152304094782540628|doi-broken-date=1 November 2024|issn=1050-9844}}</ref><ref>{{Cite journal|last=Gray|first=Robert W.|date=1995-10-01|title=Exact Transformation Equations for Fuller's World Map|url=https://utpjournals.press/doi/10.3138/1677-3273-Q862-1885|journal=Cartographica: The International Journal for Geographic Information and Geovisualization|volume=32|issue=3|pages=17–25|doi=10.3138/1677-3273-Q862-1885|issn=0317-7173}}</ref><ref>{{Cite journal|last=Crider|first=John E.|date=2008-03-01|title=Exact Equations for Fuller's Map Projection and Inverse|url=https://utpjournals.press/doi/10.3138/carto.43.1.67|journal=Cartographica: The International Journal for Geographic Information and Geovisualization|volume=43|issue=1|pages=67–72|doi=10.3138/carto.43.1.67|issn=0317-7173}}</ref><ref>[https://www.researchgate.net/publication/334307604_Dymaxion_Map_Transformations_-_Technical_White_Paper "Dymaxion Map Transformations - Technical White Paper"], Kitrick, Christopher, 2018</ref>

== Properties ==
{{multiple image
|image1  =Zeroth stellation of icosahedron.svg
|caption1=An icosahedron, the shape the world map is projected onto before unfolding
|width1  =55
|image2  =Dymaxion 2003 animation small1.webm
|caption2=The world flattens to a Dymaxion map as it unfolds into an [[icosahedron]] net with nearly contiguous land masses.
|width2  =240
}}
Though neither [[conformal map projection|conformal]] nor [[equal-area projection|equal-area]],<ref>{{cite web | url=http://pro.arcgis.com/en/pro-app/latest/help/mapping/properties/fuller.htm | title=Fuller—ArcGIS Pro {{pipe}} Documentation }}</ref> Fuller claimed that his map had several advantages over other projections for world maps.

It has less distortion of relative size of areas, most notably when compared to the [[Mercator projection]]; and less distortion of shapes of areas, notably when compared to the [[Gall–Peters projection]]. Other [[Map projection#Compromise projections|compromise projections]] attempt a similar trade-off.

More unusually, the Dymaxion map does not have any "right way up". Fuller argued that in the universe there is no "up" and "down", or "north" and "south": only "in" and "out".<ref>Fuller, ''Intuition'' (1972).</ref> Gravitational forces of the stars and planets created "in", meaning "towards the gravitational center", and "out", meaning "away from the gravitational center". He attributed the north-up-superior/south-down-inferior presentation of most other world maps to [[cultural bias]].

Fuller intended the map to be unfolded in different ways to emphasize different aspects of the world.<ref>[http://www.geni.org/globalenergy/library/buckminster_fuller/dymaxion_map/fuller-faq.shtml "Frequently Asked Questions About The Fuller Projection"], Buckminster Fuller Institute, 1992, accessed 2010-07-28</ref> Peeling the triangular faces of the icosahedron apart in one way results in an icosahedral net that shows an almost contiguous land mass comprising all of Earth's continents – not groups of continents divided by oceans. Peeling the solid apart in a different way presents a view of the world dominated by connected oceans surrounded by land.

Showing the continents as "one island earth" also helped Fuller explain, in his book ''[[Critical Path (book)|Critical Path]]'', the journeys of early seafaring people, who were in effect using [[prevailing winds]] to circumnavigate this world island.

However, the Dymaxion map can also prove difficult to use. It is, for example, confusing to describe the four [[cardinal directions]] and locate [[geographic coordinates]]. The awkward shape of the map may be counterintuitive to most people trying to use it. For example, the shortest route from Africa to South America is not obvious. Depending on how the map is projected, land masses and oceans are often divided into several pieces.

== Conformal variant ==

In 2019, Daniel "daan" Strebe developed a [[conformal map projection|conformal]] icosahedral projection, similar to the conformal projections to an octahedron by [[Oscar S. Adams]] (1928) and to a tetrahedron by [[Laurence P. Lee]] (1965), all three using [[Dixon elliptic functions]]. A conformal map preserves angles and local shapes from the sphere at the expense of increasing the scale distortion near the vertices of the icosahedron.<ref>{{cite web |url=https://www.mapthematics.com/ProjectionsList.php?Projection=309 |title= Directory of Map Projections: dymaxion-like conformal |last= Strebe|first= Daniel|website= Mapthematics|publisher= Mapthematics LLC|access-date= 2022-06-19}}</ref>

{{switcher
|[[File:tobias_jung_tissot_30_dymaxion_fuller_projection.jpg|400px]]
|Fuller projection
|[[File:tobias_jung_tissot_30_dymaxion-like_conformal_projection.jpg|400px]]
|Dymaxion-like conformal projection
}}
<small>Comparison of the Fuller projection and Strebe's Dymaxion-like
<br />conformal projection with [[Tissot's indicatrix|Tissot's indicatrices]] at 30° intervals</small>

== Influence ==

A 1967 [[Jasper Johns]] painting, ''Map (Based on Buckminster Fuller's Dymaxion Airocean World)'', depicting a Dymaxion map, hangs in the permanent collection of the [[Museum Ludwig]] in [[Cologne]].<ref>{{Cite web|url=https://www.kulturelles-erbe-koeln.de/documents/obj/05010054|title=Kulturelles Erbe Köln: Johns, Jasper, Map (Weltkarte)|website=www.kulturelles-erbe-koeln.de|language=en|access-date=2017-01-18}}</ref><ref>{{Cite web|url=https://www.kulturelles-erbe-koeln.de/documents/aus/05094087|title=Ausstellung "Ludwig goes Pop"}}</ref>

The [[World Game]], a collaborative simulation game in which players attempt to solve world problems,<ref>{{cite web|last=Richards|first=Allen|title=R. Buckminster Fuller: Designer of the Geodesic Dome and the World Game|url=http://www.motherearthnews.com/nature-and-environment/the-plowboy-interview-r-buckminster-fuller.aspx?PageId=1|publisher=Mother Earth News|access-date=19 January 2014|date=May–June 1971}}</ref><ref>{{cite web|last=Aigner|first=Hal|title=Sustaining Planet Earth: Researching World Resources|url=http://www.motherearthnews.com/nature-and-environment/world-resources-zmaz70ndzgoe.aspx|publisher=Mother Earth News|access-date=19 January 2014|date=November–December 1970}}</ref> is played on a 70-by-35-foot Dymaxion map.<ref>{{cite news|last=Perry|first=Tony|title=This Game Anything but Child's Play : Buckminster Fuller's creation aims to fight the real enemies of mankind: starvation, disease and illiteracy|url=https://www.latimes.com/archives/la-xpm-1995-10-02-me-52516-story.html|newspaper=The Los Angeles Times|access-date=19 January 2014|date=October 2, 1995}}</ref>

In 2013, to commemorate the 70th anniversary of the publication of the Dymaxion map in ''Life'' magazine, the Buckminster Fuller Institute announced the "Dymax Redux", a competition for graphic designers and visual artists to re-imagine the Dymaxion map.<ref name="DYMAX REDUX Winner">{{cite web|title=DYMAX REDUX Winner|url=http://www.bfi.org/mapcontest|publisher=The Buckminster Fuller Institute|access-date=21 January 2014|archive-date=2 February 2014|archive-url=https://web.archive.org/web/20140202093941/http://www.bfi.org/mapcontest|url-status=dead}}</ref><ref>{{cite web|last=Campbell-Dollaghan|first=Kelsey|title=7 Brilliant Reinventions of Buckminster Fuller's Dymaxion Map|url=https://gizmodo.com/7-brilliant-reinventions-of-buckminster-fullers-dymaxi-867929593|website=Gizmodo|access-date=21 January 2014|date=July 22, 2013}}</ref> The competition received over 300 entries from 42 countries.<ref name="DYMAX REDUX Winner"/>

The H3 hierarchical [[discrete global grid|global grid]] implemented by [[Uber]] uses an icosahedron oriented in Dymaxion orientation, then further subdivided into hexagons.<ref>{{cite web |url=https://www.uber.com/blog/h3/ |website=Uber Blog |last=Brodsky |first=Isaac |title=H3: Uber's Hexagonal Hierarchical Spatial Index |date=June 27, 2018 }} {{pb}} {{cite web |title= Overview of the H3 Geospatial Indexing System |website=H3 documentation |publisher=Uber |url=https://h3geo.org/docs/core-library/overview/ }}</ref>

In 2020, a collaborative effort by thousands of [[Minecraft]] players, the [[Build the Earth]] project, used Strebe's conformal variant as a projection for building a 1:1 scale representation of the Earth inside the game.<ref>{{Cite web |last=Karel |first=Daniel |date=March 10, 2022 |title=The 2,731-Person Project to Build New York City in Minecraft |url=https://www.curbed.com/2022/03/global-minecraft-team-building-new-york-city.html |access-date=2024-09-03 |website=Curbed }}</ref>

==Gallery==

<gallery mode=packed>
Image:Dymaxion map ocean2.png|This icosahedral net shows connected [[ocean]]s surrounding [[Antarctica]].
Image:World map of prehistoric human migrations.jpg|Example of use illustrating [[early human migrations]] according to [[Mitochondrial DNA|mitochondrial]] [[population genetics]] (numbers are [[millennia]] before present)
Image:Fuller_projection_with_largest_countries.svg|Dymaxion map of the world with the 30 [[List of countries and dependencies by area|largest countries and territories by total area]], roughly to scale
</gallery>

==See also==
* [[List of map projections]]
* [[Authagraph projection]], inspired by Fuller, 1999
* [[Peirce quincuncial projection]], 1879
* [[Polyhedral map projection]], earliest known is by Leonardo da Vinci, 1514

==References==
{{Reflist}}

== External links ==
{{Commons category|Fuller projection}}
* [https://www.bfi.org/about-fuller/big-ideas/dymaxion-map/ Fuller Map homepage]
* [https://web.archive.org/web/20161218223035/http://www.westnet.com/~crywalt/unfold.html Dymaxion Project Animation]
* [http://www.rwgrayprojects.com/rbfnotes/maps/graymap1.html Icosahedron and Fuller maps]
* [http://teczno.com/faumaxion-II/ Dynamically generated maps based on the Dymaxion projection]

{{Map projections}}
{{Buckminster Fuller}}
{{Authority control}}

[[Category:Buckminster Fuller]]
[[Category:Map projections]]
[[Category:1943 introductions]]