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 of an icosahedron. The resulting map is heavily interrupted in order to reduce shape and size distortion compared to other world maps, 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 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. Fuller applied for a patent in the United States in February 1944 for the cuboctahedron projection, which was issued in January 1946.
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 contiguous land masses. References today to the Fuller projection or Dymaxion usually indicate this version.
Projection of each triangle
Unlike other polyhedral map projections, 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 circles on the Earth (modeled as a sphere), and then the scale diminishes toward the middle of the triangle. The transformation process was formally mathematically defined in 1978.
Properties
Though neither conformal nor equal-area, 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 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". 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. 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, 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 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.
Comparison of the Fuller projection and Strebe's Dymaxion-like
conformal projection with Tissot's indicatrices at 30° intervals
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.
The World Game, a collaborative simulation game in which players attempt to solve world problems, is played on a 70-by-35-foot Dymaxion map.
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. The competition received over 300 entries from 42 countries.
The H3 hierarchical global grid implemented by Uber uses an icosahedron oriented in Dymaxion orientation, then further subdivided into hexagons.
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.
Gallery
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![image]()
This icosahedral net shows connected oceans surrounding Antarctica. -
![image]()
Example of use illustrating early human migrations according to mitochondrial population genetics (numbers are millennia before present) -
![image]()
Dymaxion map of the world with the 30 largest countries and territories by total area, roughly to scale
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
- "Life Presents R. Buckminster Fuller's Dymaxion World". LIFE. 1 March 1943. pp. 41–55.
- U.S. patent 2,393,676
- Fuller, Ideas and Integrities (1969 ed., p. 139).
- Gray, Robert W. (1994-01-01). "Fuller's Dymaxion Map". Cartography and Geographic Information Systems. 21 (4): 243–246. doi:10.1559/152304094782540628 (inactive 1 November 2024). ISSN 1050-9844.
{{cite journal}}: CS1 maint: DOI inactive as of November 2024 (link) - Gray, Robert W. (1995-10-01). "Exact Transformation Equations for Fuller's World Map". Cartographica: The International Journal for Geographic Information and Geovisualization. 32 (3): 17–25. doi:10.3138/1677-3273-Q862-1885. ISSN 0317-7173.
- Crider, John E. (2008-03-01). "Exact Equations for Fuller's Map Projection and Inverse". Cartographica: The International Journal for Geographic Information and Geovisualization. 43 (1): 67–72. doi:10.3138/carto.43.1.67. ISSN 0317-7173.
- "Dymaxion Map Transformations - Technical White Paper", Kitrick, Christopher, 2018
- "Fuller—ArcGIS Pro | Documentation".
- Fuller, Intuition (1972).
- "Frequently Asked Questions About The Fuller Projection", Buckminster Fuller Institute, 1992, accessed 2010-07-28
- Strebe, Daniel. "Directory of Map Projections: dymaxion-like conformal". Mapthematics. Mapthematics LLC. Retrieved 2022-06-19.
- "Kulturelles Erbe Köln: Johns, Jasper, Map (Weltkarte)". www.kulturelles-erbe-koeln.de. Retrieved 2017-01-18.
- "Ausstellung "Ludwig goes Pop"".
- Richards, Allen (May–June 1971). "R. Buckminster Fuller: Designer of the Geodesic Dome and the World Game". Mother Earth News. Retrieved 19 January 2014.
- Aigner, Hal (November–December 1970). "Sustaining Planet Earth: Researching World Resources". Mother Earth News. Retrieved 19 January 2014.
- Perry, Tony (October 2, 1995). "This Game Anything but Child's Play : Buckminster Fuller's creation aims to fight the real enemies of mankind: starvation, disease and illiteracy". The Los Angeles Times. Retrieved 19 January 2014.
- "DYMAX REDUX Winner". The Buckminster Fuller Institute. Archived from the original on 2 February 2014. Retrieved 21 January 2014.
- Campbell-Dollaghan, Kelsey (July 22, 2013). "7 Brilliant Reinventions of Buckminster Fuller's Dymaxion Map". Gizmodo. Retrieved 21 January 2014.
- Brodsky, Isaac (June 27, 2018). "H3: Uber's Hexagonal Hierarchical Spatial Index". Uber Blog. "Overview of the H3 Geospatial Indexing System". H3 documentation. Uber.
- Karel, Daniel (March 10, 2022). "The 2,731-Person Project to Build New York City in Minecraft". Curbed. Retrieved 2024-09-03.
External links
- Fuller Map homepage
- Dymaxion Project Animation
- Icosahedron and Fuller maps
- Dynamically generated maps based on the Dymaxion projection
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 of an icosahedron The resulting map is heavily interrupted in order to reduce shape and size distortion compared to other world maps but the interruptions are chosen to lie in the ocean The world on a Dymaxion projection with 15 graticuleDymaxion projection with Tissot s indicatrix of deformation 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 HistoryThe March 1 1943 edition of 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 Fuller applied for a patent in the United States in February 1944 for the cuboctahedron projection which was issued in January 1946 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 contiguous land masses References today to the Fuller projection or Dymaxion usually indicate this version Projection of each triangleUnlike other polyhedral map projections 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 circles on the Earth modeled as a sphere and then the scale diminishes toward the middle of the triangle The transformation process was formally mathematically defined in 1978 PropertiesAn icosahedron the shape the world map is projected onto before unfolding source source source source The world flattens to a Dymaxion map as it unfolds into an icosahedron net with nearly contiguous land masses Though neither conformal nor equal area 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 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 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 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 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 variantIn 2019 Daniel daan Strebe developed a 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 Fuller projectionDymaxion like conformal projection Comparison of the Fuller projection and Strebe s Dymaxion like conformal projection with Tissot s indicatrices at 30 intervalsInfluenceA 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 The World Game a collaborative simulation game in which players attempt to solve world problems is played on a 70 by 35 foot Dymaxion map 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 The competition received over 300 entries from 42 countries The H3 hierarchical global grid implemented by Uber uses an icosahedron oriented in Dymaxion orientation then further subdivided into hexagons 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 GalleryThis icosahedral net shows connected oceans surrounding Antarctica Example of use illustrating early human migrations according to mitochondrial population genetics numbers are millennia before present Dymaxion map of the world with the 30 largest countries and territories by total area roughly to scaleSee alsoList of map projections Authagraph projection inspired by Fuller 1999 Peirce quincuncial projection 1879 Polyhedral map projection earliest known is by Leonardo da Vinci 1514References Life Presents R Buckminster Fuller s Dymaxion World LIFE 1 March 1943 pp 41 55 U S patent 2 393 676 Fuller Ideas and Integrities 1969 ed p 139 Gray Robert W 1994 01 01 Fuller s Dymaxion Map Cartography and Geographic Information Systems 21 4 243 246 doi 10 1559 152304094782540628 inactive 1 November 2024 ISSN 1050 9844 a href wiki Template Cite journal title Template Cite journal cite journal a CS1 maint DOI inactive as of November 2024 link Gray Robert W 1995 10 01 Exact Transformation Equations for Fuller s World Map Cartographica The International Journal for Geographic Information and Geovisualization 32 3 17 25 doi 10 3138 1677 3273 Q862 1885 ISSN 0317 7173 Crider John E 2008 03 01 Exact Equations for Fuller s Map Projection and Inverse Cartographica The International Journal for Geographic Information and Geovisualization 43 1 67 72 doi 10 3138 carto 43 1 67 ISSN 0317 7173 Dymaxion Map Transformations Technical White Paper Kitrick Christopher 2018 Fuller ArcGIS Pro Documentation Fuller Intuition 1972 Frequently Asked Questions About The Fuller Projection Buckminster Fuller Institute 1992 accessed 2010 07 28 Strebe Daniel Directory of Map Projections dymaxion like conformal Mapthematics Mapthematics LLC Retrieved 2022 06 19 Kulturelles Erbe Koln Johns Jasper Map Weltkarte www kulturelles erbe koeln de Retrieved 2017 01 18 Ausstellung Ludwig goes Pop Richards Allen May June 1971 R Buckminster Fuller Designer of the Geodesic Dome and the World Game Mother Earth News Retrieved 19 January 2014 Aigner Hal November December 1970 Sustaining Planet Earth Researching World Resources Mother Earth News Retrieved 19 January 2014 Perry Tony October 2 1995 This Game Anything but Child s Play Buckminster Fuller s creation aims to fight the real enemies of mankind starvation disease and illiteracy The Los Angeles Times Retrieved 19 January 2014 DYMAX REDUX Winner The Buckminster Fuller Institute Archived from the original on 2 February 2014 Retrieved 21 January 2014 Campbell Dollaghan Kelsey July 22 2013 7 Brilliant Reinventions of Buckminster Fuller s Dymaxion Map Gizmodo Retrieved 21 January 2014 Brodsky Isaac June 27 2018 H3 Uber s Hexagonal Hierarchical Spatial Index Uber Blog Overview of the H3 Geospatial Indexing System H3 documentation Uber Karel Daniel March 10 2022 The 2 731 Person Project to Build New York City in Minecraft Curbed Retrieved 2024 09 03 External linksWikimedia Commons has media related to Fuller projection Fuller Map homepage Dymaxion Project Animation Icosahedron and Fuller maps Dynamically generated maps based on the Dymaxion projection
