Polyhedral map projection

Short description: Type of map projection

Buckminster Fuller's Dymaxion map.

A polyhedral map projection is a map projection based on a spherical polyhedron. Typically, the polyhedron is overlaid on the globe, and each face of the polyhedron is transformed to a polygon or other shape in the plane. The best-known polyhedral map projection is Buckminster Fuller's Dymaxion map. When the spherical polyhedron faces are transformed to the faces of an ordinary polyhedron instead of laid flat in a plane, the result is a polyhedral globe.[1]

Often the polyhedron used is a Platonic solid or Archimedean solid. However, other polyhedra can be used: the AuthaGraph projection makes use of a polyhedron with 96 faces, and the myriahedral projection allows for an arbitrary large number of faces.[2] Although interruptions between faces are common, and more common with an increasing number of faces, some maps avoid them: the Lee conformal projection only has interruptions at its border, and the AuthaGraph projection scales its faces so that the map fills a rectangle without internal interruptions. Some projections can be tesselated to fill the plane, the Lee conformal projection among them.

To a degree, the polyhedron and the projection used to transform each face of the polyhedron can be considered separately, and some projections can be applied to differently shaped faces. The gnomonic projection transforms the edges of spherical polyhedra to straight lines, preserving all polyhedra contained within a hemisphere, so it is a common choice. The Snyder equal-area projection can be applied to any polyhedron with regular faces.[3] The projection used in later versions of the Dymaxion map can be generalized to other equilateral triangular faces,[4] and even to certain quadrilaterals.[5]

Polyhedral map projections are useful for creating discrete global grids, as with the quadrilateralized spherical cube and Icosahedral Snyder Equal Area (ISEA) grids.[6]

## Contents

* 1 History
* 2 Gallery
* 3 See also
* 4 References
* 5 External links

## History

The earliest known polyhedral projection is the octant projection developed by Leonardo da Vinci or his associate around 1514, which transforms the faces of an octahedron to Reuleaux triangles.[1]

Christian Gottlieb Reichard created a polyhedral globe based on the cube in 1803. An icosahedral globe appeared in 1851. Polyhedral globes cheaply constructed from cardboard were popular for a time in Europe.[1]

Projections based on dihedra begin appearing with the Peirce quincuncial projection in 1879, Guyou hemisphere-in-a-square projection in 1887, and Adams hemisphere-in-a-square projection in 1925. Although the dihedra are not traditional polyhedra they are spherical polyhedra, and the methods used in these projections are also used in other polyhedral projections. In the same work as the hemisphere-in-a-square projection, Adams created maps depicting the entire globe in a rhombus, hexagon, and hexagram.[7][8]

Bernard J. S. Cahill invented the "butterfly map", based on the octahedron, in 1909. This was generalized into the Cahill–Keyes projection in 1975 and the Waterman butterfly projection in 1996. Cahill's work was also influential on Fuller's Dymaxion maps: Fuller's first version, based on a cuboctahedron, was published in 1943, and his second, based on an icosahedron, was published in 1954.[1]

In 1965, Wellman Chamberlin (also known for his Chamberlin trimetric projection) and Howard E. Paine of the National Geographic Society designed a polyhedral map based on the 12 equal pentagon faces of a dodecahedron. 20 years later, Chamberlin and Paine used that polyhedral map in "Global Pursuit", a board game intended to teach geography to children.[9][10]

The quadrilateralized spherical cube was devised in 1975 for the Cosmic Background Explorer project.[11][12]

The Traveller (1977) and GURPS Space (1987) role-playing games use an icosahedral projection for planetary maps with a hexagon grid.

## Gallery

* Octant projection

* Cahill's butterfly map

* Cahill–Keyes projection

* Waterman butterfly projection

* Lee conformal world on a tetrahedron

* Peirce quincuncial

* Guyou hemisphere-in-a-square projection

* Adams hemisphere-in-a-square projection

## See also

* HEALPix, which is not strictly a polyhedral map projection
* List of map projections

## References

1. ↑ 1.0 1.1 1.2 1.3 Pędzich, Paweł (2016-12-01). "Image of the World on polyhedral maps and globes" (in en). Polish Cartographical Review 48 (4): 197–210. doi:10.1515/pcr-2016-0014. ISSN 2450-6966. Bibcode: 2016PCRv...48..197P.
2. ↑ van Wijk, Jarke J. (2008). "Unfolding the Earth: Myriahedral Projections" (in en). The Cartographic Journal 45 (1): 32–42. doi:10.1179/000870408X276594. ISSN 0008-7041. Bibcode: 2008CartJ..45...32V. http://www.tandfonline.com/doi/full/10.1179/000870408X276594.
3. ↑ Snyder, John P (1992-03-01). "An Equal-Area Map Projection For Polyhedral Globes". Cartographica: The International Journal for Geographic Information and Geovisualization 29 (1): 10–21. doi:10.3138/27H7-8K88-4882-1752. ISSN 0317-7173. https://utpjournals.press/doi/10.3138/27H7-8K88-4882-1752.
4. ↑ 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. https://utpjournals.press/doi/10.3138/carto.43.1.67.
5. ↑ Crider, John E. (2009-01-01). "A Geodesic Map Projection for Quadrilaterals". Cartography and Geographic Information Science 36 (2): 131–147. doi:10.1559/152304009788188781. ISSN 1523-0406. Bibcode: 2009CGISc..36..131C. https://doi.org/10.1559/152304009788188781.
6. ↑ Sahr, Kevin; White, Denis; Kimerling, A.J. (2003). "Geodesic discrete global grid systems". Cartography and Geographic Information Science 30 (2): 121–134. doi:10.1559/152304003100011090. Bibcode: 2003CGISc..30..121S. http://webpages.sou.edu/~sahrk/sqspc/pubs/gdggs03.pdf.
7. ↑ Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps. United States Coast and Geodetic Survey Special Publication No. 112. U.S. Government Printing Office. https://geodesy.noaa.gov/library/pdfs/Special_Publication_No_112.pdf.
8. ↑ Conformal Projections Based on Elliptic Functions. Cartographica Monographs. 16. Toronto: B. V. Gutsell, York University. 1976. ISBN 0-919870-16-3. https://archive.org/details/conformalproject0000leel. Supplement No. 1 to The Canadian Cartographer 13.
9. ↑ Scheel, Eugene (May 19, 2002). "With Intellect and Artistry, Wellman Chamberlin Created a World of His Own". The Washington Post. https://www.washingtonpost.com/archive/local/2002/05/19/with-intellect-and-artistry-wellman-chamberlin-created-a-world-of-his-own/627dbea8-370a-4b16-83cd-46e6388a9dfb/.
10. ↑ "Global Pursuit (1987)" (in en-US). https://boardgamegeek.com/boardgame/3676/global-pursuit.
11. ↑ Chan, F.K.; O'Neill, E. M. (1975). Feasibility Study of a Quadrilateralized Spherical Cube Earth Data Base (CSC - Computer Sciences Corporation, EPRF Technical Report 2-75) (Technical report). Monterey, California: Environmental Prediction Research Facility.
12. ↑ O'Neill, E. M. (1976). Extended Studies of a Quadrilateralized Spherical Cube Earth Data Base (PDF) (Technical report). Monterey, California: Environmental Prediction Research Facility. Archived (PDF) from the original on May 7, 2019.

## External links

* Unfolding the Earth: Myriahedral Projections — an interactive visualization of myriahedral projections.

* v
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Map projection

* History
* List
* Portal

By surface

| Cylindrical|

| Mercator-conformal|

* Gauss–Krüger
* Transverse Mercator
* Oblique Mercator

|

Equal-area|

* Balthasart
* Behrmann
* Gall–Peters
* Hobo–Dyer
* Lambert
* Smyth equal-surface
* Trystan Edwards

* Cassini
* Central
* Equirectangular
* Gall stereographic
* Gall isographic
* Miller
* Space-oblique Mercator
* Web Mercator

Pseudocylindrical|

* Eckert II
* Eckert IV
* Eckert VI
* Equal Earth
* Goode homolosine
* Kavrayskiy VII
* Mollweide
* Sinusoidal
* Tobler hyperelliptical
* Wagner VI
* Winkel I and II

Conical|

* Albers
* Equidistant
* Lambert conformal

Pseudoconical|

* Bonne
* Bottomley
* Polyconic
* American
* Werner

Azimuthal
(planar)|

| General perspective|

* Gnomonic
* Orthographic
* Stereographic

|

* Equidistant
* Lambert equal-area

Pseudoazimuthal|

* Aitoff
* Hammer
* Wiechel
* Winkel tripel

By metric

| Conformal|

* Adams hemisphere-in-a-square
* Gauss–Krüger
* Guyou hemisphere-in-a-square
* Lambert conformal conic
* Mercator
* Peirce quincuncial
* Stereographic
* Transverse Mercator

|

Equal-area|

| Bonne|

* Sinusoidal
* Werner

|

Bottomley|

* Sinusoidal
* Werner

Cylindrical|

* Balthasart
* Behrmann
* Gall–Peters
* Hobo–Dyer
* Lambert cylindrical equal-area
* Smyth equal-surface
* Trystan Edwards

Tobler hyperelliptical|

* Collignon
* Mollweide

* Albers
* Briesemeister
* Eckert II
* Eckert IV
* Eckert VI
* Equal Earth
* Goode homolosine
* Hammer
* Lambert azimuthal equal-area
* Quadrilateralized spherical cube
* Strebe 1995

Equidistant in
some aspect|

* Conic
* Equirectangular
* Sinusoidal
* Two-point
* Werner

Gnomonic|

* Gnomonic

Loxodromic|

* Loximuthal
* Mercator

Retroazimuthal
(Mecca or Qibla)|

* Craig
* Hammer
* Littrow

By construction

| Compromise|

* Chamberlin trimetric
* Kavrayskiy VII
* Miller cylindrical
* Natural Earth
* Robinson
* Van der Grinten
* Wagner VI
* Winkel tripel

|

Hybrid|

* Goode homolosine
* HEALPix

Perspective|

| Planar|

* Gnomonic
* Orthographic
* Stereographic

|

* Central cylindrical

Polyhedral|

* Cahill Butterfly
* Cahill–Keyes M-shape
* Dymaxion
* ISEA
* Quadrilateralized spherical cube
* Waterman butterfly

See also

* Interruption (map projection)
* Latitude
* Longitude
* Tissot's indicatrix
* Map projection of the tri-axial ellipsoid

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