**AuthaGraph** is an approximately equal-area world map projection invented by Japanese architect Hajime Narukawa ^{ [1] } in 1999.^{ [2] } The map is made by equally dividing a spherical surface into 96 triangles, transferring it to a tetrahedron while maintaining area proportions, and unfolding it onto a rectangle. The map substantially preserves sizes and shapes of all continents and oceans while it reduces distortions of their shapes, as inspired by the Dymaxion map. The projection does not have some of the major distortions of the Mercator projection, like the expansion of countries in far northern latitudes, and allows for Antarctica to be displayed accurately and in whole.^{ [3] } Triangular world maps are also possible using the same method. The name is derived from "authalic" and "graph".^{ [3] }

The method used to construct the projection ensures that the 96 regions of the sphere that are used to define the projection each have the correct area, but the projection does not qualify as equal-area because the method does not control area at infinitesimal scales or even within those regions.

The AuthaGraph world map can be tiled in any direction without visible seams. From this map-tiling, a new world map with triangular, rectangular or a parallelogram's outline can be framed with various regions at its center. This tessellation allows for depicting temporal themes, such as a satellite's long-term movement around the earth in a continuous line.^{ [4] }

In 2011 the AuthaGraph mapping projection was selected by the Japanese National Museum of Emerging Science and Innovation (Miraikan) as its official mapping tool.^{ [5] } In October 2016, the AuthaGraph mapping projection won the 2016 Good Design Grand Award from the Japan Institute of Design Promotion.^{ [6] }

In geometry, a **cuboctahedron** is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is the only radially equilateral convex polyhedron.

**Cartography** is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.

The **Gall–Peters projection** is a rectangular map projection that maps all areas such that they have the correct sizes relative to each other. Like any equal-area projection, it achieves this goal by distorting most shapes. The projection is a particular example of the cylindrical equal-area projection with latitudes 45° north and south as the regions on the map that have no distortion.

A **map** is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.

The **Mercator projection** is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator, but accelerates with latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

In geometry, the **rhombicuboctahedron**, or **small rhombicuboctahedron**, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each one. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

In cartography, a **map projection** is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the surface of the globe into locations on a plane. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. Every distinct map projection distorts in a distinct way, by definition. The study of map projections is the characterization of these distortions. There is no limit to the number of possible map projections. Projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, and manifolds. However, "map projection" refers specifically to a cartographic projection.

The **Robinson projection** is a map projection of a world map which shows the entire world at once. It was specifically created in an attempt to find a good compromise to the problem of readily showing the whole globe as a flat image.

A **world map** is a map of most or all of the surface of Earth. World maps form a distinctive category of maps due to the problem of projection. Maps by necessity distort the presentation of the earth's surface. These distortions reach extremes in a world map. The many ways of projecting the earth reflect diverse technical and aesthetic goals for world maps.

**Nicolas Auguste Tissot** was a 19th-century French cartographer, who in 1859 and 1881 published an analysis of the distortion that occurs on map projections. He devised Tissot's indicatrix, or *distortion circle*, which when plotted on a map will appear as an ellipse whose elongation depends on the amount of distortion by the map at that point. The angle and extent of the elongation represents the amount of angular distortion of the map. The size of the ellipse indicates the amount that the area is distorted.

In computer graphics, **cube mapping** is a method of environment mapping that uses the six faces of a cube as the map shape. The environment is projected onto the sides of a cube and stored as six square textures, or unfolded into six regions of a single texture. The cube map is generated by first rendering the scene six times from a viewpoint, with the views defined by a 90 degree view frustum representing each cube face.

The **Lambert azimuthal equal-area projection** is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the **Lambert zenithal equal-area projection**.

The **Shibaura Institute of Technology**, abbreviated as *Shibaura kōdai*, is a private university of Technology in Japan, with campuses located in Tokyo and Saitama. Established in 1927 as the Tokyo Higher School of Industry and Commerce, it was chartered as a university in 1949.

The **National Museum of Emerging Science and Innovation**, simply known as the * Miraikan*, is a museum created by Japan's Science and Technology Agency.

**HEALPix**, an acronym for **H**ierarchical **E**qual **A**rea iso**L**atitude **Pix**elisation of a 2-sphere, refers to either an algorithm for pixelisation of the 2-sphere or to the associated class of map projections. The pixelisation algorithm was devised in 1997 by Krzysztof M. Górski at the Theoretical Astrophysics Center in Copenhagen, Denmark, and first published as a preprint in 1998.

The **Cahill–Keyes projection** is a polyhedral compromise map projection first proposed by Gene Keyes in 1975. The projection is a refinement of an earlier 1909 projection by Cahill. The projection was designed to achieve a number of desirable characteristics, namely symmetry of component maps (octants), scalability allowing the map to continue to work well even at high resolution, uniformity of geocells, metric-based joining edges, minimized distortion compared to a globe, and an easily understood orientation to enhance general usability and teachability.

A **Discrete Global Grid** (**DGG**) is a mosaic which covers the entire Earth's surface. Mathematically it is a space partitioning: it consists of a set of non-empty regions that form a partition of the Earth's surface. In a usual grid-modeling strategy, to simplify position calculations, each region is represented by a point, abstracting the grid as a set of region-points. Each region or region-point in the grid is called a **cell**.

**Hajime Narukawa** is a Japanese architect. He was born in 1971 in Kawasaki-City, Kanagawa and lives and practices in Tokyo.

The **latitudinally equal-differential polyconic projection** (等差分纬线多圆锥投影) is a polyconic map projection in use since 1963 in mainland China. Maps on this projection are produced by China's State Bureau of Surveying and Mapping and other publishers. Its original method of construction has not been preserved, but a mathematical approximation has been published.

In map projections, an **interruption** is any place where the globe has been split. All map projections are interrupted at at least one point. Typical world maps are interrupted along an entire meridian. In that typical case, the interruption forms an east/west boundary, even though the globe has no boundaries.

- ↑ "鳴川肇 – Hajime Narukawa".
*ist2010.jp*. Archived from the original on 5 November 2016. Retrieved 7 February 2019. - ↑ "A new accurate world « Exploraciones".
*exploraciones.cl*. Archived from the original on 2 April 2015. Retrieved 29 March 2015. - 1 2 "AuthaGraph オーサグラフ 世界地図". Archived from the original on 13 December 2018. Retrieved 7 February 2019.
- ↑ "ICC Online – Archive – 2009 – Open Space 2009 – Works".
*ntticc.or.jp*. Retrieved 7 February 2019. - ↑ "The National Museum of Emerging Science and Innovation (Miraikan)".
*jst.go.jp*. Retrieved 7 February 2019. - ↑ "World Map Projection [AuthaGraph World Map] – Good Design Grand Award" . Retrieved 7 February 2019.

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