Adams World in a Square II Projection
May 26,2026

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Introduction

Adams World in a Square II Projection is a conformal map projection developed by American mathematician Oscar Sherman Adams of the U.S. Coast and Geodetic Survey and presented in 1925. It is one of two projections (alongside Adams World in a Square I) that map the entire sphere onto a perfect square. In Adams's original design, the equator and central meridian are displayed as the diagonals of the square, with the poles projecting as points in opposite corners. The projection preserves conformality everywhere except at the four corners of the square. A notable property is its ability to be tessellated or mosaicked, enabling the creation of continuous repeating patterns. This projection (or a very similar one) was used in 1979 by Athelstan Spilhaus, with assistance from Robert Hanson and Ervin Schmid of the National Geodetic Survey, for his world ocean map.

Projection Basic

The Adams World in a Square II Projection consists of the following main characteristics:

Classification: Conformal (equal-angle) projection; tessellating capable
Graticule: Equator and central meridian are straight lines forming diagonals of the square; antimeridian projects as a straight line bent at the equator, forming the square outline; other meridians are complex curves; parallels are complex curves, unequally spaced along the central meridian and concave toward the nearest pole
Poles: Projected as points located in opposite corners of the square
Symmetry: Symmetric across both the equator and the central meridian
Mathematical Basis: Derived using elliptic functions and lemniscate integrals, with coordinates computed via Legendre's table of elliptic integrals for modulus k=1/2k=1/2

Pros

Strict conformality: As a conformal projection, Adams Square II preserves local angles and shapes with high fidelity. Small features maintain their correct form, making the projection theoretically sound for applications requiring local geometric accuracy. Conformality only fails at the four corner points.
Compact square format: Unlike rectangular or circular world maps, this projection fits the entire Earth into a perfect square. The symmetrical, compact format is visually striking and can be aesthetically appealing for decorative, artistic, or symbolic cartographic applications.
Seamless tessellation capability: A unique and beneficial property is that the projection can be tessellated or mosaicked—individual squares can be tiled infinitely in both horizontal and vertical directions without gaps or overlaps, creating a continuous repeating pattern of the entire Earth. This property was exploited by Spilhaus for his world ocean map.
Uninterrupted representation: Unlike interrupted projections such as the Goode Homolosine, the Adams Square II presents the entire Earth's surface in a single, continuous square without cuts or breaks, preserving spatial continuity across the globe.
Modern software support: The projection is available as a standard option in modern GIS platforms, including ArcGIS Pro (version 2.5 and later) and ArcGIS Desktop (version 10.8 and later), with ellipsoidal equations developed by Esri. PROJ also supports it under the alias adams_ws2.

Cons

Extreme area distortion: Being conformal, the projection severely exaggerates areas—particularly near the four corners of the square and along the antimeridian. Polar regions are hugely exaggerated in size, making area comparisons completely unreliable.
Severe shape distortion at high latitudes: While conformality holds locally, the overall appearance of continents far from the equator becomes highly distorted. The familiar shapes of continents near the poles and along the antimeridian are stretched beyond recognition, limiting the projection's utility for general-reference mapping.
Conformality failure at corners: Although the projection is conformal nearly everywhere, conformality fails at the four corners of the square (the poles and the intersections of the antimeridian with the equator). Distortion increases progressively toward these corner points.
Complex mathematical construction: The projection requires elliptic functions, lemniscate integrals, and Jacobi elliptic transformations for its computation. Adams's original 1929 publication details a method involving series expansions that converge slowly, requiring "considerable" computation and careful handling of multiple arc functions.
Limited practical use: According to official documentation, "this projection has no practical use aside from designing a world map in a square or to tessellate or mosaic a large, flat surface". It is not suitable for navigation, measurement, or analytical GIS workflows.
Poor intuitive readability: For most viewers unfamiliar with advanced conformal projections, the Adams Square II map is difficult to interpret. The placement of poles at opposite corners (rather than the top and bottom) and the unfamiliar graticule pattern with the equator running diagonally disorient users, making it unsuitable for educational or general-audience mapping.

Application Scenario

The Adams World in a Square II Projection occupies an extremely specialized niche in cartography. Its primary use is for novelty world maps and decorative cartography where the aesthetic appeal of a square format outweighs geometric accuracy. The projection's most notable practical application was Athelstan Spilhaus's world ocean map in 1979, which used a square conformal projection (or close variant) to emphasize the continuity of the world's oceans while minimizing interruptions. The projection's ability to tessellate has been explored for mosaicking large flat surfaces and for creating repeating patterns in artistic and symbolic contexts. In theoretical and mathematical cartography, Adams Square II serves as an elegant example of conformal mapping using elliptic functions and lemniscate integrals, and it remains of historical interest to those studying the work of Oscar S. Adams at the U.S. Coast and Geodetic Survey. However, for any practical geographic application—including navigation, general-reference atlas production, area analysis, climate mapping, or GIS data management—the Adams Square II projection is almost never the appropriate choice. It is superseded by more practical conformal projections (e.g., Mercator for navigation) or compromise projections (e.g., Robinson, Winkel Tripel) for world map display.