Why Map Projections Matter: The Impossible Task of Flattening a Sphere

Every map you have ever seen is a lie — a necessary one. The Earth is a three-dimensional oblate spheroid, and projecting its curved surface onto a flat sheet inevitably introduces distortion. Cartographers have developed hundreds of projections, each trading off accuracy in one area (shape, area, distance, or direction) for accuracy in another. While most projections attempt to cover the entire globe, a fascinating subset deliberately leaves parts of the world incomplete or intentionally distorts the familiar layout for specific purposes. These incomplete and unique projections reveal deep insights into both the mathematics of cartography and the creative ways humans visualize their world.

Understanding these projections requires first grasping the four fundamental types of distortion: shape (conformality), area (equivalence), distance (equidistance), and direction (azimuthality). No single projection can preserve all four simultaneously. Incomplete projections embrace this limitation by abandoning the goal of a global view, focusing instead on a region, a thematic message, or an artistic statement.

For a thorough primer on map projection basics, the Esri ArcGIS documentation provides an excellent technical overview.

The Philosophy and Practicality of Incomplete Map Projections

Incomplete map projections intentionally exclude large portions of the Earth’s surface. They are not errors — they are deliberate design choices that reduce distortion for the area of interest. By cutting away unnecessary regions, the projection can preserve shape and scale more faithfully for the targeted geography.

Azimuthal Projections: Point of View

Azimuthal (or zenithal) projections project the globe onto a plane tangent at a single point. All points on the map are shown as they would appear from that central point, making them ideal for polar maps or for showing great-circle routes from a chosen city. The distortion increases radially outward, so cartographers often limit the map to one hemisphere. Common variants include the Azimuthal Equidistant (used for UN flags and polar map, preserving distance from the center) and the Lambert Azimuthal Equal-Area (preserving area, widely used for continent maps).

The azimuthal family shows how limiting the map to a half-sphere yields a projection with minimal distortion near the center — a tradeoff few global projections can match.

Interrupted (Cut) Projections: A Patchwork of Accuracy

An interrupted projection breaks the globe into several lobes or gores that are then flattened separately, like peeling an orange and laying the peel flat. The cuts fall in oceans or uninhabited areas, so that landmasses remain largely intact and suffer less distortion. The most famous example is the Goode Homolosine Projection, which splits the world into six interrupted lobes. It is an equal-area projection, meaning it preserves the relative sizes of countries — a crucial feature for thematic maps showing population or land use.

The Cahill-Keyes Butterfly Projection (and its newer variants) takes a different approach: it uses octahedral goring to produce a map that can be folded into a globe-like shape. Interrupted projections trade visual continuity for accuracy in area and shape, making them popular in textbooks and atlases.

Regional Projections: Focusing on One Continent

Many maps are designed to depict only a single continent or country with minimal distortion. For instance, the Albers Equal-Area Conic projection is often used for the United States and Canada because it preserves area accuracy across mid-latitude regions. The Transverse Mercator is standard for narrow north-south countries like Chile. By ignoring the rest of the world, these projections can use optimal parameters for the region — a practice that extends even to national mapping systems like the Universal Transverse Mercator (UTM) grid.

Unique and Artistic Map Projections: Breaking the Mold

Beyond incomplete projections lies a realm of truly unconventional designs. These maps often challenge the viewer to see global relationships differently, sometimes sacrificing practicality for conceptual impact.

The Dymaxion Map: Unfolding the Globe

Invented by Buckminster Fuller in 1943, the Dymaxion map projects the Earth onto a polyhedron (a cuboctahedron) and then unfolds the faces into a flat net. The result is a nearly continuous map with minimal shape and area distortion compared to traditional projections. Fuller intended it as a tool for seeing the world without political bias — his map shows the continents as a single landmass surrounded by ocean. The Dymaxion map has been called “the world’s least distorted map” and is still used today in educational and environmental contexts. A detailed explanation is available at the Buckminster Fuller Institute.

Peirce Quincuncial Projection: The World in a Square

Developed by Charles Sanders Peirce in 1879, the Peirce quincuncial projection is a conformal (shape-preserving) projection that maps the entire sphere onto a square. It uses a complex mathematical transformation that results in a single square with the South Pole at the center and the North Pole split into four corners. The distortion is extreme near the corners, but the pattern resembles a “quincunx” (a cross of five points). It is rarely used for practical navigation but remains a favorite among mathematicians and map enthusiasts for its elegant symmetry.

Waterman Butterfly Projection

Steve Waterman’s butterfly projection (1996) extends the idea of gored maps by dividing the globe into eight symmetrical sections that resemble butterfly wings. It is based on a truncated octahedron and offers low areal distortion while maintaining a striking visual pattern. The butterfly layout emphasizes the Pacific and Atlantic oceans as central, providing an unconventional view of continental relationships. It is used in some posters and software as an artistic alternative to the Robinson or Winkel Tripel projections.

The Winkel Tripel and the AuthaGraph: Modern Innovations

While not incomplete, two modern projections deserve mention for their unique compromises. The Winkel Tripel, created by Oswald Winkel in 1921, averages the coordinates of the Aitoff and Equirectangular projections to produce a map that balances area and shape distortion — it is the standard used by the National Geographic Society since 1998. The AuthaGraph projection, invented by Hajime Narukawa in 1999, uses a tetrahedral goring method to create an equal-area map with very low shape distortion; its unique layout allows the map to be tessellated with no visible seams. The AuthaGraph won the Good Design Grand Award in 2016 and is considered one of the most accurate flat maps of the globe. More details can be found in the AuthaGraph official site.

Practical Applications: Why We Use Incomplete and Unique Projections

These projections are not just academic curiosities. They serve real-world needs across fields.

The Azimuthal Equidistant projection is used for radio antenna range maps and for showing distances from a central point on air-traffic control displays. The Universal Transverse Mercator (UTM) system uses 60 narrow zones, each with its own Transverse Mercator projection — effectively a collection of incomplete projections that cover only 6° bands. The US National Grid and military grid reference systems rely on this approach. For global analysis, the Equal Earth projection (2018) was designed as an alternative to the Robinson, offering equal-area accuracy with a familiar look; many GIS professionals now adopt it for world maps in thematic contexts.

Education and Atlases

Textbooks frequently use interrupted projections like the Goode Homolosine to teach about continental sizes without the bias of the Mercator projection, which massively exaggerates the area of high-latitude countries. Educational mapmakers choose incomplete projections to ensure that students see accurate comparisons of landmass — for instance, that Greenland is not bigger than Africa. The Dymaxion map is sometimes used in geography classrooms to spark discussions about map bias.

Art and Graphic Design

Map projections have become a canvas for artistic expression. The Waterman Butterfly and Peirce Quincuncial projections appear in posters, logos, and generative art. The Myriahedral projection and HEALPix (often used in cosmology) show how projection mathematics can create visually stunning patterns. Some artists intentionally distort maps to create surreal landscapes or to comment on geopolitical power dynamics.

Controversies and Criticisms of Incomplete Projections

Not all incomplete projections are well received. Critics argue that interrupting the map induces cognitive confusion — viewers may not realize the ocean is continuous. The Gall-Peters projection, though equal-area and not technically incomplete, sparked heated debate because its rectilinear shape and severe east-west stretching made it unpopular even though it corrected the area distortions of the Mercator. Incomplete projections that cut through landmasses (like some early interrupted designs) are avoided because they misrepresent political boundaries. Modern cartographers follow best practices: cut through oceans only, or use sufficiently small interruptions that the human eye can mentally reconnect the pieces.

The Future: Interactive Maps and Dynamic Projections

Digital mapping has revolutionized how we think about projections. Online platforms like Google Maps and Mapbox use Web Mercator for its ease of tiling, but users can now switch to different projections on the fly. The new Jason Davies map transition tool demonstrates this beautifully, allowing real-time morphing between dozens of projections. In an era of interactive globes and virtual reality, the concept of an “incomplete” projection may become less relevant — users can rotate a 3D globe to see any region without distortion. Yet for static maps and printed atlases, the clever design of incomplete and unique projections remains essential for communicating accurate spatial information.

As cartographic technology evolves, we may see more projections that blend artistic creativity with mathematical precision. The long history of map projections — from Ptolemy’s conic to Narukawa’s AuthaGraph — shows that there is no single “best” way to flatten the Earth. Incomplete and unique projections remind us that every map is a choice, and that sometimes leaving part of the world unseen allows the rest to be seen more clearly.