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): How well the projection preserves the local shapes of features.
  • Area (Equivalence): Whether the relative sizes of regions are accurate.
  • Distance (Equidistance): How true distances are preserved, usually along certain lines.
  • Direction (Azimuthality): The preservation of accurate compass bearings from a central point.

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. By selectively omitting or fragmenting parts of the globe, they can better preserve desired properties in the areas that matter most.

For a thorough primer on map projection basics, the Esri ArcGIS documentation provides an excellent technical overview, explaining how different projections serve various geographic and analytical needs.

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. This approach allows cartographers to tailor maps for specific applications, from navigation to thematic visualization.

Azimuthal Projections: Point of View

Azimuthal (or zenithal) projections project the globe onto a plane tangent at a single point. Imagine placing a light source inside a transparent globe and projecting the surface features onto a flat screen touching the globe at one spot. All points on the map are shown as they would appear from that central point, making these projections ideal for polar maps or for illustrating great-circle routes emanating from a chosen city.

The distortion increases radially outward from the center, so cartographers often limit the map to one hemisphere—typically 180 degrees of longitude—to keep distortion manageable. Some common azimuthal variants include:

  • Azimuthal Equidistant Projection: Preserves accurate distances from the central point to any other location. This projection is famously used in the United Nations emblem and for polar maps. Because distances from the center are true, it is practical for radio range maps and air-traffic control.
  • Lambert Azimuthal Equal-Area Projection: Preserves area, making it ideal for continent-level thematic maps where size comparisons are crucial. It is often used to depict regions like Africa or Australia with minimal area distortion.

By limiting the map to a half-sphere, azimuthal projections yield a representation with minimal distortion near the center — a tradeoff few global projections can match. This focused perspective is invaluable for applications requiring precise angular or distance relationships radiating from a single point.

Interrupted (Cut) Projections: A Patchwork of Accuracy

Interrupted projections break the globe into several lobes or gores, which are then flattened separately—akin to peeling an orange and laying the peel flat without stretching. The cuts are strategically placed in oceans or sparsely inhabited areas so that landmasses remain largely intact and suffer less distortion. This approach allows for more accurate area and shape preservation across continents.

The most famous example is the Goode Homolosine Projection, developed by John Paul Goode in 1923. It splits the world into six interrupted lobes, combining a sinusoidal projection near the equator with a Mollweide projection at higher latitudes. This equal-area projection preserves the relative sizes of countries, making it particularly useful for thematic maps showing population density, land use, or ecological data.

The Cahill-Keyes Butterfly Projection is a modern reinterpretation of interrupted projections. It uses octahedral goring to produce a map that can be folded into a globe-like shape, resembling a butterfly when unfolded. This design balances minimal distortion with visual appeal and is gaining popularity in educational and artistic contexts.

Interrupted projections trade visual continuity for accuracy in area and shape, making them popular in textbooks, atlases, and thematic mapping. Despite their segmented appearance, they provide a more truthful representation of the Earth's surface than many continuous projections.

Regional Projections: Focusing on One Continent or Country

Many maps are designed to depict only a single continent or country with minimal distortion by ignoring the rest of the world. Regional projections optimize parameters specifically for the geography of interest, thereby improving accuracy in shape, area, or distance within the targeted zone.

Some notable examples include:

  • Albers Equal-Area Conic Projection: Widely used for the United States and Canada, this projection preserves area accuracy across mid-latitude regions and is ideal for large countries with east-west orientation. It uses two standard parallels to minimize distortion within the chosen latitude band.
  • Transverse Mercator Projection: Employed for narrow north-south countries like Chile and Norway, it preserves shape and scale along a central meridian with distortion increasing away from it.
  • Universal Transverse Mercator (UTM) System: Divides the globe into 60 longitudinal zones, each 6° wide, applying a Transverse Mercator projection within each. This system is widely used for military and civilian mapping due to its high accuracy over small regions.

By focusing on smaller geographic extents, regional projections enable detailed and accurate mapping for navigation, land management, and resource planning. They exemplify how incomplete coverage can be a strength rather than a limitation.

Unique and Artistic Map Projections: Breaking the Mold

Beyond incomplete projections lies a realm of truly unconventional and artistic designs. These maps often challenge the viewer to see global relationships differently, sometimes sacrificing conventional practicality for conceptual impact, aesthetics, or philosophical statements about the world.

The Dymaxion Map: Unfolding the Globe

Invented by Buckminster Fuller in 1943, the Dymaxion map projects the Earth onto a polyhedron known as a cuboctahedron, which is then unfolded into a flat net. Unlike traditional projections that try to fit the globe into a rectangle or oval, the Dymaxion map’s segmented shape reveals the continents as one nearly continuous landmass surrounded by ocean, emphasizing Earth's connectedness.

Fuller intended the Dymaxion map as a tool for seeing the world without political bias — it avoids placing any continent at the center and does not show "up" or "down" in a conventional sense. This map minimizes both shape and area distortion compared to many other projections, making it one of the most accurate flat maps of the globe. It is still used today in educational and environmental contexts to encourage holistic thinking about global issues.

For an in-depth exploration, visit the Buckminster Fuller Institute, which preserves Fuller's legacy and provides resources on the Dymaxion map’s design and applications.

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 perfect square. This feat is accomplished through a complex mathematical transformation involving elliptic functions and conformal mapping theory.

The map positions the South Pole at the center and splits the North Pole into four corners, creating a pattern resembling a quincunx—a cross arrangement of five points. While distortion becomes extreme near the corners, the projection preserves local shapes remarkably well elsewhere.

Though rarely used for practical navigation, the Peirce quincuncial projection remains a favorite among mathematicians and map enthusiasts due to its elegant symmetry and unique visual appeal. It also inspires modern artistic representations of global data.

Waterman Butterfly Projection

Steve Waterman’s butterfly projection, introduced in 1996, extends the idea of gored maps by dividing the globe into eight symmetrical sections resembling butterfly wings. This projection 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 features, providing an unconventional view of continental relationships. It offers a fresh perspective on global geography, highlighting connections across oceans often marginalized in traditional projections.

Due to its artistic design and balanced distortion, the Waterman butterfly projection is used in posters, educational materials, and some mapping software as an alternative to more conventional projections like Robinson or Winkel Tripel.

The Winkel Tripel and the AuthaGraph: Modern Innovations in Projection

While not incomplete, two modern projections deserve mention for their innovative approaches to balancing distortion:

  • Winkel Tripel Projection: Created by Oswald Winkel in 1921, this projection averages the coordinates of the Aitoff and Equirectangular projections to produce a map that balances area and shape distortion. Its pleasing visual balance and reduced extreme distortions led the National Geographic Society to adopt it as their standard world map projection in 1998.
  • AuthaGraph Projection: Invented by Japanese architect Hajime Narukawa in 1999, this projection 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 endlessly without visible seams, making it suitable for innovative display methods, including 3D models and interactive media. 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 on the AuthaGraph official site, which provides interactive demonstrations and technical explanations.

Practical Applications: Why We Use Incomplete and Unique Projections

These projections are not just academic curiosities. They serve real-world needs across a variety of fields, from navigation and geographic information systems (GIS) to education and the arts.

The Azimuthal Equidistant projection is widely used for mapping radio antenna range and for visualizing distances on air-traffic control displays. Its unique property of preserving distances from a central point makes it invaluable for these purposes.

The Universal Transverse Mercator (UTM) system exemplifies the use of incomplete projections in GIS. By dividing the globe into 60 narrow zones, each with its own Transverse Mercator projection, it effectively creates a series of localized maps that minimize distortion within each zone. This approach is essential for national mapping systems like the US National Grid and for military grid reference systems worldwide.

For global thematic mapping, the Equal Earth projection (introduced in 2018) is gaining traction as an alternative to the Robinson projection. It offers equal-area accuracy while maintaining a familiar appearance, making it popular among GIS professionals for representing global data sets such as climate or population.

Education and Atlases

Textbooks frequently employ interrupted projections like the Goode Homolosine to teach students about continental sizes without the bias of the Mercator projection, which greatly exaggerates the area of high-latitude countries like Greenland and Russia. By using incomplete projections, educators ensure that learners gain a more accurate understanding of the relative scale of landmasses worldwide.

The Dymaxion map is sometimes incorporated into geography classrooms and environmental studies to spark discussions about map bias and to encourage a more holistic, interconnected view of the planet.

Art and Graphic Design

Map projections have become a canvas for artistic expression. Projections like the Waterman Butterfly and Peirce Quincuncial appear in posters, logos, and generative art projects. Artists and designers exploit the mathematical foundations of projections like Myriahedral and HEALPix (commonly used in cosmology) to create visually stunning and thought-provoking patterns.

Some artists intentionally distort maps to create surreal landscapes or to comment on geopolitical power dynamics, challenging viewers to reconsider their assumptions about geography and cultural dominance. These artistic projections expand the traditional role of maps beyond navigation to become tools of storytelling and critique.

Controversies and Criticisms of Incomplete Projections

Not all incomplete projections are universally appreciated. Critics argue that interrupting the map induces cognitive confusion — viewers may not realize that the oceans are continuous or that the “gaps” represent artificial seams rather than natural boundaries. This can hinder intuitive understanding, especially for casual map users.

The Gall-Peters projection, though equal-area and not strictly incomplete, sparked heated debate because its rectilinear shape and severe east-west stretching made it unpopular despite correcting the Mercator’s area distortions. The controversy highlighted how map aesthetics and cultural perception can influence acceptance as much as technical accuracy.

Incomplete projections that cut through landmasses (as some early interrupted designs did) are generally avoided today because they misrepresent political boundaries and can confuse geopolitical understanding. Modern cartographers adhere to best practices by placing interruptions only in oceans or using sufficiently small gaps so the human eye can mentally reconnect the pieces.

The Future: Interactive Maps and Dynamic Projections

Digital mapping has revolutionized how we think about projections, enabling dynamic, interactive experiences that overcome many limitations of static maps. Online platforms like Google Maps and Mapbox primarily use Web Mercator for its ease of tiling and compatibility with web technologies, but users now have the option to switch projections on the fly.

The Jason Davies map transition tool beautifully demonstrates this capability by allowing real-time morphing between dozens of projections. This interactivity enables users to appreciate the strengths and weaknesses of each projection firsthand and to choose the most appropriate one for their needs.

In an era of interactive globes, augmented reality, 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. However, for static maps, printed atlases, and certain thematic contexts, 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 and truthfully.