Waterman Butterfly
The Waterman Butterfly unfolds the globe like origami: instead of stretching the world into a rectangle, it cuts the oceans and lays the continents into eight wings. Landmasses keep almost true proportions, and the price — sliced oceans — is visible at a glance.
An unfolded globe that cuts the oceans to save the continents
The idea is as old as Bernard Cahill's 1909 butterfly map: project the sphere onto an octahedron, then unfold it flat. In 1996 Steve Waterman derived his variant from the geometry of the Waterman polyhedra, choosing the shape and cuts so they run through water. The result is a map on which Africa, the Americas, and Eurasia stay in one piece and look natural.
This projection shows that flattening a sphere always demands a choice: where to put the tear. Mercator hides it in stretched poles; the Waterman Butterfly shows it openly as sliced oceans. For a student it is a great lesson — the 'orange peel' cannot be flattened without a cut, so the question is not whether, but where.
Global Cartographic Grid
Distortion Properties
| Property | Characteristic |
|---|---|
| Area | ⚖️CompromiseLow distortion (continents keep close-to-true area proportions) |
| Shape | ❌DistortedLandmasses keep natural shapes; distortion is pushed into the oceans and cut lines |
| Distances | ❌DistortedUnreliable globally (the map is interrupted into eight lobes) |
| Angles & Directions | ✅PreservedPreserved locally near facet centers, worse toward the lobe edges |
| Continuity | ❌DistortedInterrupted (oceans are cut, the world unfolds like butterfly wings) |
History & Origin
Created in 1996 by Steve Waterman from the geometry of the Waterman polyhedra (a dense sphere packing). It builds on Bernard J. S. Cahill's earlier 1909 "butterfly map", which unfolded the globe into eight triangular octahedral lobes. Waterman arranged the cuts to run through the oceans and spare the continents.
Applications
Reference and educational maps, posters, and "unfolded globe" visualizations that show every continent at once with little distortion. Unsuitable for navigation or distance measurement.
How to read this map
Think of it as a flattened globe box: the continents are the intact panels, and the oceans are the folds that had to be cut so everything could lie flat.
- Read the continents as on a globe — their shape and size are close to true.
- Do not measure distances across the oceans; that is where the cuts run.
- Notice that the same ocean appears on two sides of the wings.
- Compare the size of Africa and Greenland with Mercator — here the illusion disappears.
What you gain and lose
Waterman sacrifices ocean continuity and distance fidelity to preserve continental shape and area. It is a good map of land, a poor map of the sea.
Posters, reference maps, and lessons about flattening the globe and about every map cutting something.
Navigation, measuring sea distances, and maps of oceans or currents.
✦ How do different countries look in this projection?
Analyze shape distortions of 5 countries in this cartographic projection and test them in the sandbox.
Brazil stays whole and shows South America's fair scale.
Test on map →
Russia spans several lobes — a good test of reading the cuts.
Test on map →
Australia stays continuous and natural in shape.
Test on map →
Greenland returns to a reasonable size, without the Mercator myth.
Test on map →
Poland gives a familiar European reference near a wing's center.
Test on map →Facts worth remembering
- Its direct ancestor is Bernard Cahill's 1909 butterfly map.
- The name comes from the Waterman polyhedra derived from sphere packing.
- The projection maps the sphere onto an octahedron unfolded into eight triangular lobes.
Keep reading about maps that reshape intuition
Frequently Asked Questions
Mostly for reference and education — it shows every continent at once with little shape and area distortion. It is an "unfolded globe", not a navigation map.
Because the globe is projected onto an octahedron and then unfolded into eight triangular lobes. The cuts run through the oceans, so the wings form the characteristic butterfly shape while the continents stay intact.
Not in the strict mathematical sense, but its area distortion is very small. Continents keep close-to-true proportions, far fairer than on a Mercator map.
Do not measure distances or courses across the oceans — the map is interrupted, so oceans and sea routes are cut apart. Use Mercator for navigation and an equal-area projection for measuring area.
Cahill (1909) first unfolded the globe into eight octahedral lobes. Waterman (1996) derived his arrangement from the geometry of the Waterman polyhedra, giving slightly different proportions and cut lines while keeping the same butterfly idea.