Every map projection distorts. Mercator preserves angles but inflates areas. Peters preserves areas but distorts shapes. The choice between projections is a choice between which distortions to accept, and that choice depends on the map's purpose. There is no optimal projection — only optimal tradeoffs.
Recent work applies dimensionality reduction to the space of map projections themselves, representing projections as points in a low-dimensional space where proximity indicates similarity in distortion characteristics. The result is a map of maps: a visualization showing how hundreds of known projections relate to each other, clustered by the distortion patterns they share.
The insight: projections that were invented centuries apart by different mathematicians for different purposes often cluster together, revealing shared distortion structures that weren't apparent from their mathematical formulations. The projections look different in their equations but produce similar patterns of area and angle distortion. The mathematical diversity masks a smaller underlying set of distortion families.
This matters practically. When choosing a projection for a specific application — say, a climate visualization where area accuracy matters — the map of maps shows not just which projection minimizes area distortion but which family of projections shares that property, offering alternatives with different secondary characteristics (shape preservation, computational simplicity, aesthetic appeal).
The broader lesson is about understanding solution spaces. When a problem has many known solutions (projections, algorithms, designs), mapping the relationships between solutions reveals structure invisible from examining solutions individually. Two projections might seem equally good by area distortion alone but occupy very different positions in the full distortion space, meaning they handle edge cases (polar regions, narrow landmasses, oceanic areas) differently. The map of maps makes these hidden tradeoffs visible.