Science Friday

How Math Helps Us Map The World

October 16, 2025

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  • Creating a flat map from the curved Earth is mathematically impossible without distortion, as proven by Gauss, meaning every map is an imperfect representation. 
  • The Mercator projection, while distorting size (e.g., making Greenland look much larger than South America), was mathematically designed to preserve angles, making it historically useful for sailors navigating with a compass. 
  • Modern digital navigation tools, like phone maps, can degrade our innate sense of direction, and complex logistical problems like package delivery rely heavily on solving variations of the traveling salesman problem using proprietary algorithms. 

Segments

Map Distortion and Mercator
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(00:01:02)
  • Key Takeaway: All maps are inherently distorted because translating the curved Earth onto a flat surface mathematically requires tearing or distortion.
  • Summary: Every map is distorted because translating a curved sphere like Earth onto a flat sheet is mathematically impossible without distortion, a fact proven by Gauss in the 19th century. The Mercator projection preserves shapes and angles, which was crucial for sailors using a compass, but severely distorts the relative sizes of landmasses, making areas near the poles appear disproportionately large. For instance, Greenland is shown as nearly the size of South America when it is actually eight times smaller.
Guest Background and Map Types
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(00:05:40)
  • Key Takeaway: Subway maps prioritize connectivity and station order over accurate spatial distance, leading to potential misjudgments of travel time on foot.
  • Summary: Dr. Paulina Rowińska developed an interest in maps through childhood geography enthusiasm before studying mathematics. She highlights that subway maps are designed to efficiently show lines and station sequence, intentionally ignoring actual distances between stops. Using a subway map to plan walking routes can be misleading, as visually equal segments on the map might represent vastly different walking times.
Navigation Technology Impact
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(00:08:36)
  • Key Takeaway: Over-reliance on phone-based navigation systems may be causing a measurable decline in human spatial navigation skills.
  • Summary: The shift from physical maps to digital navigation on phones means people often stop thinking about where they are going or how to navigate independently. Research suggests that constantly following turn-by-turn directions leads to people becoming worse at navigating. The physical act of using a paper map, like cycling with one in London, can foster a better understanding of the city’s layout.
Measuring Coastlines
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(00:10:26)
  • Key Takeaway: The measured length of a coastline is scale-dependent; zooming in reveals more detail (nooks and crannies), resulting in a longer measurement.
  • Summary: Measuring a coastline is difficult because coastlines are not smooth lines, especially in areas like Norway with fjords. The measured length changes based on the scale of the map used; a larger scale map shows more detail and thus a longer perimeter. Comparing coastline lengths is only meaningful if both measurements are taken using the exact same scale.
Mapping the Ocean Floor
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(00:12:00)
  • Key Takeaway: Mapping the ocean floor is significantly more challenging than land mapping due to water obscuring satellite views and the sheer inaccessibility of the vast deep ocean.
  • Summary: Mapping the ocean floor is challenging because water blocks detailed satellite imaging, meaning only large features like underwater mountains are easily caught. Direct measurement is required, which is difficult given the ocean’s vastness and inaccessibility. Until the mid-20th century, many people incorrectly assumed the ocean floor was uniformly flat.
Logistics and Traveling Salesman
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(00:13:24)
  • Key Takeaway: Package delivery efficiency relies on solving the traveling salesman problem daily, a mathematically complex task where companies seek ‘good enough’ routes rather than the absolute best.
  • Summary: The traveling salesman problem—finding the most efficient route to visit multiple destinations without repetition—is actively used by logistics companies like FedEx and UPS. Because the daily set of deliveries changes, finding the absolute optimal route is computationally impossible, so companies rely on proprietary, ‘good enough’ algorithms to save time and money. The success of these delivery services is directly tied to the mathematical efficiency of these secret routing algorithms.
Poor Map Design and Favorites
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(00:15:45)
  • Key Takeaway: Maps that overload information or misrepresent data proportions, such as standard election maps, are poorly designed, while maps showing undersea geography are highly valued.
  • Summary: Poorly designed maps often try to show too much information at once, obscuring the primary purpose. Election maps that color states based on winner suggest area size equals voting power, which misrepresents the data; maps where area is proportional to votes are preferred. A favorite map mentioned is an old National Geographic world map that beautifully visualizes the geography of the ocean bottom.