Mapmatics: How We Navigate the World Through Numbers

Mapmatics: How We Navigate the World Through Numbers

Introduction

I have always liked maps.

There is something strangely comforting about looking at a map. A complicated world suddenly becomes a collection of lines, colours, names and numbers. Countries have borders. Cities have dots. Roads connect one place to another. Mountains become shapes. Oceans become empty spaces.

Everything seems organised.

But recently, while reading Mapmatics: How We Navigate the World Through Numbers by Paulina Rowińska, I started thinking about maps very differently.

What if a map is not simply a picture of the world?

What if it is a mathematical interpretation of the world?

And even more importantly:

What if the way a map represents the world influences the way we understand the world itself?

That is the fascinating idea behind Mapmatics.

Rowińska combines mathematics, geography, cartography, history, technology and politics to explore how maps work — and how deeply they influence our everyday lives. The book asks questions that sound surprisingly ordinary at first: How do we measure the Earth? How does a delivery driver find the most efficient route? Why do some countries look bigger on maps than they really are? How can maps influence elections? And how can we map things that we cannot even see?

The answers are much more mathematical than I expected.

And they are also much more political.

Chapter 1: How to Describe the Earth

Before we can draw a map, we need to answer a basic question:

What exactly are we mapping?

We usually imagine the Earth as a sphere. But the real Earth is not a perfect mathematical sphere. It is slightly flattened at the poles and bulges around the equator.

To describe locations precisely, humans developed systems based on numbers — especially latitude and longitude.

Instead of saying:

“The restaurant is somewhere north of the city centre.”

we can give its coordinates.

Suddenly, a physical place becomes a mathematical point.

This is one of the most powerful ideas behind modern geography: we can transform physical space into numerical data.

Once a location becomes data, it can be measured, compared, calculated and processed by computers.

And that is where mathematics begins to change the way we experience geography.

Today, when I open a map application on my phone, I rarely think about the extraordinary mathematical system underneath that little blue dot.

I simply think:

“Where am I?”

But behind that question are coordinates, geometry, satellites, algorithms and enormous amounts of spatial data.

The world has become something that can be calculated.

Chapter 2: How to Make a Map

This is where things become particularly interesting.

The Earth is three-dimensional.

A traditional map is two-dimensional.

So how do we put a curved planet onto a flat surface?

The answer is through something called a map projection.

And there is a problem.

We cannot flatten a sphere without distorting something.

If we try to preserve distances, we may distort areas.

If we preserve areas, we may distort shapes.

If we preserve angles, we may distort size.

There is no perfect projection.

One of the most famous examples is the Mercator projection, originally developed for navigation. It became extremely influential because it is useful for certain navigational purposes.

But it also dramatically enlarges areas near the poles.

This means that when we look at a familiar world map, our visual impression of the relative size of countries can be misleading.

And this leads to one of the most important lessons in Mapmatics:

A map does not have to be “wrong” to distort reality.

Every map makes choices.

The better question is not:

“Is this map accurate?”

but:

“What kind of accuracy was this map designed to preserve?”

That is a much more interesting question.

Chapter 3: How to Measure a Line

One of the strangest things I learned from this book is that even something as apparently simple as measuring a line can become complicated.

Imagine measuring a coastline.

You take a ruler and measure it.

You get a number.

But now imagine using a shorter ruler.

You can follow more of the coastline's twists and turns.

The measured coastline becomes longer.

Use an even smaller ruler and you capture even more detail.

The result can become longer still.

This is related to the fascinating mathematical idea of fractals and the complexity of natural shapes.

A coastline is not a simple geometric line.

It contains details within details.

And this raises a surprisingly philosophical question:

How much does the measurement depend on the way we measure?

We often assume that every object has one objective numerical value waiting for us to discover.

But sometimes, the measurement itself depends on the scale and method we choose.

That idea extends far beyond geography.

It makes me think about how often we believe numbers are completely objective when, in reality, numbers can depend on definitions, assumptions and measurement systems.

Chapter 4: How to Navigate

This chapter brings mathematics into something much more familiar:

finding the best route.

Imagine a delivery driver who has hundreds of packages to deliver in one day.

The problem is not simply finding a road from A to B.

The driver needs to determine an efficient sequence:

A → B → C → D → E...

But once the number of destinations becomes large, the possibilities become enormous.

This is where mathematics and computer science become essential.

A road network can be represented as a graph.

Places become nodes.

Roads become connections between those nodes.

Algorithms can then calculate possible routes according to different criteria.

Shortest distance.

Shortest time.

Lowest cost.

Fewest transfers.

Or perhaps the route that avoids traffic.

This made me look at Google Maps differently.

When I ask my phone:

“How do I get there?”

I am not simply asking it to show me a road.

I am asking a computer to solve a mathematical optimisation problem.

And there is another interesting point.

The “best” route does not necessarily exist in isolation.

The answer depends on what we mean by best.

The shortest route may not be the fastest.

The fastest route may not be the cheapest.

The cheapest route may not be the safest.

So even an apparently simple question such as

“What is the best way to get there?”

actually requires us to define what “best” means.

Chapter 5: How to Simplify a Map

One of my favourite ideas in the book is that sometimes a map becomes more useful by becoming less accurate.

Consider a subway map.

If you look at an underground railway map, the stations are often not positioned according to their exact geographical locations.

Lines may be straightened.

Distances may be exaggerated or reduced.

Angles may be simplified.

Yet the map works beautifully.

Why?

Because the purpose of a subway map is not to tell us the precise geographical position of every station.

It is to answer a different question:

Where should I go, and where should I change trains?

This is an important lesson about information design.

A good map does not necessarily show everything.

It shows the information that matters for its purpose.

This idea appears everywhere in modern life.

A good diagram simplifies.

A good graph simplifies.

A good interface simplifies.

A good explanation simplifies.

But simplification always involves a choice.

And therefore we should ask:

What has been removed?

Sometimes the information that disappears is irrelevant.

But sometimes it is precisely the information we needed.

Chapter 6: How to Shape Society

This is where Mapmatics becomes much more than a book about geography.

Maps can have political consequences.

One example is electoral maps.

The way voting districts are drawn can influence political representation.

This is where the concept of gerrymandering becomes important.

By changing the boundaries of electoral districts, political groups can sometimes create arrangements that give one party an advantage over another.

The map is no longer simply describing political reality.

It can help create political reality.

This completely changes how we should think about maps.

A border on a map can look objective.

A line is just a line.

But who drew that line?

Why was it drawn there?

Who benefits from it?

Who loses?

These are political questions hidden inside what looks like geography.

And this is one of the most powerful ideas in the book:

Maps can be instruments of power.

Historically, maps have been used to define territories, claim land, organise populations and represent political authority.

Today, digital maps may look neutral because they are generated by software.

But software also follows rules.

And rules are designed by people.

So the question of power has not disappeared.

It has simply moved into algorithms and databases.

Chapter 7: How to Save a Life

Maps can also do something much more positive:

they can save lives.

A classic example is the nineteenth-century cholera outbreak in London and the work of physician John Snow.

Rather than looking only at individual cases, Snow examined where the cases occurred geographically.

When the cases were plotted on a map, a pattern emerged.

The spatial concentration of illnesses helped point toward a contaminated water source.

This demonstrates the enormous power of spatial data.

Imagine having a list:

Person A — sick.

Person B — sick.

Person C — sick.

Person D — sick.

It is difficult to see the relationship.

Now put those people on a map.

Suddenly we can see that they are concentrated in the same area.

The map reveals something that the raw list hides.

This principle remains extremely important today.

Maps can help researchers understand:

disease outbreaks
pollution
natural disasters
population movements
crime patterns
access to healthcare
emergency response

In other words, geography can turn data into evidence.

Chapter 8: How to Map the Invisible

Perhaps the most fascinating idea in the book is that we can map things that we cannot see.

We cannot simply open a window and look inside the Earth.

We cannot see seismic waves travelling through the planet.

We cannot directly observe every structure beneath our feet.

Yet scientists can use measurements to construct models of what lies beneath the surface.

Earthquakes, for example, produce seismic waves that travel through the Earth.

By studying how these waves behave, scientists can infer information about the planet's interior.

This is a beautiful example of mathematical reasoning.

We observe something we can measure.

We analyse the pattern.

Then we use mathematics to infer something we cannot directly observe.

So a map does not necessarily have to represent something visible.

It can represent a model.

And this idea extends into many areas of science.

We map invisible fields.

We map atmospheric systems.

We map underground structures.

We map networks.

We map probability.

We map relationships.

In this sense, “mapping” becomes much broader than drawing places on paper.

It becomes a way of making invisible structures understandable.

Afterword: How to Keep Up with Change

Traditional maps can give us the impression that the world is stable.

A printed map shows roads, cities, borders and rivers as if they were permanent.

But the real world is constantly changing.

Cities expand.

Roads are built.

Coastlines move.

Populations migrate.

Political borders change.

Weather changes.

Climate changes.

And traffic changes from one minute to the next.

Digital maps are therefore fundamentally different from the maps our grandparents might have used.

They can incorporate time.

A modern map can tell us not only:

“Where is this?”

but also:

“What is happening here right now?”

Traffic applications are a simple example.

The road itself may not have changed.

But the information attached to it has.

This means that modern cartography is increasingly about dynamic data rather than static pictures.

We are no longer simply mapping the world.

We are mapping a world in motion.

What Maps Really Tell Us

By the end of Mapmatics, I felt that the subject was no longer really “maps”.

It was about how humans represent reality.

A map is a model.

And every model simplifies reality.

It chooses some information and ignores other information.

It makes certain relationships visible while making others invisible.

That means we should never look at a map completely passively.

We can ask:

Who made it?

What was its purpose?

What does it preserve?

What does it distort?

What has been left out?

What assumptions are hidden inside it?

And perhaps most importantly:

How does this representation influence the way I think?

This is particularly relevant now that so much of our understanding of the world comes through digital maps.

We follow the blue line on our phones.

We trust the estimated arrival time.

We accept the suggested route.

We look at satellite images.

We search for places.

We compare distances.

We let algorithms decide what is “nearby”.

Without realising it, we are constantly navigating a mathematical representation of reality.

Why I Found This Book So Interesting

What surprised me most about Mapmatics is how many different subjects are hiding inside the idea of a map.

At first, I thought I was going to read a book about geography and mathematics.

Instead, I found myself reading about:

mathematics
history
navigation
computer science
science
politics
medicine
technology
and even philosophy.

The book also changed the way I think about the word “objective.”

We often trust numbers because numbers appear neutral.

A map looks scientific.

An algorithm looks mathematical.

A coordinate looks precise.

But precision does not necessarily mean neutrality.

Someone decided what should be measured.

Someone decided which variables mattered.

Someone designed the algorithm.

Someone decided what the map should show.

This does not mean that maps are useless or that mathematics cannot tell us objective things.

Quite the opposite.

Mathematics makes maps incredibly powerful.

But power is exactly why we should understand how these systems work.

A Simple Way to Remember the Book

After reading Mapmatics, I think there are four questions worth remembering whenever we look at a map:

1. Projection

How has reality been transformed?

The three-dimensional Earth has become a two-dimensional representation.

Something has to change.

2. Scale

At what level are we looking?

A city map and a world map cannot show the same amount of detail.

3. Simplification

What has been removed?

Every useful map leaves something out.

4. Algorithm

What rules determine what I see?

Increasingly, maps are not drawn entirely by humans.

They are produced and modified by algorithms.

These four ideas explain much more than cartography.

They are also useful ways to think about statistics, charts, artificial intelligence and data visualisation.

Final Thought

Before reading Mapmatics, I thought of maps as tools for finding places.

Now I think of them as arguments about reality.

A map tells us:

“This is what matters.”

And because it cannot show everything, it also quietly tells us:

“This is what you don't need to see.”

That does not make maps bad.

It makes them human.

We need simplification because reality is too complicated to carry around in our heads. We need coordinates because the world is too large to describe with words alone. We need algorithms because modern networks are too complicated for us to calculate manually.

But perhaps the lesson is not to stop trusting maps.

It is to become a more thoughtful map reader.

The next time Google Maps gives me a route, I might still follow the blue line.

But I will probably pause for a second and think:

Why this route?

What does the algorithm know?

What does it not know?

And when I look at a world map, I will remember that the shapes and sizes in front of me are not the world itself.

They are a mathematical interpretation of it.

And perhaps that is the most fascinating thing about Mapmatics:

We don't simply use numbers to navigate the world.

Numbers help us decide what the world looks like.