Shape

How many holes does a straw have? Geometry doesn’t just answer such quirky questions—it reveals the invisible math woven into everyday life.

1. Geometry Exists Anywhere There's a Notion of Distance

Geometry goes far beyond shapes and physical places; it can be applied to any system where distance matters. At its core, geometry is about measuring relationships, and that relationship can be defined in countless ways. For example, the "crow-fly" metric on a map measures a straight line between two points, while an alphabetical city list could form a different kind of geometry.

Even non-physical concepts can be placed within geometric frameworks. Consider actors: their "geometry" can be mapped using the costar distance, which calculates the shortest chain of movies linking one actor to another. Another surprising example of geometry involves a shuffled deck of cards. Every possible arrangement of the 52 cards can be thought of as a point in space, and the "distance" between arrangements can be measured by the number of riffle shuffles needed to reach one state from another.

Geometry opens up an almost limitless field of exploration. Whether it’s mapping actors’ costar connections, designing games, or reshuffling decks of cards, geometry teaches us that distance applies to almost everything.

Examples

• Using alphabetical city names as a unique measure of distance.
• Calculating how many links connect George Reeves and Keanu Reeves in movies.
• Exploring the vast number of arrangements in a deck of cards via shuffling.

2. Geometry Resolves Paradoxes by Balancing Intuition and Logic

When grappling with puzzles like "How many holes does a straw have?" our intuition might conflict with logical reasoning. Geometry allows us to refine and sometimes overhaul such intuitions. Depending on how "holes" are defined, answers about a straw can vary. But topology, a branch of geometry, provides a clear method to determine the most logical perspective.

Topology simplifies objects without destroying their defining structure—by shortening a straw to a flat band, we see it's essentially an annulus, a shape that has one hole. The same logic can be applied to pants (which topologically have two holes) or even complex, seemingly paradoxical objects like an inflated balloon, which might seem like it starts with "negative one holes" to account for its behavior post-popping.

This method of combining intuition with structured reasoning helps solve riddles, offering conclusions that are not only logical but also explanatory.

Examples

• Topologically reducing pants into a double-annulus shape with two holes.
• Explaining an inflated balloon’s "negative one holes" conceptually.
• Refining intuition with logic to answer quirky puzzles like the straw’s holes.

3. The Random Walk Principle Explains How Things Move Through Space

The random walk, a concept studied by Sir Ronald Ross, explains the seemingly erratic movement of objects or creatures, like mosquitoes. Ross observed that despite random choices in direction, a mosquito’s most likely final resting place remains the point where it started. This is because randomness tends to balance out over many steps, similar to how flipping a coin repeatedly averages out to 50 percent heads and 50 percent tails.

This mathematical principle finds applications far beyond mosquito paths. For example, stock prices often follow a random walk, with financial instruments fluctuating in seemingly unpredictable ways only to return to their starting points. Similarly, random walks even apply to molecules in physics and poetry, explaining how patterns repeat over time despite random movement.

Every seemingly chaotic system, from finance to nature, often displays surprisingly predictable random-walk patterns when viewed geometrically.

Examples

• A mosquito’s flight path likely returns it to where it started.
• Stock market fluctuations often settle back to initial pricing levels.
• Molecules in liquids follow random patterns that mimic random walks.

4. Markov Chains Link the Future to the Present

Markov chains, another mathematical concept, describe scenarios where an outcome depends on the current state rather than any prior history. Consider a mosquito that only flies between two bogs: it tends to stay in its current bog if blood is plentiful. Over time, the frequency of its visits to each bog stabilizes mathematically, even though its movements appear unpredictable.

Markov’s studies uncovered that variations in how letters (like vowels and consonants) follow one another can even define writing styles. For instance, a deeper mathematical pattern distinguishes Pushkin’s verses from Aksakov’s prose. Markov chains, therefore, underlie not just natural movements but also cultural patterns like language.

This method of studying transitions helps us make sense of systems where the present strongly influences the next step, bridging randomness and causality.

Examples

• Mosquitoes settling on a fixed pattern of bog-hopping probabilities.
• Analyzing vowels and consonants to differentiate Russian authors’ writing styles.
• Using Markov chains to study human decision making based on current states.

5. Pandemics Spread Through Geometric Patterns of Growth

Ross also applied geometry to pandemics, revealing how diseases spread exponentially through populations. Early stages show slow increases in cases, followed by seemingly explosive growth. But this growth eventually slows as the virus runs out of new hosts. This pattern—governed by geometric progression—reflects the principles behind exponential growth and decay.

Using these ideas to calculate the variable R0, scientists can determine how contagious a disease is. When R0 is below 1, infections decline; above 1, they rise. These concepts also apply to less harmful "contagions," like rumors, which mirror geometric growth and eventual saturation.

Understanding pandemics geometrically provides clearer, data-driven views into both their rise and potential decline.

Examples

• Tracking COVID-19 deaths as they exponentially increased before slowing.
• Using geometric progression to model how diseases spread.
• Connecting mathematical models of disease to the spread of rumors.

6. Games Reflect Predictable Geometries

Games like checkers, chess, and Nim can be broken down into geometric patterns resembling tree diagrams. In these games, every possible outcome branches out from a starting root. By analyzing these branches, players can often predict outcomes based on strategies that eliminate losing options.

For instance, analyzing a simple game like Nim reveals that certain starting arrangements guarantee victory or defeat, depending on who plays first and how moves are responded to. This same method applies to more complex games like checkers—where perfect play ensures specific predictable outcomes, such as draws.

Breaking games into geometric patterns allows us to identify winning strategies, making abstract ideas tangible.

Examples

• Analyzing Nim positions to see where losses are unavoidable.
• Showing how perfect checkers play leads to guaranteed draws.
• Using tree diagrams to visually map all possible game outcomes.

7. Machines Learn by Minimizing Errors Using Gradient Descent

Gradient descent is a step-by-step process machines use to improve their learning. Imagine teaching a computer what a cat looks like. By adjusting its "guessing strategy" step by step, the machine refines its approach until it achieves the lowest possible error rate, turning data into knowledge.

This learning process involves testing possible strategies, measuring errors (called wrongness scores), and iteratively reducing those errors. Over time, machines "learn" correct patterns by adjusting their guesses until they align with reality.

Just like a mountaineer climbing by assessing the steepest path upward, gradient descent helps machines find the point of least error by learning from their environment.

Examples

• Feeding computers thousands of labeled images to teach pattern recognition.
• Using wrongness scores to measure and improve machine-learned strategies.
• Applying gradient descent in artificial intelligence for medical diagnoses.

8. Math Has Been Manipulated for Political Advantage

In modern politics, geometry has intersected with partisanship through gerrymandering—manipulating voting districts to favor one party. In Wisconsin, Republicans used precise algorithms to create a map maximizing their voter efficiency, giving them disproportionate control of the state legislature in 2018 despite statewide Democratic victories.

By calculating efficiency gaps (the difference in wasted votes between parties), researchers uncovered glaring inequities. Advanced geometric tools offer ways to expose such manipulations, even if correcting them remains politically daunting.

Understanding how math distorts representation can lead to better-informed attempts to create fair voting systems.

Examples

• Wisconsin Republicans maintaining legislative dominance by designing districts.
• Using efficiency gaps to measure fair versus skewed district maps.
• Analyzing voting patterns with geometric tools to highlight gerrymandering.

9. Computer-Generated Maps Fight Gerrymandering

Modern software can create unbiased maps of voting districts to counteract gerrymandering. These maps consider factors like contiguity and legal requirements while avoiding human biases. To detect unfair maps, statistical ensembles compare actual district maps against thousands of randomly generated alternatives.

For example, in one study of Wisconsin’s maps, computer-generated ensembles showed that the extreme GOP victory in 2012 was unlikely based on unbiased maps—a clear sign of gerrymandering. This method helps mathematically prove unfairness and offers tools to challenge biased districts.

By combining computers with geometry, we can make elections fairer and more transparent.

Examples

• Pennsylvania court rejecting "Goofy kicks Donald Duck" gerrymandered map.
• Using random map ensembles to expose partisan bias in Wisconsin districts.
• Applying geometric principles to improve fairness in voting systems.

Takeaways

1. Use geometric reasoning to approach real-world puzzles and complex problems logically.
2. Advocate for transparency in elections by supporting unbiased, computer-generated district maps.
3. Apply tools like random walks and Markov chains to better understand patterns in systems like language or stocks.