In Pursuit of the Unknown Summary

How have equations shaped the way we understand our world and even the very fabric of reality? This book explores equations as the unseen architects of our modern lives.

1. The Pythagorean Theorem Shapes How We Navigate the World

The Pythagorean theorem, expressed as a² + b² = c², is a cornerstone of geometry, enabling calculations involving right-angled triangles. While seemingly simple, its implications have revolutionized mapping, geography, and our understanding of mathematical space. This theorem allows any polygon to be broken down into right-angled triangles, making precise measurements feasible.

Initially used for surveying land, the theorem served as the foundation for triangulation—the creation of networks of triangles to measure distances and angles over vast terrains. This principle supports modern mapping technologies like GPS, where satellite data depends on the general principles of Pythagoras’s work.

Additionally, generalized forms of the theorem allow for measuring curved spaces like spheres. By applying triangles to spheres, we discovered Earth’s spherical shape. Today, researchers use these advanced methods to attempt to understand the shape and curvature of the entire universe.

Examples

• Navigational maps rely on networks of triangles derived from the theorem.
• Satellites apply these principles in positioning and GPS systems.
• Astronomers measure cosmic and planetary curvature using adaptations of the theorem.

2. Euler’s Formula Brings Beauty and Simplicity to 3D Shapes

Euler’s formula for polyhedra, expressed as F - E + V = 2, offers a profound way to connect the faces (F), edges (E), and vertices (V) of three-dimensional shapes. This formula gave rise to topology, a branch of geometry that studies objects based on their connectivity rather than their angles or lengths.

Euler’s formula makes it possible to determine whether two seemingly different shapes are topologically equivalent by simple counting. If two objects give the same result from this formula (2 for polyhedra), then they share the same topology, no matter how stretched or deformed one appears compared to the other.

This principle has found applications far beyond geometry, such as in biology. For example, mathematicians analyze the twists and curls of DNA to study its role in cell division—a biological process that must accommodate the complex interweaving of the DNA double helix.

Examples

• Classifying different polyhedra using simple arithmetic steps.
• Ensuring DNA replication processes are understood correctly using topological principles.
• Deforming shapes like spheres and toruses to explore their mathematical similarities.

3. The Bell Curve Organizes Chaos in Statistics and Medicine

The normal distribution, often visualized as a bell curve, reveals predictable patterns in datasets filled with apparent randomness. This concept emerged when mathematicians realized many social and natural phenomena—like heights or weights—cluster around an average with symmetrical variation on either side.

Used extensively in social science, the 19th-century Belgian sociologist Adolphe Quetelet uncovered the bell curve in data on crime rates, birth weights, and human behavior. Surprisingly, these patterns suggested collective human behavior follows mathematical rules, even if individual actions seem chaotic.

Medical researchers depend on the bell curve for evaluating the effectiveness of new treatments. For instance, they ensure a drug’s observed success rate doesn’t merely appear successful due to chance. The bell curve helps weed out errors and reliability issues, ensuring confidence in statistical conclusions.

Examples

• Quetelet observed bell-curved patterns when analyzing human height and crime statistics.
• Medical trials assess drug effectiveness using the normal distribution.
• Manufacturers use it to evaluate variations in product quality.

4. Water Flow Becomes Predictable Thanks to Navier-Stokes

The Navier-Stokes equation makes it possible to predict how fluids like air and water move. These flows are essential for everything from how blood circulates in the human body to the aerodynamic designs of cars and planes. Created in the 19th century, this equation unlocked a new understanding of fluid motion and its applications.

Engineers today simulate airflow around objects, such as car bodies, using algorithms derived from Navier-Stokes. These methods drastically minimize drag, leading to more fuel-efficient vehicles. Airplane wings, too, owe their shape to equations defining the movement of air.

This equation contributes significantly to modern surgery, such as calculating blood flow during complex procedures. Similarly, climate scientists rely on these mathematical principles for ocean and atmospheric simulations to model and predict climate change implications.

Examples

• Car and aircraft manufacturers improve designs based on fluid dynamics simulations.
• Surgeons calculate blood flow for critical operations like bypass surgeries.
• Climate scientists model changing ocean currents and air pressures to forecast weather.

5. Newton’s Law of Gravitation Makes the Cosmos Predictable

Newton’s law of gravitation (F = G * m₁ * m₂ / r²) provides a universal explanation for how objects attract one another. It underlies everything from planetary orbits to satellite trajectories. By observing a falling apple, Newton realized this same force explained objects on Earth and celestial phenomena.

His work helped map the motion of planets and predict events like lunar eclipses with high accuracy. Even modern space exploration owes much to Newton’s discovery. By calculating gravitational forces, engineers devised efficient routes for space missions like Apollo.

The principles of this law continue to support vital technologies. GPS systems, satellite communications, and planetary rovers apply Newtonian equations to chart precise routes and operations.

Examples

• Predicting planetary alignment based on gravitational forces.
• Designing the precise orbits used in moon missions such as Apollo.
• Calibrating artificial satellites used for communications and positioning systems.

6. The Theory of Relativity Changed Fundamental Ideas about Space-Time

Before Einstein, space and time were seen as separate entities. Relativity intertwined them into a four-dimensional fabric where gravity bends space-time. This concept replaced Newtonian theories for understanding cosmic forces.

Relativity predicts phenomena like time dilation, where time moves slower closer to heavy objects like Earth due to gravitational pull. Technology adapts to these principles; for instance, GPS satellites compensate for time discrepancies caused by both their speed and differences in Earth’s gravity.

On a larger scale, relativity explains black holes and the Big Bang. Researchers also study expanding galaxies to understand how such cosmic events align with Einstein’s predictions.

Examples

• GPS systems depend on relativity for timing corrections.
• Black holes were first predicted through Einstein’s modifications to gravity.
• Universal expansion aligns closely with General Relativity principles.

7. Schrödinger’s Equation: The Quantum World Explained

Schrödinger’s equation transforms our view of matter by modeling tiny particles as probabilistic waves instead of solid objects. It’s the foundation of quantum mechanics, revealing the mysterious wave-particle duality of things like electrons and light.

At this microscopic level, particles act oddly—they can exist in multiple states simultaneously or maintain past "ghost" positions until observed. Schrödinger's equation helps researchers predict particle behavior in substances like semiconductors, used in memory chips across devices.

Today’s modern gadgets, from phones to kitchen appliances, owe their innovation to quantum theory, modeled mathematically through this revolutionary equation.

Examples

• Memory chips in electronics are designed based on quantum mechanics.
• Physicists explain atoms’ observed dual particle-wave nature using Schrödinger’s results.
• Electron behavior in semiconductors is modeled through quantum formulation.

8. Chaos Theory Shows Order Can Arise from Disorder

Chaos theory describes systems that follow precise rules but produce seemingly random outcomes. A food processor exemplifies such behavior: its blades spin predictably, but the resulting food mixture is chaotic. This principle applies to natural systems like weather or animal populations.

Scientists now understand the limits of prediction in chaotic systems. For example, meteorologists use chaos-based simulations for weather forecasting to account for unpredictable variables. While daily weather predictions aren’t perfect, long-term forecasting has improved.

Applications extend to processes in industries such as food production, pharmaceuticals, and more—where knowing the “mixing” dynamics helps refine results.

Examples

• Food processors demonstrate chaos through blade interaction with food particles.
• Population modeling reveals chaotic fluctuations in animal groups generation by generation.
• Meteorology uses chaos simulations to refine weather forecasts daily.

9. The Black-Scholes Formula Transformed Finance—for Better and Worse

The Black-Scholes formula calculates the value of financial derivatives like options, aiding traders in assessing risks and setting prices. This mathematical breakthrough revolutionized finance, turning derivatives into essential tools for global trade.

However, overreliance on this method encouraged risky investments prior to the 2008 financial crisis. Layers of increasingly complex and interconnected derivatives were unaccounted for, heightening systemic market vulnerabilities.

While the formula made modern stock and commodity trading highly efficient, the lesson remains clear: proper scrutiny is necessary before applying math to unpredictable conditions.

Examples

• Trading options became widespread thanks to Black-Scholes calculations.
• Misuse of derivative-based equations triggered the 2008 global financial crisis.
• Investors use the formula to price options during normal market stability.

Takeaways

1. Apply mathematical basics, like the Pythagorean theorem, when tackling practical projects like home renovations or measuring landscapes.
2. Be cautious when using simplified mathematical models for complex problems, especially in investing or forecasting markets.
3. Look for patterns or predictability in complex systems—chaos theory often reveals underlying order even within unpredictable environments.