Fascinated by the intricate dance between mathematics and real-world phenomena? Curious about the deep roots of gerrymandering in American politics and how it's evolving with technology? What about exploring how Abraham Lincoln's persuasive skills were greatly influenced by geometry? Dive into the captivating world of mathematics as it weaves its way through politics, pandemics, AI and even games like checkers!

Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else

In 1864, Reverend J.P. Gulliver of Connecticut shared a conversation with Abraham Lincoln about how the president developed his persuasive skills. Lincoln credited his rhetorical prowess to geometry, particularly Euclid's Elements. He struggled with the term "demonstrate" and eventually isolated himself to study geometry until he could confidently understand and apply it. This rigorous self-education helped Lincoln build a foundation for clear and logical thinking.

Lincoln's dedication to Euclid mirrored his broader belief in rigorous, step-by-step reasoning. His partner, William Herndon, described Lincoln's late-night study sessions and his intense attempt to solve the ancient problem of "squaring the circle," which is constructing a square with the same area as a given circle using only a straightedge and compass. Despite the impossibility of the task, Lincoln's effort demonstrated his commitment to understanding complex problems.

Lincoln's use of Euclidean principles was evident in his famous speeches, like the Gettysburg Address, where he referred to the United States as "dedicated to the proposition that all men are created equal." This concept of a "proposition" reflects Euclidean logic, where truths are derived from self-evident axioms.

The influence of Euclid extended beyond Lincoln. Thomas Jefferson, a math enthusiast, also valued geometry in forming democratic principles. He believed the foundational principles of geometry paralleled those of a free society.

The teaching of geometry evolved over time, with Euclid's methods initially dominating education. However, as the field progressed, educators sought more engaging and intuitive approaches. Inventional Geometry, for example, encouraged self-discovery and problem-solving, reflecting a shift towards a more interactive learning experience.

Geometry's role in education was debated, with some advocating for its logical rigor and others emphasizing its practical applications. The balance between logic and intuition, as highlighted by mathematicians like Henri Poincaré, remains crucial. Geometry teaches not only the discipline of proof but also the importance of understanding and intuition in solving problems.

Ultimately, Lincoln's embrace of Euclidean geometry exemplifies the broader value of mathematical thinking in developing clear, honest, and rigorous reasoning—a lesson that remains relevant in education and beyond.

Did Euclid's Geometry Shape Abraham Lincoln's Legendary Rhetoric?

How many holes does a straw have? It's a question that has perplexed the internet and sparked heated debates. Some say zero holes, others one, and some insist on two. Each perspective seems obvious to its adherents, but upon closer examination, none are free of complications.

Let's break it down. The zero-hole argument claims that rolling a rectangle into a straw doesn't create holes. But this overlooks the concept that objects can have holes without substance being removed, like a bagel, formed by joining dough ends together.

The two-hole theory faces its own challenges. If a straw has two holes, where does one hole end and the other begin? Consider a slice of swiss cheese; we don’t count holes separately on each side. Or imagine filling in the bottom of a straw, turning it into a cup. Does a cup have a hole? As it morphs from cup to ashtray, at what point does the hole disappear?

The one-hole theory seems most popular. Yet, calling the mouth and anus the same hole is unsettling. Even if we accept this, arguments persist. For instance, if making a new hole should increase the number of holes, how many holes are in a pair of pants? Sewing the waist shut turns them into a denim straw. If pants originally had three holes, sewing one shut should leave two, not one.

Topology, a branch of geometry, provides a solution. It suggests the straw has one hole, but this hole exists in a way that can be perceived from different points. Pants, when flattened into a thong, show two holes. This aligns with the understanding that pants are essentially two straws.

In the world of mathematics, sometimes we must accept counterintuitive notions for broader explanatory power. For instance, a balloon has –1 holes before being popped, allowing consistency with the principle that adding a hole increases the count by one.

Ultimately, debates like these reveal the beauty of mathematical thinking and the flexibility of definitions in geometry. Through topology, we understand shapes and spaces in ways that transcend everyday intuition, allowing us to navigate complex dimensions and abstract concepts. The exploration of holes in straws and pants exemplifies the intriguing interplay between intuition and formalism in mathematics.

Internet's Heated Debate: How Many Holes Does a Straw Have?

Back at the 1904 St. Louis exposition, Sir Ronald Ross, the "Mosquito Man" famous for discovering that malaria was carried by the anopheles mosquito, gave a lecture on "The Logical Basis of the Sanitary Policy of Mosquito Reduction." Ross introduced the concept of the random walk by explaining how mosquitoes wander randomly, affecting malaria control efforts. His insights laid the groundwork for the theory of the random walk, which would later influence fields like physics, finance, and poetry.

Ross's idea was simple: mosquitoes wander randomly, not following a direct path. He used this to calculate the chances of mosquitoes spreading malaria after being born at a certain point and wandering until they die. His calculations showed how far mosquitoes are likely to travel, a concept we now understand through simulations and mathematical analyses.

Interestingly, Ross's work also ties into polling and statistics. For example, the randomness of mosquito movements mirrors the randomness in polling data, where the outcomes of small samples can predict larger trends if the samples are unbiased. However, just as winds can sway mosquitoes, biases can affect poll results.

Ross's insights extended beyond biology. They influenced financial models, like those of Louis Bachelier, who applied random walk theory to stock prices, and Albert Einstein, who used it to explain Brownian motion.

In another fascinating application, mathematician Andrei Markov used the random walk concept to develop Markov chains, which model sequences where the next state depends only on the current state. Markov even applied this to analyze the structure of Russian poetry, showing how probabilistic models could capture linguistic patterns.

These ideas continue to impact modern fields, from market analysis to the development of artificial intelligence. The random walk and Markov chains illustrate the profound connection between seemingly random processes and predictable patterns, demonstrating how mathematical principles can illuminate the complexities of the world around us.

How Did Mosquito Man Discoveries Influence Modern Science and Finance?

Marion Franklin Tinsley was an unparalleled champion in the game of checkers, far exceeding the dominance of Serena Williams in tennis or Babe Ruth in baseball. Tinsley, a math professor from Tallahassee, Florida, learned the game from a family boarder, Mrs. Kershaw, and honed his skills under the tutelage of world champion Asa Long. 

He won his first U.S. title in 1954 and the world championship a year later, a title he held for most of the next four decades. Tinsley’s style was not aggressive; he simply outplayed everyone, losing only three tournament games from 1951 to 1990.

In 1992, Tinsley faced a new kind of opponent: Chinook, a computer program developed by Jonathan Schaeffer’s team at the University of Alberta. Despite losing a couple of games to Chinook, Tinsley maintained his championship status. However, during a rematch in 1994, Tinsley resigned due to health issues, and Chinook was declared the world champion. Tinsley passed away eight months later.

Checkers, like Nim and other games, can be represented by a tree of possible moves and outcomes. While Nim's strategy can be mapped out relatively simply, checkers is far more complex. Despite this complexity, Schaeffer's team eventually proved that perfect play in checkers results in a draw, ending the era of human supremacy in the game.

This shift reflects a broader phenomenon in games and mathematics: the use of computational power to navigate complex decision spaces. In Go, for example, the sheer number of possible game states makes solving the game through brute force impractical. Instead, modern AI like AlphaGo uses advanced algorithms and trial and error to develop strategies that rival or surpass human expertise.

This method of trial and error, combined with sophisticated scoring functions, allows these programs to approximate the perfect strategy without having to compute every possible move. It’s a testament to the evolving intersection of mathematics, computer science, and artificial intelligence, demonstrating that sometimes, the journey through countless trials is more valuable than the quest for perfection.

Marion Tinsley vs. Chinook: The Checkers Showdown That Revolutionized AI

Meredith Broussard, an NYU professor, recently explained artificial intelligence on TV, emphasizing that it's not about killer robots but just math. Let’s dive deeper.

Imagine you're a mountaineer without a map, trying to reach the summit. You assess the slope around you and choose the steepest upward path. This process, called gradient descent, is a core idea in machine learning. It’s a method of trial and error—taking small steps and choosing the one that offers the most progress. This strategy helps computers learn, such as identifying cats in images by tweaking their approach to reduce errors.

A classic example is teaching a computer to distinguish between cats and non-cats using a massive set of images. The computer starts with a strategy, adjusts it through gradient descent, and improves by minimizing wrongness. This technique applies to various tasks, from predicting movie ratings to guiding self-driving cars.

However, gradient descent can get trapped in local optima—suboptimal points where no small change improves the outcome. To overcome this, we use methods like stochastic gradient descent, which introduces randomness to explore different possibilities and avoid getting stuck.

Machine learning also involves complex structures like neural networks, which mimic the brain's neurons. These networks consist of layers and nodes, each processing inputs and passing outputs to the next layer. Adjusting the weights on the connections between nodes fine-tunes the network’s performance.

For instance, GPT-3, a language model with 175 billion parameters, uses neural networks to generate human-like text. Despite the vast number of possible strategies, gradient descent helps find effective ones.

Understanding AI's math helps demystify it. It's about finding the best path, adjusting strategies, and learning from data. While the algorithms can seem complex, the underlying principles are rooted in simple trial and error, paving the way for remarkable advancements.

Can NYU's Meredith Broussard Demystify AI's Math Magic for Us?

Mathematics is hard—a fact often softened for students, but acknowledging it can foster learning. Robin Gottlieb, a master teacher, emphasizes that labeling a lesson as "easy" when it's not misleads students into doubting their abilities. Acknowledging math's difficulty encourages students to ask questions and explore, realizing that difficulty is part of learning. 

A compelling story involves Frank Nelson Cole, who silently proved a complex equation after "three years of Sundays," demonstrating that even long, difficult work can result in a short, elegant proof. This story illustrates that difficulty isn't just in proving truths but in discovering them.

In mathematics, the challenge isn't just solving problems but understanding why something is true. Andrew Granville shares an anecdote about Cole, highlighting the different kinds of difficulty in math. 

Mathematicians like Lisa Piccirillo can tackle famously tough problems, like the Conway knot, with elegant solutions, suggesting that sometimes, the challenge lies in realizing the simplicity of the solution. 

AI and neural networks, like the one Mark Hughes used to study knots, can assist in mathematical exploration but struggle with human intuition and creativity. Machines excel in computation but lack the nuanced understanding humans bring to problem-solving.

Mathematics isn't only about solving equations; it involves understanding patterns and phenomena, as seen in Ronald Ross's work on disease modeling. Ross's approach to understanding epidemics mathematically was groundbreaking, turning unpredictable plagues into quantifiable phenomena, much like Newton's laws did for comets.

Ross's ambition and mathematical passion were evident in his work, even as he struggled with recognition and personal grievances. He collaborated with Hilda Hudson, whose deep mathematical insights complemented his vision, forming a partnership that advanced the mathematical understanding of epidemics.

Mathematical models of epidemics, like the SIR model, offer insights into disease spread and control. While they can't predict the future, they help us understand trends and make informed decisions. They also find applications beyond epidemiology, like tracking how rumors spread on social media.

In mathematics, the idea that "what happens today will happen tomorrow" underpins models across various fields, from disease spread to celestial dynamics, demonstrating the universality and power of mathematical thought.

Mastering Math: Insights from Robin Gottlieb, Frank Nelson Cole, and Lisa Piccirillo

In May 2020, the White House's optimistic "cubic fit" model for COVID-19 deaths, initiated by Kevin Hassett, was mocked for its inaccuracy. This misstep echoed Hassett's overoptimistic 1999 stock market prediction in "Dow 36,000." 

The lesson here is about understanding mathematical fallacies rather than just distrusting one person. To grasp this, we can look back at the British rinderpest outbreak of the 1860s, where statistician William Farr countered the prevailing fear of exponential growth with his own prediction of a decline, based on mathematical modeling. His prediction was controversial but ultimately correct, demonstrating the complexity of modeling epidemics.

Farr's method involved intricate calculations, akin to those used centuries earlier for astronomical predictions, like calculating square roots through successive approximations. These methods, revived by Charles Babbage's Difference Engine, illustrate the challenges of extrapolation—predicting beyond known data points.

Fast forward to 2020, both Hassett's and Justin Wolfers' predictions on COVID-19 death rates fell victim to this extrapolation trap, illustrating the danger of relying on rigid models. Farr's and later John Brownlee's work showed that epidemics rarely follow a neat bell curve, contrary to some interpretations of "Farr’s Law."

In predictive modeling, balance is key. While some models fit past patterns without understanding underlying mechanisms, others, like Ronald Ross’s differential equations, attempt to reverse-engineer the dynamics. Modern examples, like Youyang Gu's COVID-19 model, successfully integrate data-driven machine learning with mechanistic insights.

The lesson is that all models are approximations. They can't perfectly predict the future but can guide us by indicating trends. As Zeynep Tufekci aptly noted, "Coronavirus Models Aren’t Supposed to Be Right." They are tools for understanding and decision-making, not crystal balls. In modeling, as in life, a blend of past wisdom and current innovation is essential for navigating uncertainty.

Did Kevin Hassett's COVID-19 Prediction Echo His Infamous Dow 36,000 Blunder?

In 1977, a Dutch math team at the International Mathematical Olympiad presented a fascinating sequence puzzle to their British counterparts. This sequence, known as the "Look-and-Say," intrigued mathematicians like John Conway, who discovered its unique properties and linked it to 92 "atoms" controlling its behavior. Although seemingly a simple amusement, it led to insights into mathematical progression and eigenvalues.

Eigenvalues are a recurring theme in understanding patterns, like the growth of the Look-and-Say sequence or the spread of diseases. In modeling pandemics, they help us comprehend the progression of infections, even when divided across regions like the Dakotas, or within age groups. This understanding aids in predicting and controlling epidemics, showing how the dynamics of disease spread can resemble Fibonacci numbers or other mathematical sequences.

The Golden Ratio, another fascinating mathematical concept, emerges in various patterns, including Fibonacci numbers. This ratio, symbolized by φ, has captivated thinkers through history for its appearance in geometry, nature, and even stock market theories like Elliott Wave Theory. The intricate dance between mathematics and real-world phenomena continues to inspire, whether it's in predicting market trends or understanding the spread of a virus.

Mathematics also plays a crucial role in modern technology. Google's search algorithm, for instance, relies on eigenvalues to rank web pages effectively, transforming random walks on the internet into organized information retrieval, akin to the random walks seen in Monopoly or viral spread models.

In quantum mechanics, the concept of eigenvalues explains the uncertainty principle, highlighting how particles cannot simultaneously have well-defined position and momentum. This principle, rooted in the noncommutative nature of certain operations, underscores the deep connections between mathematics and the fundamental laws of physics.

The rich tapestry of mathematics is woven with sequences, ratios, and values that resonate across disciplines, illustrating the beauty and utility of this language of the universe. Whether in understanding the growth of a sequence, modeling a pandemic, or exploring the quantum world, mathematics provides a framework for exploring the mysteries around us.

From Conway's Puzzle to Google's Algorithm: The Universal Power of Eigenvalues

In the early days of Markov's random walk theory, Hungarian mathematicians like George Pólya and Florian Eggenberger applied it to various phenomena, including the spread of diseases like smallpox and scarlet fever. 

Eggenberger's thesis, "The Contagion of Probability," described how diseases spread in a manner akin to a random walk across a grid, which he likened to the streets of Manhattan. This model revealed that the spread of disease was not as fast as one might assume because it accounted for spatial constraints—infected individuals mostly spread the disease to their immediate neighbors.

Historically, pandemics like the Black Death moved slowly across Europe, taking years to reach various regions. However, the 1872 North American horse flu epizootic spread unevenly, influenced by the geometry of the newly built railroads, which allowed the disease to hop from city to city rather than spreading in a smooth wave. This highlighted how human infrastructure, like railroads, could alter the "geometry" of disease spread.

In modern times, the COVID-19 pandemic demonstrated the complexity of global connectivity, with the virus rapidly spreading across continents via air travel. This reflects a "wrinkled" world geometry, where traditional boundaries are compressed, similar to the concept explored in Madeline L'Engle's "A Wrinkle in Time," where space is bent to allow faster travel.

Networks, whether of people or molecules, often exhibit "small-world" properties, where most nodes are connected by short paths. This concept was famously explored by Stanley Milgram in the "small-world experiment," which inspired the idea of "six degrees of separation" and later the "six degrees of Kevin Bacon" game. Modern social networks like Facebook reveal even shorter average paths between users, underscoring our interconnectedness.

This interconnectedness poses challenges, particularly in the spread of diseases, where a few long-distance connections can turn local outbreaks into global pandemics. The mathematical study of these networks helps us understand and predict the dynamics of such spreads. The geometry of our world, once vast and insurmountable, is now a tight-knit web of connections, a reality that both fascinates and challenges us in the face of global events.

How Did George Pólya's Theories Predict the Spread of COVID-19 Across Our 'Wrinkled' World?

On November 6, 2018, Wisconsin Democrats had reasons to celebrate. They won the governorship, re-elected their Senate candidate, and took key state offices. Yet, despite this success, Republicans maintained control of the Wisconsin State Assembly and even gained a seat in the state senate. This paradox stemmed from how the legislative district lines were drawn, a process heavily influenced by Republicans to secure their advantage, known as gerrymandering. 

The roots of gerrymandering in American politics are deep, tracing back to the early days of the republic. The term itself originates from a district drawing in Massachusetts aimed at favoring a political party, although the practice predates even that. With the advent of technology, gerrymandering has evolved into a precise science, allowing for the creation of district maps that can almost guarantee a party's dominance, irrespective of the actual vote count.

Mathematicians and political scientists have proposed various ways to measure and combat gerrymandering. The efficiency gap, for instance, quantifies how effectively a party's votes are utilized in winning seats. Another approach involves using computer-generated simulations to create thousands of random district maps, providing a benchmark for what nonpartisan districting might look like.

Despite these advancements, the legal system has struggled to address the problem. The U.S. Supreme Court, in a significant decision, ruled that partisan gerrymandering is a political issue beyond the scope of the federal courts. This decision leaves the door open for continued manipulation of district maps by partisan interests, unless reform can be achieved through state constitutions or public pressure.

The awareness and understanding of gerrymandering have grown, partly thanks to media coverage and public engagement. While the road to fairer districting may be fraught with legal and political challenges, the push for transparency and mathematical methods offers a glimmer of hope for a more equitable electoral process.

Wisconsin's 2018 Election: A Victory Shadowed by Gerrymandering

Abraham Lincoln's persuasive skills were greatly influenced by his study of Euclidean geometry, which helped him develop a foundation for logical thinking. This dedication to rigorous reasoning was evident in his speeches and beliefs. The role of geometry in education has evolved, shifting towards more engaging and intuitive methods, balancing logical rigor with practical applications.

Mathematics often reveals deep insights through seemingly simple questions, like the debate over the number of holes in a straw. Topology, a branch of geometry, offers unique perspectives on such problems, highlighting the importance of abstract thinking in understanding shapes and spaces.

The concept of the random walk, introduced by Sir Ronald Ross, has applications in various fields, from epidemiology to finance. This idea illustrates the connection between random processes and predictable patterns, showcasing the power of mathematical principles.

In the realm of games, the dominance of Marion Tinsley in checkers and the subsequent rise of computer programs like Chinook demonstrate the intersection of human skill and computational power. This shift reflects broader trends in using AI and algorithms to navigate complex decision spaces.

Artificial intelligence, often misunderstood, relies on mathematical principles like gradient descent to learn and improve. Understanding these concepts helps demystify AI and underscores the importance of trial and error in developing effective strategies.

Mathematics is inherently challenging, and acknowledging this difficulty can foster a deeper understanding and appreciation for the subject. Stories of mathematicians solving complex problems illustrate that both the journey and the discovery are integral to mathematical exploration.

Modeling epidemics mathematically, as seen in Ronald Ross's work, helps us understand and predict disease spread. These models, while not perfect, guide decision-making by indicating trends and patterns.

The story of gerrymandering in American politics highlights how mathematical methods can measure and combat electoral manipulation. Despite legal challenges, the push for transparency and fair districting continues, offering hope for a more equitable electoral process.

Overall, mathematics provides a framework for exploring complex phenomena, from disease spread to political fairness, revealing patterns and principles that guide our understanding of the world.

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