The blog post explores the interconnectedness of various measurement systems and mathematical concepts, examining potential historical links that are likely coincidental. The author notes the near equivalence of a meter to a royal cubit times the golden ratio, and how this relates to the dimensions of the Great Pyramid of Giza. While acknowledging the established historical definition of the meter based on Earth's circumference, the post speculates on whether ancient Egyptians might have possessed a sophisticated understanding of these relationships, potentially incorporating the golden ratio and Earth's dimensions into their construction. However, the author ultimately concludes that the observed connections are likely due to mathematical happenstance rather than deliberate design.
This interactive visualization explains Markov chains by demonstrating how a system transitions between different states over time based on predefined probabilities. It illustrates that future states depend solely on the current state, not the historical sequence of states (the Markov property). The visualization uses simple examples like a frog hopping between lily pads and the changing weather to show how transition probabilities determine the long-term behavior of the system, including the likelihood of being in each state after many steps (the stationary distribution). It allows users to manipulate the probabilities and observe the resulting changes in the system's evolution, providing an intuitive understanding of Markov chains and their properties.
HN users largely praised the visual clarity and helpfulness of the linked explanation of Markov Chains. Several pointed out its educational value, both for introducing the concept and for refreshing prior knowledge. Some commenters discussed practical applications, including text generation, Google's PageRank algorithm, and modeling physical systems. One user highlighted the importance of understanding the difference between "Markov" and "Hidden Markov" models. A few users offered minor critiques, suggesting the inclusion of absorbing states and more complex examples. Others shared additional resources, such as interactive demos and alternative explanations.
Terry Tao's blog post discusses the recent proof of the three-dimensional Kakeya conjecture by Hong Wang and Joshua Zahl. The conjecture states that any subset of three-dimensional space containing a unit line segment in every direction must have Hausdorff dimension three. While previous work, including Tao's own, established lower bounds approaching three, Wang and Zahl definitively settled the conjecture. Their proof utilizes a refined multiscale analysis of the Kakeya set and leverages polynomial partitioning techniques, building upon earlier advances in incidence geometry. The post highlights the key ideas of the proof, emphasizing the clever combination of existing tools and innovative new arguments, while also acknowledging the remaining open questions in higher dimensions.
HN commenters discuss the implications of the recent proof of the three-dimensional Kakeya conjecture, praising its elegance and accessibility even to non-experts. Several highlight the significance of "polynomial partitioning," the technique central to the proof, and its potential applications in other areas of mathematics. Some express excitement about the possibility of tackling higher dimensions, while others acknowledge the significant jump in complexity this would entail. The clear exposition of the proof by Tao is also commended, making the complex subject matter understandable to a broader audience. The connection to the original Kakeya needle problem and its surprising implications for analysis are also noted.
This post explores the complexities of representing 3D rotations, contrasting quaternions with other methods like rotation matrices and Euler angles. It highlights the issues of gimbal lock and interpolation difficulties inherent in Euler angles, and the computational cost of rotation matrices. Quaternions, while less intuitive, offer a more elegant and efficient solution. The post breaks down the math behind quaternions, explaining how they represent rotations as points on a 4D hypersphere, and demonstrates their advantages for smooth interpolation and avoiding gimbal lock. It emphasizes the practical benefits of quaternions in computer graphics and other applications requiring 3D manipulation.
HN users generally praised the article for its clear explanation of quaternions and their application to 3D rotations. Several commenters appreciated the visual approach and interactive demos, finding them helpful for understanding the concepts. Some discussed alternative representations like rotation matrices and axis-angle, comparing their strengths and weaknesses to quaternions. A few users pointed out the connection to complex numbers and offered additional resources for further exploration. One commenter mentioned the practical uses of quaternions in game development and other fields. Overall, the discussion highlighted the importance of quaternions as a tool for representing and manipulating rotations in 3D space.
The blog post details a formal verification of the standard long division algorithm using the Dafny programming language and its built-in Hoare logic capabilities. It walks through the challenges of representing and reasoning about the algorithm within this formal system, including defining loop invariants and handling edge cases like division by zero. The core difficulty lies in proving that the quotient and remainder produced by the algorithm are indeed correct according to the mathematical definition of division. The author meticulously constructs the necessary pre- and post-conditions, and elaborates on the specific insights and techniques required to guide the verifier to a successful proof. Ultimately, the post demonstrates the power of formal methods to rigorously verify even relatively simple, yet subtly complex, algorithms.
Hacker News users discussed the application of Hoare logic to verify long division, with several expressing appreciation for the clear explanation and visualization of the algorithm. Some commenters debated the practical benefits of formal verification for such a well-established algorithm, questioning the likelihood of uncovering unknown bugs. Others highlighted the educational value of the exercise, emphasizing the importance of understanding foundational algorithms. A few users delved into the specifics of the chosen proof method and its implications. One commenter suggested exploring alternative verification approaches, while another pointed out the potential for applying similar techniques to other arithmetic operations.
The Simons Institute for the Theory of Computing at UC Berkeley has launched "Stone Soup AI," a year-long research program focused on collaborative, open, and decentralized development of foundation models. Inspired by the folktale, the project aims to build a large language model collectively, using contributions of data, compute, and expertise from diverse participants. This open-source approach intends to democratize access to powerful AI technology and foster greater transparency and community ownership, contrasting with the current trend of closed, proprietary models developed by large corporations. The program will involve workshops, collaborative coding sprints, and public releases of data and models, promoting open science and community-driven advancement in AI.
HN commenters discuss the "Stone Soup AI" concept, which involves prompting LLMs with incomplete information and relying on their ability to hallucinate missing details to produce a workable output. Some express skepticism about relying on hallucinations, preferring more deliberate methods like retrieval augmentation. Others see potential, especially for creative tasks where unexpected outputs are desirable. The discussion also touches on the inherent tendency of LLMs to confabulate and the need for careful evaluation of results. Several commenters draw parallels to existing techniques like prompt engineering and chain-of-thought prompting, suggesting "Stone Soup AI" might be a rebranding of familiar concepts. A compelling point raised is the potential for bias amplification if hallucinations consistently fill gaps with stereotypical or inaccurate information.
Terence Tao's blog post explores how "landscape functions," a mathematical tool from optimization and computer science, could improve energy efficiency in buildings. He explains how these functions can model the complex interplay of factors affecting energy consumption, such as appliance usage, weather conditions, and occupancy patterns. By finding the "minimum" of the landscape function, one can identify the most energy-efficient operating strategy for a given building. Tao suggests that while practical implementation presents challenges like data acquisition and model complexity, landscape functions offer a promising theoretical framework for bridging the "green gap" – the disparity between predicted and actual energy savings in buildings – and ultimately reducing electricity costs for consumers.
HN commenters are skeptical of the practicality of applying the landscape function to energy optimization. Several doubt the computational feasibility, pointing out the complexity and scale of the power grid. Others question the focus on mathematical optimization, suggesting that more fundamental issues like transmission capacity and storage are the real bottlenecks. Some express concerns about the idealized assumptions in the model, and the lack of consideration for real-world constraints. One commenter notes the difficulty of applying abstract mathematical tools to complex real-world systems, and another suggests exploring simpler, more robust approaches. There's a general sentiment that while the math is interesting, its impact on lowering electricity costs is likely minimal.
Modular forms, complex functions with extraordinary symmetry, are revolutionizing how mathematicians approach fundamental problems. These functions, living in the complex plane's upper half, remain essentially unchanged even after being twisted and stretched in specific ways. This unusual resilience makes them powerful tools, weaving connections between seemingly disparate areas of math like number theory, analysis, and geometry. The article highlights their surprising utility, suggesting they act as a "fifth fundamental operation" akin to addition, subtraction, multiplication, and division, enabling mathematicians to perform calculations and uncover relationships previously inaccessible. Their influence extends to physics, notably string theory, and continues to expand mathematical horizons.
HN commenters generally expressed appreciation for the Quanta article's accessibility in explaining a complex mathematical concept. Several highlighted the connection between modular forms and both string theory and the monster group, emphasizing the unexpected bridges between seemingly disparate areas of math and physics. Some discussed the historical context of modular forms, including Ramanujan's contributions. A few more technically inclined commenters debated the appropriateness of the "fifth fundamental operation" phrasing, arguing that modular forms are more akin to functions or tools built upon existing operations rather than a fundamental operation themselves. The intuitive descriptions provided in the article were praised for helping readers grasp the core ideas without requiring deep mathematical background.
A Penn State student has refined a century-old math theorem known as the Kutta-Joukowski theorem, which calculates the lift generated by an airfoil. This refined theorem now accounts for rotational and unsteady forces acting on airfoils in turbulent conditions, something the original theorem didn't address. This advancement is significant for the wind energy industry, as it allows for more accurate predictions of wind turbine blade performance in real-world, turbulent wind conditions, potentially leading to improved efficiency and design of future turbines.
HN commenters express skepticism about the impact of this research. Several doubt the practicality, pointing to existing simulations and the complex, chaotic nature of wind making precise calculations less relevant. Others question the "100-year-old math problem" framing, suggesting the Betz limit is well-understood and the research likely focuses on a specific optimization problem within that context. Some find the article's language too sensationalized, while others are simply curious about the specific mathematical advancements made and how they're applied. A few commenters provide additional context on the challenges of wind farm optimization and the trade-offs involved.
This post provides a gentle introduction to stochastic calculus, focusing on the Ito integral. It explains the motivation behind needing a new type of calculus for random processes like Brownian motion, highlighting its non-differentiable nature. The post defines the Ito integral, emphasizing its difference from the Riemann integral due to the non-zero quadratic variation of Brownian motion. It then introduces Ito's Lemma, a crucial tool for manipulating functions of stochastic processes, and illustrates its application with examples like geometric Brownian motion, a common model in finance. Finally, the post briefly touches on stochastic differential equations (SDEs) and their connection to partial differential equations (PDEs) through the Feynman-Kac formula.
HN users generally praised the clarity and accessibility of the introduction to stochastic calculus. Several appreciated the focus on intuition and the gentle progression of concepts, making it easier to grasp than other resources. Some pointed out its relevance to fields like finance and machine learning, while others suggested supplementary resources for deeper dives into specific areas like Ito's Lemma. One commenter highlighted the importance of understanding the underlying measure theory, while another offered a perspective on how stochastic calculus can be viewed as a generalization of ordinary calculus. A few mentioned the author's background, suggesting it contributed to the clear explanations. The discussion remained focused on the quality of the introductory post, with no significant dissenting opinions.
This post discusses the second part of Grant Sanderson's (3Blue1Brown) interview with mathematician Terence Tao, focusing on the cosmic distance ladder. It explains how astronomers determine distances to increasingly far-off celestial objects, building upon previously established measurements. The video delves into standard candles like Cepheid variables and Type Ia supernovae, highlighting their role in measuring vast distances. It also explores the inherent uncertainties and challenges involved in these methods, including the difficulty in calibrating measurements and potential sources of error that propagate as distances increase. Finally, the post touches on the "tension" in cosmology related to discrepancies in measurements of the Hubble constant, which describes the universe's expansion rate.
Hacker News users discuss the second part of Grant Sanderson's (3Blue1Brown) video with Terence Tao on the cosmic distance ladder, generally praising its clarity and accessibility. Several commenters highlight the effective use of visualizations to explain complex concepts, particularly redshift and standard candles. Some express appreciation for Tao's ability to explain advanced topics simply, while others note the video's effectiveness in conveying the uncertainties and iterative nature of scientific measurement. A few commenters mention the surprising role of type Ia supernovae in measuring distances, and one points out the clever historical analogy to measuring the height of Mount Everest. The overall sentiment is positive, with many finding the video both educational and engaging.
The post explores the mathematical puzzle of representing any integer using four twos and a limited set of operations. It demonstrates how combining operations like addition, subtraction, multiplication, division, square roots, factorials, decimals, and concatenation, alongside techniques like logarithms and the gamma function (a generalization of the factorial), allows for expressing a wide range of integers. The author showcases examples and discusses the challenges of representing larger numbers, particularly prime numbers, due to the increasing complexity of the required expressions. The ultimate goal isn't a formal proof, but rather a practical exploration of the expressive power of combining these mathematical tools with a limited set of starting digits.
HN commenters largely focused on the limitations and expansions of the puzzle. Some pointed out that the allowed operations weren't explicitly defined, leading to debates about the validity of certain solutions, particularly the use of the square root and floor/ceiling functions. Others discussed alternative approaches, such as using logarithms or the successor function. A few commenters explored variations of the puzzle, including using different numbers or a different quantity of the given number. The overall sentiment was one of intrigue, with many appreciating the puzzle's challenge and the creativity it sparked.
The post explores the mathematical puzzle of representing any integer using four twos and a limited set of operations. It demonstrates how combining operations like addition, subtraction, multiplication, division, square roots, factorials, decimal points, and concatenation, along with concepts like double factorials and the gamma function (a generalization of the factorial), allows for creative expression of numerous integers. While acknowledging the potential for more complex representations using less common operations, the post focuses on showcasing the flexibility and surprising reach of this mathematical exercise using a relatively small toolkit of functions. It ultimately highlights the challenge and ingenuity involved in manipulating a limited set of numbers to achieve a wide range of results.
Hacker News users generally enjoyed the puzzle presented in the linked article about constructing integers using four twos. Several commenters explored alternative solutions using different mathematical operations like bitwise XOR, square roots, and logarithms, showcasing a playful engagement with the challenge. Some discussed the arbitrary nature of the "four twos" constraint, suggesting that similar puzzles could be devised with other numbers or constraints. A few comments delved into the role of such puzzles in education, highlighting their value in encouraging creative problem-solving. One commenter pointed out the similarity to the "four fours" puzzle, referencing a website dedicated to exploring its variations.
Calcverse is a collection of simple, focused online calculators built by a solo developer as a counterpoint to the current hype around AI agents. The creator emphasizes the value of small, well-executed tools that solve specific problems efficiently. The calculators currently offered on the site cover areas like loan comparisons, unit conversions, and investment calculations, with more planned for the future. The project embraces a minimalist design and aims to provide a practical alternative to overly complex software.
HN users generally praised the calculator's clean UI/UX and appreciated the developer's focus on a simple, well-executed project rather than chasing the AI hype. Several commenters suggested potential improvements or expansions, including adding more unit conversions, financial calculators, and even integrating with existing tools like Excel or Google Sheets. Some pointed out the existing prevalence of specialized online calculators, questioning the project's long-term viability. Others expressed interest in the technical implementation details, particularly the use of Qwik and Partytown. A few jokingly questioned the project's description as "just" calculators, recognizing the complexity and value in building a robust and user-friendly calculation tool.
The blog post "On Zero Sum Games (The Informational Meta-Game)" argues that while many real-world interactions appear zero-sum, they often contain hidden non-zero-sum elements, especially concerning information. The author uses poker as an analogy: while the chips exchanged represent a zero-sum component, the information revealed through betting, bluffing, and tells creates a meta-game that isn't zero-sum. This meta-game involves learning about opponents and improving one's own strategies, generating future value even within apparently zero-sum situations like negotiations or competitions. The core idea is that leveraging information asymmetry can transform seemingly zero-sum interactions into opportunities for mutual gain by increasing overall understanding and skill, thus expanding the "pie" over time.
HN commenters generally appreciated the post's clear explanation of zero-sum games and its application to informational meta-games. Several praised the analogy to poker, finding it illuminating. Some extended the discussion by exploring how this framework applies to areas like politics and social dynamics, where manipulating information can create perceived zero-sum scenarios even when underlying resources aren't truly limited. One commenter pointed out potential flaws in assuming perfect rationality and complete information, suggesting the model's applicability is limited in real-world situations. Another highlighted the importance of trust and reputation in navigating these information games, emphasizing the long-term cost of deceptive tactics. A few users also questioned the clarity of certain examples, requesting further elaboration from the author.
Mathematicians and married couple, George Willis and Monica Nevins, have solved a long-standing problem in group theory concerning just-infinite groups. After two decades of collaborative effort, they proved that such groups, which are infinite but become finite when any element is removed, always arise from a specific type of construction related to branch groups. This confirms a conjecture formulated in the 1990s and deepens our understanding of the structure of infinite groups. Their proof, praised for its elegance and clarity, relies on a clever simplification of the problem and represents a significant advancement in the field.
Hacker News commenters generally expressed awe and appreciation for the mathematicians' dedication and the elegance of the solution. Several highlighted the collaborative nature of the work and the importance of such partnerships in research. Some discussed the challenge of explaining complex mathematical concepts to a lay audience, while others pondered the practical applications of this seemingly abstract work. A few commenters with mathematical backgrounds offered deeper insights into the proof and its implications, pointing out the use of representation theory and the significance of classifying groups. One compelling comment mentioned the personal connection between Geoff Robinson and the commenter's advisor, offering a glimpse into the human side of the mathematical community. Another interesting comment thread explored the role of intuition and persistence in mathematical discovery, highlighting the "aha" moment described in the article.
The "Buenos Aires constant" is a humorous misinterpretation of mathematical notation. It stems from a misunderstanding of how definite integrals are represented. Someone saw the integral of a function with respect to x, evaluated from a to b, written as ∫ₐᵇ f(x) dx and mistakenly believed the b in the upper limit of integration was a constant multiplied by the entire integral, similar to how a coefficient might multiply a variable. They specifically misinterpreted ∫₀¹ x² dx as b times some constant and, upon calculating the integral's value of 1/3, assumed b = 1 and therefore the "Buenos Aires constant" was 3. This anecdotal observation highlights how notational conventions can be confusing if not properly understood.
Hacker News commenters discuss the arbitrary nature of the "Buenos Aires constant," pointing out that fitting any small dataset to a specific function will inevitably yield some "interesting" constant. Several users highlight that this is a classic example of overfitting and that similar "constants" can be contrived with other mathematical functions and small datasets. One commenter provides Python code demonstrating how easily such relationships can be manufactured. Another emphasizes the importance of considering the degrees of freedom when fitting a model, echoing the sentiment that finding a "constant" like this is statistically meaningless. The general consensus is that while amusing, the Buenos Aires constant holds no mathematical significance.
The post "XOR" explores the remarkable versatility of the exclusive-or (XOR) operation in computer programming. It highlights XOR's utility in a variety of contexts, from cryptography (simple ciphers) and data manipulation (swapping variables without temporary storage) to graphics programming (drawing lines and circles) and error detection (parity checks). The author emphasizes XOR's fundamental mathematical properties, like its self-inverting nature (A XOR B XOR B = A) and commutativity, demonstrating how these properties enable elegant and efficient solutions to seemingly complex problems. Ultimately, the post advocates for a deeper appreciation of XOR as a powerful tool in any programmer's arsenal.
HN users discuss various applications and interpretations of XOR. Some highlight its reversibility and use in cryptography, while others explain its role in parity checks and error detection. A few comments delve into its connection with addition and subtraction in binary arithmetic. The thread also explores the efficiency of XOR in comparison to other bitwise operations and its utility in situations requiring toggling, such as graphics programming. Some users share personal anecdotes of using XOR for tasks like swapping variables without temporary storage. A recurring theme is the elegance and simplicity of XOR, despite its power and versatility.
Terence Tao argues against overly simplistic solutions to complex societal problems, using the analogy of a chaotic system. He points out that in such systems, small initial changes can lead to vastly different outcomes, making prediction difficult. Therefore, approaches focusing on a single "root cause" or a "one size fits all" solution are likely to be ineffective. Instead, he advocates for a more nuanced, adaptive approach, acknowledging the inherent complexity and embracing diverse, localized solutions that can be adjusted as the situation evolves. He suggests that relying on rigid, centralized planning is often counterproductive, preferring a more decentralized, experimental approach where local actors can respond to specific circumstances.
Hacker News users discussed Terence Tao's exploration of using complex numbers to simplify differential equations, particularly focusing on the example of a forced damped harmonic oscillator. Several commenters appreciated the elegance and power of using complex exponentials to represent oscillations, highlighting how this approach simplifies calculations and provides a more intuitive understanding of phase shifts and resonance. Some pointed out the broader applicability of complex numbers in physics and engineering, mentioning uses in electrical circuits, quantum mechanics, and signal processing. A few users discussed the pedagogical implications, suggesting that introducing complex numbers earlier in physics education could be beneficial. The thread also touched upon the abstract nature of complex numbers and the initial difficulty some students face in grasping their utility.
Elements of Programming (2009) by Alexander Stepanov and Paul McJones provides a foundational approach to programming by emphasizing abstract concepts and mathematical rigor. The book develops fundamental algorithms and data structures from first principles, focusing on clear reasoning and formal specifications. It uses abstract data types and generic programming techniques to achieve code that is both efficient and reusable across different programming languages and paradigms. The book aims to teach readers how to think about programming at a deeper level, enabling them to design and implement robust and adaptable software. While rooted in practical application, its focus is on the underlying theoretical framework that informs good programming practices.
Hacker News users discuss the density and difficulty of Elements of Programming, acknowledging its academic rigor and focus on foundational concepts. Several commenters point out that the book isn't for beginners and requires significant mathematical maturity. The book's use of abstract algebra and its emphasis on generic programming are highlighted, with some finding it insightful and others overwhelming. The discussion also touches on the impracticality of some of the examples for real-world coding and the lack of readily available implementations in popular languages. Some suggest alternative resources for learning practical programming, while others defend the book's value for building a deeper understanding of fundamental principles. A recurring theme is the contrast between the book's theoretical approach and the practical needs of most programmers.
"Anatomy of Oscillation" explores the ubiquitous nature of oscillations in various systems, from physics and engineering to biology and economics. The post argues that these seemingly disparate phenomena share a common underlying structure: a feedback loop where a system's output influences its own input, leading to cyclical behavior. It uses the example of a simple harmonic oscillator (a mass on a spring) to illustrate the core principles of oscillation, including the concepts of equilibrium, displacement, restoring force, and inertia. The author suggests that understanding these basic principles can help us better understand and predict oscillations in more complex systems, ultimately offering a framework for recognizing recurring patterns in seemingly chaotic processes.
Hacker News users discussed the idea of "oscillation" presented in the linked Substack article, primarily focusing on its application in various fields. Some commenters questioned the novelty of the concept, arguing that it simply describes well-known feedback loops. Others found the framing helpful, highlighting its relevance to software development processes, personal productivity, and even biological systems. A few users expressed skepticism about the practical value of the framework, while others offered specific examples of oscillation in their own work, such as product development cycles and the balance between exploration and exploitation in learning. The discussion also touched upon the optimal frequency of oscillations and the importance of recognizing and managing them for improved outcomes.
A Brown University undergraduate, Noah Golowich, disproved a long-standing conjecture in data science related to the "Kadison-Singer problem." This problem, with implications for signal processing and quantum mechanics, asked about the possibility of extending certain "frame" functions while preserving their key properties. A 2013 proof showed this was possible in specific high dimensions, leading to the conjecture it was true for all higher dimensions. Golowich, building on recent mathematical tools, demonstrated a counterexample, proving the conjecture false and surprising experts in the field. His work, conducted under the mentorship of Assaf Naor, highlights the potential of exploring seemingly settled mathematical areas.
Hacker News users discussed the implications of the undergraduate's discovery, with some focusing on the surprising nature of such a significant advancement coming from an undergraduate researcher. Others questioned the practicality of the new algorithm given its computational complexity, highlighting the trade-off between statistical accuracy and computational feasibility. Several commenters also delved into the technical details of the conjecture and its proof, expressing interest in the specific mathematical techniques employed. There was also discussion regarding the potential applications of the research within various fields and the broader implications for data science and machine learning. A few users questioned the phrasing and framing in the original Quanta Magazine article, finding it slightly sensationalized.
"An Infinitely Large Napkin" introduces a novel approach to digital note-taking using a zoomable, infinite canvas. It proposes a system built upon a quadtree data structure, allowing for efficient storage and rendering of diverse content like text, images, and handwritten notes at any scale. The document outlines the technical details of this approach, including data representation, zooming and panning functionalities, and potential features like collaborative editing and LaTeX integration. It envisions a powerful tool for brainstorming, diagramming, and knowledge management, unconstrained by the limitations of traditional paper or fixed-size digital documents.
Hacker News users discuss the "infinite napkin" concept with a mix of amusement and skepticism. Some appreciate its novelty and the potential for collaborative brainstorming, while others question its practicality and the limitations imposed by the fixed grid size. Several commenters mention existing tools like Miro and Mural as superior alternatives, offering more flexibility and features. The discussion also touches on the technical aspects of implementing such a system, with some pondering the challenges of efficient rendering and storage for an infinitely expanding canvas. A few express interest in the underlying algorithm and the possibility of exploring different geometries beyond the presented grid. Overall, the reception is polite but lukewarm, acknowledging the theoretical appeal of the infinite napkin while remaining unconvinced of its real-world usefulness.
The blog post explores the surprising observation that repeated integer addition can approximate floating-point multiplication, specifically focusing on the case of multiplying by small floating-point numbers slightly greater than one. It explains this phenomenon by demonstrating how the accumulation of fractional parts during repeated addition mimics the effect of multiplication. When adding a floating-point number slightly larger than one to itself repeatedly, the fractional part grows with each addition, eventually getting large enough to increment the integer part. This stepping increase in the integer part, combined with the accumulating fractional component, closely resembles the scaling effect of multiplication by that same number. The post illustrates this relationship using both visual representations and mathematical explanations, linking the behavior to the inherent properties of floating-point numbers and their representation in binary.
Hacker News commenters generally praised the article for clearly explaining a non-obvious relationship between integer addition and floating-point multiplication. Some highlighted the practical implications, particularly in older hardware or specialized situations where integer operations are significantly faster. One commenter pointed out the historical relevance to Quake III's fast inverse square root approximation, while another noted the connection to logarithms and how this technique could be extended to other operations. A few users discussed the limitations and boundary conditions, emphasizing the approximation's validity only within specific ranges and the importance of understanding those constraints. Some commenters provided further context by linking to related concepts like the "magic number" used in the Quake III algorithm and resources on floating-point representation.
Michael Atiyah's "Mathematics in the 20th Century" provides a sweeping overview of the field's progress during that period, highlighting key trends and breakthroughs. He emphasizes the increasing abstraction and unification within mathematics, exemplified by the rise of algebraic topology and category theory. Atiyah discusses the burgeoning interplay between mathematics and physics, particularly quantum mechanics and general relativity, which spurred new mathematical developments. He also touches upon the growing influence of computers and the expansion of specialized areas, while noting the enduring importance of core subjects like analysis and geometry. The essay concludes with reflections on the evolving nature of mathematical research, acknowledging the challenges of specialization while expressing optimism for future discoveries driven by interdisciplinary connections and new perspectives.
Hacker News commenters discuss Atiyah's lecture, praising its clarity, accessibility, and broad yet insightful overview of 20th-century mathematics. Several highlight the interesting connections Atiyah draws between seemingly disparate fields, particularly geometry and physics. Some commenters reminisce about Atiyah's lectures, describing him as a brilliant and engaging speaker. Others share anecdotes or additional resources related to the topics discussed, including links to other writings by Atiyah and recommendations for further reading. A few express disappointment that the lecture doesn't delve deeper into certain areas, but the overall sentiment is one of appreciation for Atiyah's insightful and inspiring presentation.
This post advocates for clear, legible mathematical handwriting, emphasizing the importance of distinguishing similar symbols. It offers specific guidelines for writing letters (like lowercase 'x' and 'times,' 'u' and 'union,' and Greek letters), numerals (particularly distinguishing '1,' '7,' and 'I'), and other mathematical symbols (such as plus/minus, radicals, and various brackets). The author stresses vertical alignment within equations, proper spacing, and the use of serifs for improved clarity. Overall, the goal is to enhance readability and avoid ambiguity in handwritten mathematics, benefiting both the writer and anyone reading the work.
Hacker News users discuss the linked guide on mathematical handwriting, largely praising its practical advice. Several commenters highlight the importance of clear communication in mathematics, emphasizing that legible handwriting benefits both the writer and the reader. Some share personal anecdotes about struggling with handwriting and the impact it has on mathematical work. The suggestion to practice writing Greek letters resonates with many, as does the advice on spacing and distinguishing similar-looking symbols. A few commenters offer additional tips, such as using lined paper turned sideways for better vertical alignment and practicing writing on a whiteboard to improve clarity and flow. Overall, the comments reflect an appreciation for the guide's focus on the often-overlooked skill of legible mathematical writing.
This post explores the inherent explainability of linear programs (LPs). It argues that the optimal solution of an LP and its sensitivity to changes in constraints or objective function are readily understandable through the dual program. The dual provides shadow prices, representing the marginal value of resources, and reduced costs, indicating the improvement needed for a variable to become part of the optimal solution. These values offer direct insights into the LP's behavior. Furthermore, the post highlights the connection between the simplex algorithm and sensitivity analysis, explaining how pivoting reveals the impact of constraint adjustments on the optimal solution. Therefore, LPs are inherently explainable due to the rich information provided by duality and the simplex method's step-by-step process.
Hacker News users discussed the practicality and limitations of explainable linear programs (XLPs) as presented in the linked article. Several commenters questioned the real-world applicability of XLPs, pointing out that the constraints requiring explanations to be short and easily understandable might severely restrict the solution space and potentially lead to suboptimal or unrealistic solutions. Others debated the definition and usefulness of "explainability" itself, with some suggesting that forcing simple explanations might obscure the true complexity of a problem. The value of XLPs in specific domains like regulation and policy was also considered, with commenters noting the potential for biased or manipulated explanations. Overall, there was a degree of skepticism about the broad applicability of XLPs while acknowledging the potential value in niche applications where transparent and easily digestible explanations are paramount.
Noether's theorem, proven by mathematician Emmy Noether in 1915, reveals a profound connection between symmetries in nature and conservation laws. It states that every continuous symmetry in a physical system corresponds to a conserved quantity. For example, the symmetry of physical laws over time leads to the conservation of energy, and the symmetry of laws across space leads to the conservation of momentum. This theorem has become a cornerstone of modern physics, providing a powerful tool for understanding and predicting the behavior of physical systems, from classical mechanics and electromagnetism to quantum field theory and general relativity. It unified seemingly disparate concepts and drastically simplified the search for new laws of physics.
HN commenters generally praised the Quanta article for its clear explanation of Noether's theorem, with several sharing personal anecdotes about learning it. Some discussed the theorem's implications, highlighting its connection to symmetries in physics and its importance in modern theories like quantum field theory and general relativity. A few commenters delved into more technical details, mentioning Lagrangian and Hamiltonian mechanics, gauge theories, and the relationship between conservation laws and symmetries. One commenter pointed out the importance of differentiating between global and local symmetries, while others appreciated the article's accessibility even for those without a deep physics background. The overall sentiment was one of appreciation for both Noether's work and the article's elucidation of it.
Transfinite Nim, a variation of the classic game Nim, extends the concept to infinite ordinal numbers. Players take turns removing any finite, positive number of stones from a single heap, but the heaps themselves can be indexed by ordinal numbers. The game proceeds as usual, with the last player to remove stones winning. The article explores the winning strategy for this transfinite game, demonstrating that despite the infinite nature of the game, a winning strategy always exists. This strategy involves considering the bitwise XOR sum of the heap sizes (using the Cantor normal form for ordinals) and aiming to leave a sum of zero after your turn. Crucially, the winning strategy requires a player to leave only finitely many non-empty heaps after each turn. The article further explores variations of the game, including when infinitely many stones can be removed at once, demonstrating different winning conditions in these altered scenarios.
HN commenters discuss the implications and interesting aspects of transfinite Nim. Several express fascination with the idea of games with infinitely many positions, questioning the practicality and meaning of "winning" such a game. Some dive into the strategy, mentioning the importance of considering ordinal numbers and successor ordinals. One commenter connects the game to the concept of "good sets" within set theory, while another raises the question of whether Zermelo-Fraenkel set theory is powerful enough to determine the winner for all ordinal games. The surreal number system is also brought up as a relevant mathematical structure for understanding transfinite games. Overall, the comments show a blend of curiosity about the theoretical nature of the game and attempts to grasp the strategic implications of infinite play.
The website "Explorable Flexagons" offers an interactive introduction to creating and manipulating flexagons, a type of folded paper polygon that reveals hidden faces when "flexed." It provides clear instructions and diagrams for building common flexagons like the trihexaflexagon and hexahexaflexagon, along with tools to virtually fold and explore these fascinating mathematical objects. The site also delves into the underlying mathematical principles, including notations for tracking face transitions and exploring different flexing patterns. It encourages experimentation and discovery, allowing users to design their own flexagon templates and discover new flexing possibilities.
HN users generally praise the interactive flexagon explorer for its clear explanations and engaging visualizations. Several commenters share nostalgic memories of making flexagons as children, spurred by articles in Scientific American or books like Martin Gardner's "Mathematical Puzzles and Diversions." Some discuss the mathematical underpinnings of flexagons, mentioning group theory and combinatorial geometry. A few users express interest in physical construction techniques and different types of flexagons beyond the basic trihexaflexagon. The top comment highlights the value of interactive explanations, noting how it transforms a potentially dry topic into an enjoyable learning experience.
Summary of Comments ( 13 )
https://news.ycombinator.com/item?id=43207962
HN commenters largely dismiss the linked article as numerology and pseudoscience. Several point out the arbitrary nature of choosing specific measurements and units (meters, cubits) to force connections. One commenter notes that the golden ratio shows up frequently in geometric constructions, making its presence in the pyramids unsurprising and not necessarily indicative of intentional design. Others criticize the article's lack of rigor and its reliance on coincidences rather than evidence-based arguments. The general consensus is that the article presents a flawed and unconvincing argument for a relationship between these different elements.
The Hacker News post titled "The Meter, Golden Ratio, Pyramids, and Cubits, Oh My" has generated a moderate number of comments, most of which express skepticism and amusement at the original article's attempt to connect the meter to the Great Pyramid of Giza via the golden ratio and cubits.
Several commenters point out the historical inaccuracy of the claims. One commenter highlights that the meter's definition has changed over time, initially being related to the Earth's circumference and only later linked to a physical artifact. This debunks the idea of a pre-planned connection to ancient Egyptian measurements. Another commenter mentions the imprecision inherent in measuring the pyramid itself, making any exact correspondence with the meter highly improbable. The variability in historical cubit lengths is also raised, further undermining the argument for a precise relationship.
Another line of discussion centers on the perceived "pyramid inch" and its alleged relationship to British Imperial units. Commenters dismiss this connection as coincidental and highlight the convoluted logic required to arrive at such a conclusion. The tendency to find patterns where none exist is also discussed, referencing the phenomenon of pareidolia.
Some commenters approach the topic with humor, joking about the prevalence of such theories and the fascination with hidden connections. One commenter sarcastically suggests a connection between the size of their foot and the circumference of Jupiter. Another uses the opportunity to plug a book debunking similar historical myths.
A few commenters attempt to engage with the mathematical aspects, discussing the golden ratio and its properties. However, these discussions generally reinforce the skepticism towards the original article's claims, emphasizing the lack of evidence for any meaningful connection.
In summary, the comments on Hacker News largely reject the premise of the linked article. They point out historical inaccuracies, methodological flaws, and the general implausibility of the proposed connections. The overall tone is one of skepticism, occasionally tinged with humor and amusement at the article's attempts to find profound meaning in numerical coincidences.