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Approximation Game
2 months ago
- The article explores the approximation of real numbers using rational fractions with small denominators, focusing on the differences between rational and irrational numbers.
- It introduces the concept of 'good' approximations, defining 1-good (ε < 1/b) and 2-good (ε < 1/b²) approximations based on error metrics.
- Rational numbers are shown to have a limited number of 2-good approximations, with the number of such approximations capped and clustered near the beginning.
- Irrational numbers, however, have an infinite supply of 2-good approximations, as proven by Dirichlet's approximation theorem using the pigeonhole principle.
- The article provides intuitive explanations for these phenomena by discussing the construction of rational and real numbers, highlighting the uniform spacing of rationals and the gaps where irrationals reside.
- It mentions the complexity of Diophantine approximations and the study of numbers with high irrationality exponents, noting the difficulty in proving certain properties of algebraic irrationals.
Some Lotto Math
2 months ago
- A couple won two lottery prizes, each over £1 million, with odds over 24 trillion to one.
- Lottery odds calculations often multiply probabilities, but this approach has limitations.
- The UK Lotto involves selecting 6 numbers from 59, with prizes for matching 2 to 6 numbers.
- Probability calculations for matching numbers involve combinations and binomial coefficients.
- The 'bonus ball' adds complexity, affecting the odds for certain prize tiers.
- Lottery odds are often rounded to the nearest integer for presentation.
- Tools like Mathematica's Binomial function and Python's math.comb can calculate large combinations.
Algebraic topology: knots links and braids
2 months ago
- A knot is a simple closed curve in Euclidean 3-space (E(3)), and two knots are equivalent if there's an orientation-preserving homeomorphism mapping one to the other.
- Schoenflies proved in 1908 that homeomorphisms from simple closed curves in the plane (E(2)) to the unit circle (S(1)) can be extended to the entire plane, but this doesn't hold in higher dimensions.
- Wild embeddings in E(3) exist, such as Alexander's horned sphere and Antoine's necklace, where complements are not simply connected.
- Knots are typically studied as tame embeddings, like simple closed polygonal curves, to avoid wild behavior.
- Knot diagrams are projections into E(2) with over/under-crossing indications, sufficient to reconstruct the knot up to equivalence.
- Reidemeister moves define equivalence between knot diagrams: two diagrams represent the same knot if one can be transformed into the other via these moves.
- The unknot is the unique knot with a diagram without crossings.
- Knot composition involves tying two oriented knots sequentially, with the unknot acting as a zero element.
- A knot is prime if it cannot be expressed as a sum of two non-trivial knots, and every knot has a unique prime factorization.
- Seifert surfaces are used to define the genus of a knot, which is additive under composition and zero only for the unknot.
- Alternating knots have diagrams where overcrossings and undercrossings alternate, and most prime knots with ≤8 crossings are alternating.
- Knot invariants, like the fundamental group of the complement or the Jones polynomial, help distinguish inequivalent knots.
- Links are disjoint unions of knots, and trivial links have free fundamental groups for their complements.
- Braids are collections of arcs in E(3), and the braid group B(n) is generated by elementary braids with specific relations.
- Alexander's theorem states that every link is equivalent to one obtained from a braid by identifying its endpoints.
Faster Asin() Was Hiding in Plain Sight
2 months ago
- The author reflects on the importance of research before acting to save time, using their experience with PSRayTracing as an example.
- A faster approximation for the arcsine function was sought to improve the performance of PSRayTracing, leading to the exploration of Taylor series and Padé approximants.
- Initial attempts with Taylor series showed speed improvements but had significant errors outside the range [-0.8, 0.8], requiring fallback to std::asin().
- Padé approximants were then explored, offering better accuracy with less error, especially when combined with half-angle transformations for edge cases.
- A significant breakthrough came from discovering a fast arcsine approximation from Nvidia's Cg Toolkit, which was both elegant and significantly faster across various hardware.
- Benchmarking showed the Cg approximation was consistently faster than std::asin(), with speedups ranging from 1.02x to 1.89x depending on the hardware.
- The author learned the value of checking existing solutions before diving into complex optimizations, highlighting the importance of community knowledge and historical resources.
Virtual Math Museum
2 months ago
- 3DXM Virtual Math Museum provides hyperlinked explanations of all objects in 3D-XplorMath.
- Includes a collection of ATOs (Abstract Topological Objects).
- Development supported by The National Science Foundation via CCLI Grant (DUE Award # 0514781).
You gotta think outside the hypercube
2 months ago
- Introduction to the tesseract and its visualizations in media.
- Mathematical modeling of a hypercube starting from a square.
- Extension of the model to three dimensions (cube) and then to four dimensions (tesseract).
- Explanation of rotations in 2D, 3D, and 4D spaces.
- Different projection methods for rendering 4D objects in 2D: Cavalier, Cabinet, Isometric, Rectilinear one-point perspective, Fisheye perspective, and Mixed isometric + vanishing point.
- Visual examples and comparisons of tesseract renderings using various projection techniques.
- Discussion on the challenges and distortions in projecting 4D objects onto 2D screens.
- Link to MATLAB code for generating animation frames of the tesseract.
In Praise of Stupid Questions
2 months ago
- The author reflects on their habit of asking many questions, sometimes leading to humorous or awkward situations.
- They discuss the concept of 'stupid questions' and argue that while some questions may seem meaningless, they can lead to valuable insights or new mathematical discoveries.
- A personal anecdote is shared about using ChatGPT to explore a probability question related to coin tossing, leading to the discovery of a new way to estimate π/4.
- The article highlights the unexpected appearance of π in probability theory, similar to its appearance in geometric probability via Buffon's needle experiment.
- Matt Parker's experimental approach to estimating π using coin tosses is mentioned, demonstrating the practical challenges and theoretical underpinnings of such methods.
- The author concludes with lessons on the value of asking questions, even seemingly stupid ones, and the importance of persistence and curiosity in mathematical research.
Mathematics Distillation Challenge – Equational Theories
2 months ago
- Mathematical research traditionally involves small groups of professional mathematicians working on deep problems.
- A complementary approach involves collaborating with a broad community on mathematically interesting problems using modern technologies like AI.
- Examples of such collaborations include Polymath projects, collaborative formalization projects, and the Equational Theories Project (ETP).
- The ETP used Lean formalization and automated theorem provers to solve over 22 million true-false problems in universal algebra.
- A new project aims to distill the ETP's results into a concise, human-readable 'cheat sheet'.
- The project includes a competitive challenge to design a cheat sheet (≤10KB) that improves the performance of cheap AI models on true-false problems.
- Stage 1 of the challenge involves testing cheat sheets on a public set of 1200 problems (1000 easy, 200 hard).
- Top 1000 submissions will advance to Stage 2, which involves more advanced models and providing proofs or counterexamples.
- The competition is coordinated on a Zulip channel, with submissions due by April 20.
- Future challenges may focus on other mathematical problem classes and cooperative approaches.
Consensus Board Game
2 months ago
- The author shares a personal motivation for explaining consensus algorithms, stemming from early struggles with the concept.
- The post focuses on the mathematical and logical structure behind consensus algorithms, not the engineering aspects.
- A committee of five members tries to choose a color for a bike shed, illustrating the consensus problem with unreliable members.
- Simple majority voting is introduced, but issues like split votes and leader unavailability are highlighted.
- A 2D board concept is introduced where multiple concurrent votes with rotating leaders help overcome individual column failures.
- Safety conditions are added to ensure that no two columns can have conflicting majority decisions.
- The strategy involves focusing on left columns to prevent future conflicts, assuming all participants follow the same rules.
- Participants must commit to abstaining in certain columns to ensure no conflicting decisions can be made.
- The rightmost choice is preferred to align with previous decisions made by other participants.
- The abstract view simplifies the distributed system's complexity, where participants only see partial information and must communicate to infer the global state.
The Shape of Inequalities
2 months ago
- The article explores geometric representations of algebraic inequalities, focusing on the HM-AM-GM-QM inequality chain.
- It introduces visualizations like circles, semicircles, and containers to illustrate inequalities such as AM ≥ GM.
- The harmonic mean (HM) is explained with real-world examples like average speed calculations.
- The geometric mean (GM) is highlighted in contexts like stock growth and compounding.
- The arithmetic mean (AM) and quadratic mean (QM) are discussed, with QM's relevance in electrical engineering noted.
- A semicircle visualization demonstrates the hierarchy HM ≤ GM ≤ AM ≤ QM, showing their geometric relationships.
- The 'container' analogy compares areas and volumes to prove AM-GM inequality, emphasizing symmetry's role in maximizing capacity.
- A 3D version extends the container analogy to volumes, reinforcing the idea that symmetry (a cube) holds the maximum volume for a given perimeter.
- The sum of squares inequality (a² + b² + c² ≥ ab + bc + ca) is visualized using overlapping squares and rectangles.
- Nesbitt’s inequality is approached through Viviani’s Theorem, linking it to distances in an equilateral triangle.
- The article concludes by reflecting on the challenges of representing complex algebraic truths geometrically but appreciates the insights gained from such visualizations.
Mathematician who reshaped number theory wins prestigious Abel Prize
2 months ago
- Gerd Faltings wins the 2026 Abel Prize for his work in number theory.
- Faltings proved the Mordell conjecture (1922), showing Diophantine equations have finite solutions.
- His 1983 work on Diophantine equations made a significant impact in mathematics.
- Faltings previously won the Fields Medal in 1986 for his contributions.
- The Abel Prize, modeled after the Nobel, awards 7.5 million Norwegian Kroner.
- Faltings' work includes proving higher-power equations have finite rational solutions.
- His research contrasts with Pythagorean triples, which have infinite integer solutions.
- Andrew Wiles' proof of Fermat's Last Theorem is a related celebrated result.
Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize
2 months ago
- Gerd Faltings awarded the 2024 Abel Prize for lifetime achievements in mathematics.
- Faltings is renowned for proving the Mordell conjecture in 1983, now called Faltings's theorem.
- Faltings's theorem states that curves with equations of degree higher than 3 have finitely many rational points.
- The theorem is a cornerstone of arithmetic geometry, influencing many subsequent mathematical developments.
- Faltings also made significant contributions to p-adic Hodge theory and generalized his theorem to higher dimensions.
- He previously won the Fields Medal at age 32, marking a distinguished career.
- The Abel Prize committee praised Faltings as a towering figure who reshaped arithmetic geometry.
- Faltings reflects on the increasing competition in mathematics but remains content with his legacy.
Show HN: The King Wen Permutation: [52, 10, 2]
a month ago
- The I Ching's 64 hexagrams have two main orderings: binary natural order (0–63) and the King Wen sequence (~1000 BCE).
- Mapping between these orderings as a permutation in S64 reveals a cycle decomposition of [52, 10, 2] with no fixed points.
- 81% of hexagrams are part of a single cycle, a structure not previously documented in literature.
- The King Wen permutation σ ∈ S64 has unique properties explored in 22 client-side browser-based analyses.
The ancient reason there are 60 minutes in an hour
a month ago
- The French Republic attempted to decimalize time in 1793, dividing the day into 10 hours, each with 100 minutes, but the system was abandoned after causing practical difficulties.
- The Sumerians, an ancient Mesopotamian civilization, developed a base-60 (sexagesimal) number system, which influenced modern timekeeping (60 minutes in an hour, 60 seconds in a minute).
- The Babylonians adopted the Sumerian sexagesimal system and further divided time into smaller units, influencing later Greek and Hellenistic timekeeping methods.
- Ancient Egyptians were the first to divide the day into 12 hours, leading to the 24-hour day, possibly influenced by star clocks and religious practices.
- Mechanical clocks in the 12th century improved timekeeping accuracy, but it wasn't until the 18th century that minutes and seconds became commonly used in daily life.
- Atomic clocks, developed in the 20th century, redefined the second with extreme precision, using caesium-133 atoms, and now underpin modern technologies like GPS and the internet.
- The decimal time experiment in France lasted only 17 months, failing due to lack of practical advantage and disruption to existing systems, unlike the successful adoption of metric measurements.
Why mathematicians are boycotting their biggest conference
a month ago
- Mathematicians are threatening to boycott the International Congress of Mathematicians (ICM) in Philadelphia due to U.S. military actions, visa suspensions, and immigration policies.
- A petition with over 1,500 signatures calls for moving the event, citing a double standard after the 2022 ICM was moved from Russia following the invasion of Ukraine.
- The French Mathematical Society announced it would skip the event, citing safety concerns, while other societies plan to attend to uphold international collaboration.
- Historians note parallels with past boycotts, highlighting that internationalism in mathematics always involves compromises amid political conflicts.
- Organizers have not responded, leaving it unclear how the petition will impact the July event, but it demonstrates mathematicians' collective influence.

Hasty Briefsbeta

#mathematics

Approximation Game

Some Lotto Math

Algebraic topology: knots links and braids

Faster Asin() Was Hiding in Plain Sight

Virtual Math Museum

You gotta think outside the hypercube

In Praise of Stupid Questions

Mathematics Distillation Challenge – Equational Theories

Consensus Board Game

The Shape of Inequalities

Mathematician who reshaped number theory wins prestigious Abel Prize

Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize

Show HN: The King Wen Permutation: [52, 10, 2]

The ancient reason there are 60 minutes in an hour

Why mathematicians are boycotting their biggest conference