<![CDATA[MKLab - EXPOSITION]]> https://mklab.gr/ Sat, 12 Sep 2026 08:26:59 +0000 MyBB <![CDATA[The Drake equation]]> https://mklab.gr/showthread.php?tid=1921 Wed, 09 Sep 2026 23:10:56 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1921
Jørgen Veisdal’s article explains the Drake equation, proposed by astronomer Frank Drake in 1961 as a framework for estimating how many technologically advanced extraterrestrial civilizations might currently be detectable in the Milky Way. The equation is
$N = R_* \times f_p \times n_e \times f_l \times f_i \times f_c \times L$,
where $R_*$ is the rate of star formation, $f_p$ the fraction of stars with planets, $n_e$ the number of potentially habitable planets per planetary system, $f_l$ the fraction on which life develops, $f_i$ the fraction that develops intelligence, $f_c$ the fraction producing detectable communications, and $L$ the duration for which such civilizations remain detectable. The equation was never intended to give a precise answer; rather, Drake devised it to organize the major scientific uncertainties involved in the search for extraterrestrial intelligence. Early optimistic assumptions produced estimates ranging from roughly 20 to tens of millions of communicating civilizations, illustrating how sensitive the result is to the chosen parameters.
 
Astronomical observations have since improved some of these estimates. We now know that planets appear to be extremely common, and studies using Kepler data have suggested that the Milky Way may contain billions of roughly Earth-sized planets in habitable zones. However, the biological and sociological factors—$f_l$, $f_i$, $f_c$, and especially $L$—remain largely speculative because Earth provides essentially our only example of life and technological civilization. Using more modern astronomical estimates while assigning intermediate values such as $0.5$ to the unknown factors, the article obtains an illustrative estimate of about 46 communicating extraterrestrial civilizations in the Milky Way. The author stresses that this number should not be interpreted as a scientific measurement; it mainly demonstrates how assumptions propagate through the Drake equation.
 
The article concludes by connecting the Drake equation with the Fermi paradox: if potentially habitable planets are abundant and technological civilizations are reasonably common, why have we seen no convincing evidence of them? Given the age and enormous number of stars in the Galaxy, even relatively slow interstellar expansion might allow a civilization to spread across the Milky Way in a few million years. Thus the tension between large Drake-equation estimates and the absence of observable extraterrestrials leads to Fermi’s famous question: “Where is everybody?”

Key takeaways:
  • The Drake equation is a framework for estimating the number of detectable extraterrestrial civilizations in the Milky Way.
  • It is not a precise prediction; its value depends heavily on uncertain assumptions.
  • Astronomy has improved estimates for factors such as the number of stars, planets, and potentially habitable worlds.
  • The biggest uncertainties remain biological and technological: how often life, intelligence, communication technology, and long-lived civilizations arise.
  • Small changes in these uncertain factors can change the estimate from almost zero to millions of civilizations.
  • The article gives an illustrative modern estimate of about 46 communicating civilizations, but this should not be treated as a measured value.
  • The Drake equation connects naturally to the Fermi paradox: if intelligent civilizations may be common, why have we found no convincing evidence of them?

ARTICLE / ARTICLE [PDF]]]>
Jørgen Veisdal’s article explains the Drake equation, proposed by astronomer Frank Drake in 1961 as a framework for estimating how many technologically advanced extraterrestrial civilizations might currently be detectable in the Milky Way. The equation is
$N = R_* \times f_p \times n_e \times f_l \times f_i \times f_c \times L$,
where $R_*$ is the rate of star formation, $f_p$ the fraction of stars with planets, $n_e$ the number of potentially habitable planets per planetary system, $f_l$ the fraction on which life develops, $f_i$ the fraction that develops intelligence, $f_c$ the fraction producing detectable communications, and $L$ the duration for which such civilizations remain detectable. The equation was never intended to give a precise answer; rather, Drake devised it to organize the major scientific uncertainties involved in the search for extraterrestrial intelligence. Early optimistic assumptions produced estimates ranging from roughly 20 to tens of millions of communicating civilizations, illustrating how sensitive the result is to the chosen parameters.
 
Astronomical observations have since improved some of these estimates. We now know that planets appear to be extremely common, and studies using Kepler data have suggested that the Milky Way may contain billions of roughly Earth-sized planets in habitable zones. However, the biological and sociological factors—$f_l$, $f_i$, $f_c$, and especially $L$—remain largely speculative because Earth provides essentially our only example of life and technological civilization. Using more modern astronomical estimates while assigning intermediate values such as $0.5$ to the unknown factors, the article obtains an illustrative estimate of about 46 communicating extraterrestrial civilizations in the Milky Way. The author stresses that this number should not be interpreted as a scientific measurement; it mainly demonstrates how assumptions propagate through the Drake equation.
 
The article concludes by connecting the Drake equation with the Fermi paradox: if potentially habitable planets are abundant and technological civilizations are reasonably common, why have we seen no convincing evidence of them? Given the age and enormous number of stars in the Galaxy, even relatively slow interstellar expansion might allow a civilization to spread across the Milky Way in a few million years. Thus the tension between large Drake-equation estimates and the absence of observable extraterrestrials leads to Fermi’s famous question: “Where is everybody?”

Key takeaways:
  • The Drake equation is a framework for estimating the number of detectable extraterrestrial civilizations in the Milky Way.
  • It is not a precise prediction; its value depends heavily on uncertain assumptions.
  • Astronomy has improved estimates for factors such as the number of stars, planets, and potentially habitable worlds.
  • The biggest uncertainties remain biological and technological: how often life, intelligence, communication technology, and long-lived civilizations arise.
  • Small changes in these uncertain factors can change the estimate from almost zero to millions of civilizations.
  • The article gives an illustrative modern estimate of about 46 communicating civilizations, but this should not be treated as a measured value.
  • The Drake equation connects naturally to the Fermi paradox: if intelligent civilizations may be common, why have we found no convincing evidence of them?

ARTICLE / ARTICLE [PDF]]]> <![CDATA[Reflections on the Millennium Problems]]> https://mklab.gr/showthread.php?tid=1912 Tue, 08 Sep 2026 22:04:13 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1912 “Reflections on the Millennium Problems” (August 2026) argues that three famous Millennium Prize Problems—the Riemann Hypothesis, $P$ vs. $NP$, and the existence and smoothness of Navier–Stokes solutions—remain mathematically profound but have gradually become less consequential in the direct practical senses originally associated with them. 

For the Riemann Hypothesis, trillions of computed zeros of $\zeta(s)$ lie on the critical line $\operatorname{Re}(s)=\tfrac12$, so even if RH eventually fails, Trefethen argues that the resulting quantitative disturbance in the observed distribution of primes would probably be extremely small; the greater importance of a proof would be the new mathematical techniques it might unlock, particularly for the Generalized Riemann Hypothesis.

 For $P$ vs. $NP$, decades of computation have shown that worst-case exponential complexity often does not prevent difficult problems from being solved effectively in practice through heuristics, approximation algorithms, special structure, and favorable typical cases, so the distinction has become more of a fundamental organizing principle of theoretical computer science than an absolute practical barrier.

 For Navier–Stokes, increasing evidence suggests that finite-time singularities, if they exist at all, may require highly artificial and unstable initial configurations, making them unlikely to matter for ordinary physical fluid flows. Trefethen’s broader thesis is that extremely long-lived open problems may gradually lose some of their original practical “leverage” precisely because mathematics and computation learn to work successfully around them; nevertheless, their eventual solution would still be historic, mainly because of the new theories and methods developed in pursuing them. 

ARTICLE [PDF]]]> “Reflections on the Millennium Problems” (August 2026) argues that three famous Millennium Prize Problems—the Riemann Hypothesis, $P$ vs. $NP$, and the existence and smoothness of Navier–Stokes solutions—remain mathematically profound but have gradually become less consequential in the direct practical senses originally associated with them. 

For the Riemann Hypothesis, trillions of computed zeros of $\zeta(s)$ lie on the critical line $\operatorname{Re}(s)=\tfrac12$, so even if RH eventually fails, Trefethen argues that the resulting quantitative disturbance in the observed distribution of primes would probably be extremely small; the greater importance of a proof would be the new mathematical techniques it might unlock, particularly for the Generalized Riemann Hypothesis.

 For $P$ vs. $NP$, decades of computation have shown that worst-case exponential complexity often does not prevent difficult problems from being solved effectively in practice through heuristics, approximation algorithms, special structure, and favorable typical cases, so the distinction has become more of a fundamental organizing principle of theoretical computer science than an absolute practical barrier.

 For Navier–Stokes, increasing evidence suggests that finite-time singularities, if they exist at all, may require highly artificial and unstable initial configurations, making them unlikely to matter for ordinary physical fluid flows. Trefethen’s broader thesis is that extremely long-lived open problems may gradually lose some of their original practical “leverage” precisely because mathematics and computation learn to work successfully around them; nevertheless, their eventual solution would still be historic, mainly because of the new theories and methods developed in pursuing them. 

ARTICLE [PDF]]]> <![CDATA[Interested in foundations of mathematics]]> https://mklab.gr/showthread.php?tid=1886 Mon, 07 Sep 2026 23:00:58 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1886 Author: Vladimir Voevodsky
Lecture: ASC 2014, Nanyang Technological University, Singapore, 25 August 2014


Vladimir Voevodsky explains how his experience with increasingly complicated mathematical proofs led him toward computer-assisted proof verification and new foundations of mathematics. He contrasts problems such as solving a Rubik’s Cube—where a solution can be checked directly—with advanced mathematics, where even a published proof may contain hidden errors. He recounts his proof of the Milnor Conjecture, which took years to turn from the central idea into a complete rigorous argument, and then describes a more troubling example: a 1991 theorem he proved with Mikhail Kapranov concerning $\infty$-groupoids. Carlos Simpson later produced a counterexample, and in 2013 Voevodsky finally recognized that not merely the proof but the theorem itself, with their chosen definition, was false. This episode convinced him that relying solely on human checking becomes increasingly risky as mathematics grows more complicated. 

Voevodsky therefore asks whether mathematical proofs could be checked by computers in much the same way that symbolic computation verifies algebraic identities. To accomplish this, both mathematical statements and proofs must be encoded as symbolic objects—a process called formalization—so that software can verify mechanically that a proposed proof genuinely establishes its theorem. He argues that the traditional foundation of mathematics, ZFC set theory, was developed long before computers and was not designed for convenient formalization of modern mathematics. His search for a more suitable foundation led him beginning around 2006 to Univalent Foundations, connecting Martin-Löf type theory, homotopy theory, and foundations. By 2010 he believed a practical system was emerging, and the 2012–13 IAS program helped develop what became Homotopy Type Theory (HoTT). 

The broader aim is not simply to eliminate mistakes but to change how mathematicians work. Voevodsky suggests that the increasing complexity of modern mathematics makes researchers spend more time checking proofs and can make them less willing to pursue daring ideas. Reliable formal verification could transfer part of that checking burden to computers, allowing mathematicians to concentrate more on concepts and discovery. Projects such as UniMath, implemented using the Coq proof assistant, represented early steps toward this vision: large libraries in which definitions, theorems, and proofs can be verified mechanically. 

Key takeaways
  • Mathematical proofs can remain wrong for years, even when written by leading mathematicians and published in respected journals.
  • Formalization converts statements and proofs into symbolic expressions that a computer can mechanically verify.
  • Voevodsky regarded traditional ZFC foundations as poorly adapted to large-scale computer formalization, motivating his development of Univalent Foundations.
  • Homotopy Type Theory and UniMath emerged from this program, combining type theory, topology, and proof assistants in an effort to make computer-verified mathematics practical. 

ARTICLE [PDF]]]> Author: Vladimir Voevodsky
Lecture: ASC 2014, Nanyang Technological University, Singapore, 25 August 2014


Vladimir Voevodsky explains how his experience with increasingly complicated mathematical proofs led him toward computer-assisted proof verification and new foundations of mathematics. He contrasts problems such as solving a Rubik’s Cube—where a solution can be checked directly—with advanced mathematics, where even a published proof may contain hidden errors. He recounts his proof of the Milnor Conjecture, which took years to turn from the central idea into a complete rigorous argument, and then describes a more troubling example: a 1991 theorem he proved with Mikhail Kapranov concerning $\infty$-groupoids. Carlos Simpson later produced a counterexample, and in 2013 Voevodsky finally recognized that not merely the proof but the theorem itself, with their chosen definition, was false. This episode convinced him that relying solely on human checking becomes increasingly risky as mathematics grows more complicated. 

Voevodsky therefore asks whether mathematical proofs could be checked by computers in much the same way that symbolic computation verifies algebraic identities. To accomplish this, both mathematical statements and proofs must be encoded as symbolic objects—a process called formalization—so that software can verify mechanically that a proposed proof genuinely establishes its theorem. He argues that the traditional foundation of mathematics, ZFC set theory, was developed long before computers and was not designed for convenient formalization of modern mathematics. His search for a more suitable foundation led him beginning around 2006 to Univalent Foundations, connecting Martin-Löf type theory, homotopy theory, and foundations. By 2010 he believed a practical system was emerging, and the 2012–13 IAS program helped develop what became Homotopy Type Theory (HoTT). 

The broader aim is not simply to eliminate mistakes but to change how mathematicians work. Voevodsky suggests that the increasing complexity of modern mathematics makes researchers spend more time checking proofs and can make them less willing to pursue daring ideas. Reliable formal verification could transfer part of that checking burden to computers, allowing mathematicians to concentrate more on concepts and discovery. Projects such as UniMath, implemented using the Coq proof assistant, represented early steps toward this vision: large libraries in which definitions, theorems, and proofs can be verified mechanically. 

Key takeaways
  • Mathematical proofs can remain wrong for years, even when written by leading mathematicians and published in respected journals.
  • Formalization converts statements and proofs into symbolic expressions that a computer can mechanically verify.
  • Voevodsky regarded traditional ZFC foundations as poorly adapted to large-scale computer formalization, motivating his development of Univalent Foundations.
  • Homotopy Type Theory and UniMath emerged from this program, combining type theory, topology, and proof assistants in an effort to make computer-verified mathematics practical. 

ARTICLE [PDF]]]> <![CDATA[The Evolving Foundations of Math]]> https://mklab.gr/showthread.php?tid=1868 Sun, 06 Sep 2026 00:53:53 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1868 “The Evolving Foundations of Math” examines how the basic assumptions underlying mathematics have repeatedly been questioned, repaired and rebuilt from the late 19th century to the present. Beginning with Cantor’s revolutionary work on infinity and the development of rigorous set theory, the series traces the tension between intuition, formal proof and logical consistency through figures such as Cantor, Zermelo, Gödel and Grothendieck. It explores controversial alternatives such as ultrafinitism, the paradoxes created by infinity, the growing use of computer-verified proofs, and modern attempts to reconstruct areas such as topology and geometry from more fundamental concepts. The central message is that mathematics is not built on an immutable foundation: its definitions, axioms, methods of proof and even ideas about what mathematical objects should exist continue to evolve as mathematicians encounter paradoxes, new technologies and deeper structural questions. 

Articles in the series

Chapter 1 — Original Sin
  1. Long-Lost Letters Reveal the Lies That Gave Rise To Modern Math — Joseph Howlett
    Cantor, the discovery of different sizes of infinity, and newly uncovered evidence concerning the origins of his famous 1874 result.
  2. How Can Infinity Come in Many Sizes? A Visual Investigation — Mark Belan & Jordana Cepelewicz
    A visual explanation of countable and uncountable infinities.
  3. ‘Fantastic’ Proof Implies the Existence of In-Between Infinities — Natalie Wolchover
    New developments surrounding the continuum problem and possible intermediate sizes of infinity. 

Chapter 2 — Logic Versus Proof
4. In Math, Rigor Is Vital. But Are Digitized Proofs Taking It Too Far? — Leila Sloman
The history of mathematical rigor and the modern movement toward formalizing proofs with systems such as Lean.
5. How Writing Changes Mathematical Thought — John Pavlus
An interview exploring how mathematical notation influences the way mathematicians think.
6. The Jagged, Monstrous Function That Broke Calculus — Solomon Adams
How pathological functions challenged 19th-century intuition and forced mathematicians to make calculus more rigorous. 

Chapter 3 — Cut to the Core
7. Mathematicians Want to Banish Infinity. What Might They Gain? — Gregory Barber
An exploration of ultrafinitism, which questions whether infinite mathematical objects should exist at all.
8. Why Math’s Final Axiom Proved So Controversial — Gregory Barber
The difficult historical development of Zermelo-Fraenkel set theory and its axioms.
9. Gödel’s Incompleteness Proof, Explained — Natalie Wolchover
An accessible account of Gödel’s incompleteness theorems and the fundamental limits they place on formal mathematical systems.
10. How Infinity Leads to One of Math’s Strangest Paradoxes — Max G. Levy
The Banach-Tarski paradox and the counterintuitive consequences of infinity. 

Chapter 4 — A Revolution Begins
11. Two Researchers Are Rebuilding Mathematics From the Ground Up — Konstantin Kakaes
Peter Scholze and Dustin Clausen’s ambitious program to rethink fundamental concepts in topology and number theory.
12. How Alexander Grothendieck Revolutionized 20th-Century Mathematics — Konstantin Kakaes
An overview of Grothendieck’s transformative ideas in algebraic geometry.
13. Inside the Fight to Fix Geometry’s Foundations — Kevin Hartnett
A debate over whether fundamental arguments in modern geometry can truly be trusted and how they should be rebuilt.

Overall, the collection moves roughly from Cantor and infinity → rigor and formal proof → axioms and Gödel → Grothendieck and new foundations, making it essentially a compact history of how mathematicians have repeatedly reconsidered what mathematics is built from and what counts as a valid proof.

ARTICLES]]> “The Evolving Foundations of Math” examines how the basic assumptions underlying mathematics have repeatedly been questioned, repaired and rebuilt from the late 19th century to the present. Beginning with Cantor’s revolutionary work on infinity and the development of rigorous set theory, the series traces the tension between intuition, formal proof and logical consistency through figures such as Cantor, Zermelo, Gödel and Grothendieck. It explores controversial alternatives such as ultrafinitism, the paradoxes created by infinity, the growing use of computer-verified proofs, and modern attempts to reconstruct areas such as topology and geometry from more fundamental concepts. The central message is that mathematics is not built on an immutable foundation: its definitions, axioms, methods of proof and even ideas about what mathematical objects should exist continue to evolve as mathematicians encounter paradoxes, new technologies and deeper structural questions. 

Articles in the series

Chapter 1 — Original Sin
  1. Long-Lost Letters Reveal the Lies That Gave Rise To Modern Math — Joseph Howlett
    Cantor, the discovery of different sizes of infinity, and newly uncovered evidence concerning the origins of his famous 1874 result.
  2. How Can Infinity Come in Many Sizes? A Visual Investigation — Mark Belan & Jordana Cepelewicz
    A visual explanation of countable and uncountable infinities.
  3. ‘Fantastic’ Proof Implies the Existence of In-Between Infinities — Natalie Wolchover
    New developments surrounding the continuum problem and possible intermediate sizes of infinity. 

Chapter 2 — Logic Versus Proof
4. In Math, Rigor Is Vital. But Are Digitized Proofs Taking It Too Far? — Leila Sloman
The history of mathematical rigor and the modern movement toward formalizing proofs with systems such as Lean.
5. How Writing Changes Mathematical Thought — John Pavlus
An interview exploring how mathematical notation influences the way mathematicians think.
6. The Jagged, Monstrous Function That Broke Calculus — Solomon Adams
How pathological functions challenged 19th-century intuition and forced mathematicians to make calculus more rigorous. 

Chapter 3 — Cut to the Core
7. Mathematicians Want to Banish Infinity. What Might They Gain? — Gregory Barber
An exploration of ultrafinitism, which questions whether infinite mathematical objects should exist at all.
8. Why Math’s Final Axiom Proved So Controversial — Gregory Barber
The difficult historical development of Zermelo-Fraenkel set theory and its axioms.
9. Gödel’s Incompleteness Proof, Explained — Natalie Wolchover
An accessible account of Gödel’s incompleteness theorems and the fundamental limits they place on formal mathematical systems.
10. How Infinity Leads to One of Math’s Strangest Paradoxes — Max G. Levy
The Banach-Tarski paradox and the counterintuitive consequences of infinity. 

Chapter 4 — A Revolution Begins
11. Two Researchers Are Rebuilding Mathematics From the Ground Up — Konstantin Kakaes
Peter Scholze and Dustin Clausen’s ambitious program to rethink fundamental concepts in topology and number theory.
12. How Alexander Grothendieck Revolutionized 20th-Century Mathematics — Konstantin Kakaes
An overview of Grothendieck’s transformative ideas in algebraic geometry.
13. Inside the Fight to Fix Geometry’s Foundations — Kevin Hartnett
A debate over whether fundamental arguments in modern geometry can truly be trusted and how they should be rebuilt.

Overall, the collection moves roughly from Cantor and infinity → rigor and formal proof → axioms and Gödel → Grothendieck and new foundations, making it essentially a compact history of how mathematicians have repeatedly reconsidered what mathematics is built from and what counts as a valid proof.

ARTICLES]]> <![CDATA[Mathematical coincidence]]> https://mklab.gr/showthread.php?tid=1796 Thu, 03 Sep 2026 02:10:37 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1796
A mathematical coincidence occurs when two apparently unrelated mathematical expressions have values that are unexpectedly close, without an obvious theoretical reason. A simple example is
$2^{10}=1024\approx1000=10^3$.
Such coincidences often involve famous constants such as $\pi$, $e$, the golden ratio $\varphi$, or simple integers. Some arise because a rational number happens to approximate an irrational number extremely well—for example, $355/113$ approximates $\pi$ to six decimal places. Continued fractions can explain why some of these approximations are so accurate, although the deeper reason that unusually good approximations occur is not always clear. Other striking examples include $\pi^2\approx10$, $\pi^3\approx31$, and $2\pi+e\approx9$. 

Importantly, something that looks like a coincidence may eventually turn out to have a genuine mathematical explanation. A famous example is $e^\pi-\pi\approx20$, whose accuracy can be connected to identities involving the Jacobi theta function. Even more spectacular is Ramanujan's constant, $e^{\pi\sqrt{163}}$, which is extraordinarily close to an integer; this is not simply an accident but is connected with the special number-theoretic properties of the Heegner number $163$. The article also discusses coincidences in physics and measurement—for example, the speed of light being close to $3\times10^8$ m/s, Earth's gravitational acceleration being close to $10$ m/s², and the apparent sizes of the Sun and Moon being similar enough to permit total solar eclipses. Some of these depend partly on how humans defined measurement units, while others are genuinely accidental numerical relationships.

Key takeaways
  • Mathematical coincidences are unexpected near-equalities between seemingly unrelated quantities.
  • Some are genuinely accidental, while others hide deeper mathematics involving continued fractions, number theory, modular functions, or special constants.
  • Famous examples include $2^{10}\approx10^3$, $\pi^2\approx10$, $2\pi+e\approx9$, and $e^{\pi\sqrt{163}}\approx$ an integer.
  • Their main appeal is mathematical curiosity, but some coincidences also provide useful engineering approximations or occasionally lead mathematicians toward deeper structures. 

ARTICLE]]>
A mathematical coincidence occurs when two apparently unrelated mathematical expressions have values that are unexpectedly close, without an obvious theoretical reason. A simple example is
$2^{10}=1024\approx1000=10^3$.
Such coincidences often involve famous constants such as $\pi$, $e$, the golden ratio $\varphi$, or simple integers. Some arise because a rational number happens to approximate an irrational number extremely well—for example, $355/113$ approximates $\pi$ to six decimal places. Continued fractions can explain why some of these approximations are so accurate, although the deeper reason that unusually good approximations occur is not always clear. Other striking examples include $\pi^2\approx10$, $\pi^3\approx31$, and $2\pi+e\approx9$. 

Importantly, something that looks like a coincidence may eventually turn out to have a genuine mathematical explanation. A famous example is $e^\pi-\pi\approx20$, whose accuracy can be connected to identities involving the Jacobi theta function. Even more spectacular is Ramanujan's constant, $e^{\pi\sqrt{163}}$, which is extraordinarily close to an integer; this is not simply an accident but is connected with the special number-theoretic properties of the Heegner number $163$. The article also discusses coincidences in physics and measurement—for example, the speed of light being close to $3\times10^8$ m/s, Earth's gravitational acceleration being close to $10$ m/s², and the apparent sizes of the Sun and Moon being similar enough to permit total solar eclipses. Some of these depend partly on how humans defined measurement units, while others are genuinely accidental numerical relationships.

Key takeaways
  • Mathematical coincidences are unexpected near-equalities between seemingly unrelated quantities.
  • Some are genuinely accidental, while others hide deeper mathematics involving continued fractions, number theory, modular functions, or special constants.
  • Famous examples include $2^{10}\approx10^3$, $\pi^2\approx10$, $2\pi+e\approx9$, and $e^{\pi\sqrt{163}}\approx$ an integer.
  • Their main appeal is mathematical curiosity, but some coincidences also provide useful engineering approximations or occasionally lead mathematicians toward deeper structures. 

ARTICLE]]> <![CDATA[The Women Taking Math To The Next Dimension]]> https://mklab.gr/showthread.php?tid=1785 Wed, 02 Sep 2026 01:12:02 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1785 Author: Lauren J. Young
Published: November 16, 2017 — Science Friday 

The article profiles mathematicians Rebecca Goldin, Emily Riehl, and Eugenia Cheng, who challenge the common perception of mathematics as little more than calculations and formulas. They describe mathematics instead as a highly creative discipline driven by imagination, abstraction, problem solving, and the search for underlying structures. Goldin works on geometry and symmetry, Riehl studies higher-dimensional category theory, and Cheng works in category theory, which she describes as the “mathematics of mathematics.” Their examples range from the Fibonacci sequence and the Banach–Tarski paradox to non-Euclidean geometry, showing how mathematical ideas can lead far beyond what can be physically visualized. 

A recurring theme is that doing mathematics is much more social and exploratory than outsiders often imagine. Riehl emphasizes that research frequently involves collaboration and that asking questions is essential rather than a sign of weakness. Goldin says that she originally expected mathematics to become increasingly tedious, but instead discovered that advanced mathematics is playful, conceptual, and visually beautiful. Cheng similarly stresses that mathematics combines rigorous logic with creativity and imagination: because mathematicians are not restricted by physical reality, they can investigate structures of almost unlimited complexity. 

The three mathematicians also offer advice to young women considering mathematical careers. Goldin encourages them to pursue mathematics without feeling they must already have a career mapped out; Riehl urges students to ask questions freely; and Cheng argues that women do not need to imitate stereotypically male behavior in order to succeed. Overall, the article presents mathematics not as a cold collection of computations but as an imaginative, collaborative intellectual activity in which curiosity and creativity are as important as technical ability.

Key takeaways
  • Advanced mathematics is fundamentally about ideas, structures, patterns, and imagination, not merely calculation.
  • Mathematical research can be highly collaborative and social.
  • Understanding difficult mathematics often develops gradually over years rather than arriving instantly.
  • Asking questions is an important part of becoming a better mathematician.
  • The article argues against stereotypes about both what mathematics is and who can become a mathematician.

ARTICLE]]> Author: Lauren J. Young
Published: November 16, 2017 — Science Friday 

The article profiles mathematicians Rebecca Goldin, Emily Riehl, and Eugenia Cheng, who challenge the common perception of mathematics as little more than calculations and formulas. They describe mathematics instead as a highly creative discipline driven by imagination, abstraction, problem solving, and the search for underlying structures. Goldin works on geometry and symmetry, Riehl studies higher-dimensional category theory, and Cheng works in category theory, which she describes as the “mathematics of mathematics.” Their examples range from the Fibonacci sequence and the Banach–Tarski paradox to non-Euclidean geometry, showing how mathematical ideas can lead far beyond what can be physically visualized. 

A recurring theme is that doing mathematics is much more social and exploratory than outsiders often imagine. Riehl emphasizes that research frequently involves collaboration and that asking questions is essential rather than a sign of weakness. Goldin says that she originally expected mathematics to become increasingly tedious, but instead discovered that advanced mathematics is playful, conceptual, and visually beautiful. Cheng similarly stresses that mathematics combines rigorous logic with creativity and imagination: because mathematicians are not restricted by physical reality, they can investigate structures of almost unlimited complexity. 

The three mathematicians also offer advice to young women considering mathematical careers. Goldin encourages them to pursue mathematics without feeling they must already have a career mapped out; Riehl urges students to ask questions freely; and Cheng argues that women do not need to imitate stereotypically male behavior in order to succeed. Overall, the article presents mathematics not as a cold collection of computations but as an imaginative, collaborative intellectual activity in which curiosity and creativity are as important as technical ability.

Key takeaways
  • Advanced mathematics is fundamentally about ideas, structures, patterns, and imagination, not merely calculation.
  • Mathematical research can be highly collaborative and social.
  • Understanding difficult mathematics often develops gradually over years rather than arriving instantly.
  • Asking questions is an important part of becoming a better mathematician.
  • The article argues against stereotypes about both what mathematics is and who can become a mathematician.

ARTICLE]]> <![CDATA[Solving the Flat Cube]]> https://mklab.gr/showthread.php?tid=1740 Fri, 28 Aug 2026 18:30:53 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1740 Published: August 19, 2026

James Propp introduces the Flat Cube, a planar puzzle inspired by the Rubik’s Cube. Instead of small cubes, it consists of lozenges (rhombi) tiling a hexagon. Whenever three lozenges form a small hexagon, they may be rotated around its center in multiples of $60^\circ$. The goal is to restore a scrambled configuration using the fewest possible twists. This leads to an analogue of the Rubik’s Cube’s “God’s number”: the maximum, over all possible scrambled states, of the minimum number of moves required to solve the puzzle. Propp initially proves that this number must be at least 27. 

He gives two elegant visual proofs. The first adds a dimension: a lozenge tiling can be interpreted as the visible surface of a pile of unit cubes inside a $3\times3\times3$ box. A legal twist corresponds to adding or removing exactly one cube. Transforming the empty configuration into the completely filled one therefore requires at least
33=273^3=27
moves. The second proof removes a dimension: considering only the nine horizontally oriented lozenges turns them into beads sliding along one-dimensional tracks. Each of the $9$ beads must move $3$ positions, while one twist moves only one bead by one position, giving again
9×3=27.9\times3=27.
For the uncolored version, this reasoning can in fact be pushed further to show that the worst-case distance is exactly $27$. 

The article also connects the puzzle with the mathematics of lozenge tilings, plane partitions, combinatorics, and configuration spaces. Propp collaborated with puzzle designer Oskar van Deventer and Dmitry Andreev to turn the mathematical idea into an actual mechanical puzzle, called the Propp Twist. Particularly interesting are updates added two days after publication: Tom Rokicki showed that the order-$2$ Flat Cube has God’s number exactly $27$, while two arbitrary states can be as far as $30$ moves apart. More strikingly, Rui Viana subsequently proved that for the order-$3$ Flat Cube, God’s number is at least $81$, and Rokicki found a pair of states separated by 91 moves. Thus the problem is substantially richer than the original $27$-move argument suggests. 

Key takeaways
  • The Flat Cube converts a geometric tiling problem into a Rubik-like optimization problem.
  • A lozenge rotation can be interpreted either as adding/removing a cube or as moving a bead one step.
  • Both viewpoints give the beautiful lower-bound calculation $27=3^3=9\times3$.
  • The later results show that the full colored order-$3$ puzzle is considerably harder: its God’s number is already known to be at least $81$, with some pairs of states 91 moves apart. 

ARTICLE [PDF]]]> Published: August 19, 2026

James Propp introduces the Flat Cube, a planar puzzle inspired by the Rubik’s Cube. Instead of small cubes, it consists of lozenges (rhombi) tiling a hexagon. Whenever three lozenges form a small hexagon, they may be rotated around its center in multiples of $60^\circ$. The goal is to restore a scrambled configuration using the fewest possible twists. This leads to an analogue of the Rubik’s Cube’s “God’s number”: the maximum, over all possible scrambled states, of the minimum number of moves required to solve the puzzle. Propp initially proves that this number must be at least 27. 

He gives two elegant visual proofs. The first adds a dimension: a lozenge tiling can be interpreted as the visible surface of a pile of unit cubes inside a $3\times3\times3$ box. A legal twist corresponds to adding or removing exactly one cube. Transforming the empty configuration into the completely filled one therefore requires at least
33=273^3=27
moves. The second proof removes a dimension: considering only the nine horizontally oriented lozenges turns them into beads sliding along one-dimensional tracks. Each of the $9$ beads must move $3$ positions, while one twist moves only one bead by one position, giving again
9×3=27.9\times3=27.
For the uncolored version, this reasoning can in fact be pushed further to show that the worst-case distance is exactly $27$. 

The article also connects the puzzle with the mathematics of lozenge tilings, plane partitions, combinatorics, and configuration spaces. Propp collaborated with puzzle designer Oskar van Deventer and Dmitry Andreev to turn the mathematical idea into an actual mechanical puzzle, called the Propp Twist. Particularly interesting are updates added two days after publication: Tom Rokicki showed that the order-$2$ Flat Cube has God’s number exactly $27$, while two arbitrary states can be as far as $30$ moves apart. More strikingly, Rui Viana subsequently proved that for the order-$3$ Flat Cube, God’s number is at least $81$, and Rokicki found a pair of states separated by 91 moves. Thus the problem is substantially richer than the original $27$-move argument suggests. 

Key takeaways
  • The Flat Cube converts a geometric tiling problem into a Rubik-like optimization problem.
  • A lozenge rotation can be interpreted either as adding/removing a cube or as moving a bead one step.
  • Both viewpoints give the beautiful lower-bound calculation $27=3^3=9\times3$.
  • The later results show that the full colored order-$3$ puzzle is considerably harder: its God’s number is already known to be at least $81$, with some pairs of states 91 moves apart. 

ARTICLE [PDF]]]> <![CDATA[The Math Ceiling]]> https://mklab.gr/showthread.php?tid=1713 Thu, 20 Aug 2026 19:44:30 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1713 The Math Ceiling: Where’s Your Cognitive Breaking Point?
Author: Ben Orlin
Published: April 8, 2015
Source:Math with Bad Drawings 

Ben Orlin examines the common belief that every person has a natural “mathematical ceiling”—a point beyond which their cognitive ability simply cannot carry them. He contrasts two attitudes: the traditional view that some people are inherently “math people” while others are not, and the more optimistic educational view that essentially anyone can learn mathematics given sufficient effort. Orlin proposes a third explanation: many apparent mathematical ceilings may not be biological limits at all, but the accumulated consequences of gaps in earlier understanding.

He illustrates the idea with what he calls the “Law of the Broken Futon.” A futon with one small structural piece missing may initially appear perfectly usable, but as increasing weight is placed upon it, the unsupported parts gradually deform until the whole structure collapses. Mathematical learning can work the same way. A student may successfully manipulate linear equations, for example, without understanding that a graph represents all ordered pairs satisfying an equation. Procedures and memorized rules can compensate temporarily, but later topics—quadratics, trigonometry, calculus—place greater conceptual weight on that missing foundation. By then, repairing the gap is harder because the learner has constructed years of shortcuts around it. 

Orlin therefore argues that what students experience as their “cognitive breaking point” may often be educationally created rather than intellectually predetermined. Teaching students algorithms without understanding can allow them to pass exams and advance through the curriculum, while silently weakening the structure on which later mathematics depends. His central warning is especially striking: a learner who repeatedly gets correct answers without knowing why the methods work may appear successful today but is being prepared for difficulty tomorrow. 

Key takeaways
  • Mathematical difficulty is cumulative: small conceptual gaps can become major obstacles years later.
  • Procedural competence is not the same as understanding. Being able to execute an algorithm can conceal serious weaknesses.
  • Many supposed “math ceilings” may actually be consequences of earlier teaching and learning, rather than fixed limits of intelligence.
  • The educational priority should therefore be deep conceptual understanding before progression, even when this means moving more slowly through the curriculum. 

ARTICLE]]> The Math Ceiling: Where’s Your Cognitive Breaking Point?
Author: Ben Orlin
Published: April 8, 2015
Source:Math with Bad Drawings 

Ben Orlin examines the common belief that every person has a natural “mathematical ceiling”—a point beyond which their cognitive ability simply cannot carry them. He contrasts two attitudes: the traditional view that some people are inherently “math people” while others are not, and the more optimistic educational view that essentially anyone can learn mathematics given sufficient effort. Orlin proposes a third explanation: many apparent mathematical ceilings may not be biological limits at all, but the accumulated consequences of gaps in earlier understanding.

He illustrates the idea with what he calls the “Law of the Broken Futon.” A futon with one small structural piece missing may initially appear perfectly usable, but as increasing weight is placed upon it, the unsupported parts gradually deform until the whole structure collapses. Mathematical learning can work the same way. A student may successfully manipulate linear equations, for example, without understanding that a graph represents all ordered pairs satisfying an equation. Procedures and memorized rules can compensate temporarily, but later topics—quadratics, trigonometry, calculus—place greater conceptual weight on that missing foundation. By then, repairing the gap is harder because the learner has constructed years of shortcuts around it. 

Orlin therefore argues that what students experience as their “cognitive breaking point” may often be educationally created rather than intellectually predetermined. Teaching students algorithms without understanding can allow them to pass exams and advance through the curriculum, while silently weakening the structure on which later mathematics depends. His central warning is especially striking: a learner who repeatedly gets correct answers without knowing why the methods work may appear successful today but is being prepared for difficulty tomorrow. 

Key takeaways
  • Mathematical difficulty is cumulative: small conceptual gaps can become major obstacles years later.
  • Procedural competence is not the same as understanding. Being able to execute an algorithm can conceal serious weaknesses.
  • Many supposed “math ceilings” may actually be consequences of earlier teaching and learning, rather than fixed limits of intelligence.
  • The educational priority should therefore be deep conceptual understanding before progression, even when this means moving more slowly through the curriculum. 

ARTICLE]]> <![CDATA[The Reasonable Ineffectiveness of Mathematics]]> https://mklab.gr/showthread.php?tid=1708 Thu, 20 Aug 2026 19:01:53 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1708 Author: Derek Abbott
Published: October 2013
Journal:Proceedings of the IEEE, Vol. 101, No. 10, pp. 2147–2153
DOI: 10.1109/JPROC.2013.2274907 

Derek Abbott challenges Eugene Wigner’s famous claim about the “unreasonable effectiveness of mathematics” in describing nature. Rather than regarding the success of mathematics in physics as something mysterious or as evidence that mathematical objects have an independent Platonic existence, Abbott argues that mathematics is largely a human invention constructed to represent patterns and regularities that humans happen to observe. We naturally notice problems that our mathematical tools can solve, develop new mathematics when existing tools fail, and tend to remember successful mathematical models while overlooking the enormous number of unsuccessful ones. Thus, what appears to be a miraculous correspondence between mathematics and nature may partly be a form of selection bias.

Abbott develops Richard Hamming's earlier observations: we see what we are equipped to look for; we choose the mathematics appropriate to particular problems; science successfully addresses only a comparatively small subset of conceivable questions; and human cognition itself was shaped by evolution to understand phenomena occurring on approximately human spatial and temporal scales. Abbott adds two ideas of his own. First, mathematical laws function as a kind of lossy compression of reality: equations discard noise and complexity in order to give the human mind compact, usable descriptions. Second, scientific models undergo something like Darwinian selection—successful models survive and are published, while thousands of unsuccessful ideas disappear, making mathematics appear more consistently successful than it actually is.

He illustrates this with engineering, nonlinear systems, fractals, complex numbers and even the apparently elementary act of counting bananas. In the physical world, objects have fuzzy boundaries, measurements contain noise, and there are ultimately physical limits to how many objects can actually be counted or represented. Mathematical entities such as perfect circles, integers extending indefinitely, delta functions and exact real numbers therefore need not correspond to independently existing objects in nature. Abbott adopts a deliberately strong non-Platonist position: mathematical models should be regarded primarily as useful constructions rather than literal descriptions of what reality “really is.” This view has practical consequences, he argues, because treating mathematics as something humans are free to redesign may encourage better formalisms—for example, his advocacy of geometric algebra as a more unified alternative to the traditional mixture of scalars, vectors, complex numbers, quaternions, dot products and cross products.

Key takeaways
  • Mathematics works, but not miraculously: its successes may look extraordinary because we disproportionately notice successful applications.
  • Mathematical models are approximations: Abbott describes them as compressed, idealized representations of a noisy and complicated reality.
  • Mathematics may be anthropocentric: our mathematics reflects human perception, evolution, cognitive limitations and the scales at which we live.
  • The article argues against mathematical Platonism: Abbott concludes that mathematics is best viewed as a powerful human-created tool for describing regularities, not necessarily as a pre-existing structure of the universe.

ARTICLE [PDF]]]> Author: Derek Abbott
Published: October 2013
Journal:Proceedings of the IEEE, Vol. 101, No. 10, pp. 2147–2153
DOI: 10.1109/JPROC.2013.2274907 

Derek Abbott challenges Eugene Wigner’s famous claim about the “unreasonable effectiveness of mathematics” in describing nature. Rather than regarding the success of mathematics in physics as something mysterious or as evidence that mathematical objects have an independent Platonic existence, Abbott argues that mathematics is largely a human invention constructed to represent patterns and regularities that humans happen to observe. We naturally notice problems that our mathematical tools can solve, develop new mathematics when existing tools fail, and tend to remember successful mathematical models while overlooking the enormous number of unsuccessful ones. Thus, what appears to be a miraculous correspondence between mathematics and nature may partly be a form of selection bias.

Abbott develops Richard Hamming's earlier observations: we see what we are equipped to look for; we choose the mathematics appropriate to particular problems; science successfully addresses only a comparatively small subset of conceivable questions; and human cognition itself was shaped by evolution to understand phenomena occurring on approximately human spatial and temporal scales. Abbott adds two ideas of his own. First, mathematical laws function as a kind of lossy compression of reality: equations discard noise and complexity in order to give the human mind compact, usable descriptions. Second, scientific models undergo something like Darwinian selection—successful models survive and are published, while thousands of unsuccessful ideas disappear, making mathematics appear more consistently successful than it actually is.

He illustrates this with engineering, nonlinear systems, fractals, complex numbers and even the apparently elementary act of counting bananas. In the physical world, objects have fuzzy boundaries, measurements contain noise, and there are ultimately physical limits to how many objects can actually be counted or represented. Mathematical entities such as perfect circles, integers extending indefinitely, delta functions and exact real numbers therefore need not correspond to independently existing objects in nature. Abbott adopts a deliberately strong non-Platonist position: mathematical models should be regarded primarily as useful constructions rather than literal descriptions of what reality “really is.” This view has practical consequences, he argues, because treating mathematics as something humans are free to redesign may encourage better formalisms—for example, his advocacy of geometric algebra as a more unified alternative to the traditional mixture of scalars, vectors, complex numbers, quaternions, dot products and cross products.

Key takeaways
  • Mathematics works, but not miraculously: its successes may look extraordinary because we disproportionately notice successful applications.
  • Mathematical models are approximations: Abbott describes them as compressed, idealized representations of a noisy and complicated reality.
  • Mathematics may be anthropocentric: our mathematics reflects human perception, evolution, cognitive limitations and the scales at which we live.
  • The article argues against mathematical Platonism: Abbott concludes that mathematics is best viewed as a powerful human-created tool for describing regularities, not necessarily as a pre-existing structure of the universe.

ARTICLE [PDF]]]> <![CDATA[Can π generate itself?]]> https://mklab.gr/showthread.php?tid=1689 Mon, 17 Aug 2026 21:01:16 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1689
Summary

The paper “Can $\pi$ Generate Itself? A Monte Carlo Analysis of 314 Trillion Digits” by Alessandro Razeto and Nicola Rossi investigates a fascinating self-referential question: can the decimal digits of $\pi$ themselves be treated as pseudorandom data and used to estimate $\pi$? Using the record dataset of 314 trillion digits of $\pi$ computed by the end of 2025, the authors convert sequences of these digits into numerical samples and apply the standard Monte Carlo method for estimating the area of a circle. 

Their statistical analysis finds that the available digits display a remarkably high degree of randomness, providing empirical support for random-like behavior beyond simple tests of digit frequencies, while emphasizing that true independence cannot hold because $\pi$ is a deterministic number. After optimizing how the digits are mapped into Monte Carlo points, the experiment successfully reconstructs $\pi$ to approximately $\pi \approx 3.141593$. 

The result does not prove that $\pi$ is normal or that its digits are genuinely random; rather, it provides an unusually large-scale empirical demonstration that the known digits behave sufficiently like random numbers for $\pi$ to be used, in a playful sense, to numerically “generate itself.” (arxiv.org)


Read the paper on arXiv]]>
Summary

The paper “Can $\pi$ Generate Itself? A Monte Carlo Analysis of 314 Trillion Digits” by Alessandro Razeto and Nicola Rossi investigates a fascinating self-referential question: can the decimal digits of $\pi$ themselves be treated as pseudorandom data and used to estimate $\pi$? Using the record dataset of 314 trillion digits of $\pi$ computed by the end of 2025, the authors convert sequences of these digits into numerical samples and apply the standard Monte Carlo method for estimating the area of a circle. 

Their statistical analysis finds that the available digits display a remarkably high degree of randomness, providing empirical support for random-like behavior beyond simple tests of digit frequencies, while emphasizing that true independence cannot hold because $\pi$ is a deterministic number. After optimizing how the digits are mapped into Monte Carlo points, the experiment successfully reconstructs $\pi$ to approximately $\pi \approx 3.141593$. 

The result does not prove that $\pi$ is normal or that its digits are genuinely random; rather, it provides an unusually large-scale empirical demonstration that the known digits behave sufficiently like random numbers for $\pi$ to be used, in a playful sense, to numerically “generate itself.” (arxiv.org)


Read the paper on arXiv]]> <![CDATA[The Fall of English Literature [Christodoulou]]]> https://mklab.gr/showthread.php?tid=1591 Sat, 15 Aug 2026 14:19:51 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1591 The Fall of English Literature 
by [Daisy Christodoulou]

Daisy Christodoulou examines the striking decline of English Literature as an academic subject alongside the rise of mathematics. Using A-level trends as an example, she notes that maths and English Literature have effectively exchanged positions in popularity. Rather than blaming changes to England’s curriculum, she argues that two broader forces are more convincing explanations: economics and declining reading habits. Students increasingly view education as an investment and therefore gravitate toward subjects such as mathematics that appear to provide stronger employment and financial returns. At the same time, recreational reading has weakened, with smartphones and social media competing for both time and sustained attention.

Importantly, Christodoulou does not present the rise of mathematics as something regrettable. She argues that mathematics has earned its popularity: statistical and quantitative literacy are increasingly essential in modern life, while mathematics also possesses intellectual beauty comparable to that of great literature. But she makes the reverse argument for literature. Literature is not merely culturally beautiful; reading and writing are extraordinarily useful technologies for communicating and, more fundamentally, for thinking. Writing allows people to develop ideas beyond the limits of working memory. This leads to one of the article's most interesting observations about AI: if we routinely delegate writing to artificial intelligence, we risk delegating part of the thinking process itself.

Christodoulou therefore rejects the idea that humanities education should respond to declining attention spans simply by replacing long books with extracts, videos and audio. Instead, she proposes emphasizing the value of demanding, sustained reading precisely because it is difficult. Students have demonstrated that they are willing to tackle challenging subjects such as mathematics and physics when they understand their value; the humanities must similarly demonstrate both their utility and beauty. Nevertheless, she remains pessimistic, suggesting that English Literature could eventually become culturally marginal in Britain, much as Classics has become—a particularly significant change given the historic role of writers such as Shakespeare and Dickens in British cultural life.
 
Key takeaways
  • English Literature is declining while mathematics is rising, reflecting broader changes in education and student priorities.
  • Declining reading and digital distraction may be as important as curriculum design in explaining the humanities' difficulties.
  • Writing is a tool for thinking, making excessive outsourcing of writing to AI potentially intellectually costly.
  •  The answer may not be to make literature easier, but to show students why difficult, sustained reading is valuable—both practically and culturally. 

ARTICLE]]> The Fall of English Literature 
by [Daisy Christodoulou]

Daisy Christodoulou examines the striking decline of English Literature as an academic subject alongside the rise of mathematics. Using A-level trends as an example, she notes that maths and English Literature have effectively exchanged positions in popularity. Rather than blaming changes to England’s curriculum, she argues that two broader forces are more convincing explanations: economics and declining reading habits. Students increasingly view education as an investment and therefore gravitate toward subjects such as mathematics that appear to provide stronger employment and financial returns. At the same time, recreational reading has weakened, with smartphones and social media competing for both time and sustained attention.

Importantly, Christodoulou does not present the rise of mathematics as something regrettable. She argues that mathematics has earned its popularity: statistical and quantitative literacy are increasingly essential in modern life, while mathematics also possesses intellectual beauty comparable to that of great literature. But she makes the reverse argument for literature. Literature is not merely culturally beautiful; reading and writing are extraordinarily useful technologies for communicating and, more fundamentally, for thinking. Writing allows people to develop ideas beyond the limits of working memory. This leads to one of the article's most interesting observations about AI: if we routinely delegate writing to artificial intelligence, we risk delegating part of the thinking process itself.

Christodoulou therefore rejects the idea that humanities education should respond to declining attention spans simply by replacing long books with extracts, videos and audio. Instead, she proposes emphasizing the value of demanding, sustained reading precisely because it is difficult. Students have demonstrated that they are willing to tackle challenging subjects such as mathematics and physics when they understand their value; the humanities must similarly demonstrate both their utility and beauty. Nevertheless, she remains pessimistic, suggesting that English Literature could eventually become culturally marginal in Britain, much as Classics has become—a particularly significant change given the historic role of writers such as Shakespeare and Dickens in British cultural life.
 
Key takeaways
  • English Literature is declining while mathematics is rising, reflecting broader changes in education and student priorities.
  • Declining reading and digital distraction may be as important as curriculum design in explaining the humanities' difficulties.
  • Writing is a tool for thinking, making excessive outsourcing of writing to AI potentially intellectually costly.
  •  The answer may not be to make literature easier, but to show students why difficult, sustained reading is valuable—both practically and culturally. 

ARTICLE]]> <![CDATA[Why Human Giants Defy the Laws of Physics]]> https://mklab.gr/showthread.php?tid=1584 Thu, 13 Aug 2026 19:47:00 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1584 Why Human Giants Defy the Laws of Physics

Summary

The article explains Galileo’s “square-cube law” and why truly giant humans could not survive with ordinary human proportions: as an organism grows, its weight (and volume) increases with the cube of its size, while the strength of its bones and muscles increases only roughly with the square, because strength depends on cross-sectional area. Thus, a human scaled up tenfold would weigh about 1,000 times as much but have bones only about 100 times stronger, eventually causing the skeleton to collapse under its own weight. 

The same principle explains why large animals such as elephants and dinosaurs evolved proportionally thicker limbs, while very small animals can be remarkably strong relative to their size and experience forces such as surface tension and electrostatic attraction more strongly. The article concludes that physics places fundamental limits on biological size and helps explain why organisms must change their proportions as they evolve toward larger or smaller scales.


ARTICLE / ARCHIVE]]> Why Human Giants Defy the Laws of Physics

Summary

The article explains Galileo’s “square-cube law” and why truly giant humans could not survive with ordinary human proportions: as an organism grows, its weight (and volume) increases with the cube of its size, while the strength of its bones and muscles increases only roughly with the square, because strength depends on cross-sectional area. Thus, a human scaled up tenfold would weigh about 1,000 times as much but have bones only about 100 times stronger, eventually causing the skeleton to collapse under its own weight. 

The same principle explains why large animals such as elephants and dinosaurs evolved proportionally thicker limbs, while very small animals can be remarkably strong relative to their size and experience forces such as surface tension and electrostatic attraction more strongly. The article concludes that physics places fundamental limits on biological size and helps explain why organisms must change their proportions as they evolve toward larger or smaller scales.


ARTICLE / ARCHIVE]]> <![CDATA[Elevators]]> https://mklab.gr/showthread.php?tid=1577 Wed, 12 Aug 2026 17:23:04 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1577 Elevators

Summary

The article “Elevators” explores the surprisingly complex algorithms used to control elevator systems and minimize passenger waiting times. It begins with simple strategies such as SCAN and LOOK, then examines how multiple elevators are coordinated and how performance can be measured using wait-time distributions such as the median (p50) and 90th percentile (p90). 

It shows that traffic patterns—especially morning rush hour—strongly affect performance, and introduces smarter systems such as Otis’s Relative System Response (RSR) algorithm, which continually reassigns elevators based on factors like estimated arrival time, passenger load, direction, and avoiding elevator bunching. 

Interestingly, the simulations show that sophisticated algorithms do not always win: LOOK can outperform RSR at high traffic levels or in smaller buildings, while destination-dispatch systems can sometimes perform worse because they sacrifice flexibility. Overall, the article demonstrates through interactive simulations that elevator optimization is a trade-off between complexity, flexibility, traffic patterns, and simplicity.

ARTICLE]]> Elevators

Summary

The article “Elevators” explores the surprisingly complex algorithms used to control elevator systems and minimize passenger waiting times. It begins with simple strategies such as SCAN and LOOK, then examines how multiple elevators are coordinated and how performance can be measured using wait-time distributions such as the median (p50) and 90th percentile (p90). 

It shows that traffic patterns—especially morning rush hour—strongly affect performance, and introduces smarter systems such as Otis’s Relative System Response (RSR) algorithm, which continually reassigns elevators based on factors like estimated arrival time, passenger load, direction, and avoiding elevator bunching. 

Interestingly, the simulations show that sophisticated algorithms do not always win: LOOK can outperform RSR at high traffic levels or in smaller buildings, while destination-dispatch systems can sometimes perform worse because they sacrifice flexibility. Overall, the article demonstrates through interactive simulations that elevator optimization is a trade-off between complexity, flexibility, traffic patterns, and simplicity.

ARTICLE]]> <![CDATA[The mathematics of a total solar eclipse]]> https://mklab.gr/showthread.php?tid=1576 Wed, 12 Aug 2026 17:15:19 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1576 The mathematics of a total solar eclipse

Summary

This Royal Astronomical Society worksheet, “The Mathematics of a Total Solar Eclipse,” uses the geometry of solar eclipses to teach mathematical concepts to Years 10–11 students. It explains how the Moon can completely cover the Sun because their apparent sizes are similar, introducing similar triangles, proportionality, and scale calculations. 

Students first solve problems involving missing sides, then use real measurements of the Earth, Moon, and Sun to estimate the Earth–Sun distance, the diameters of the Sun and Earth, and the size of the Sun’s image in a pinhole camera. The worksheet also demonstrates how mathematical models based on rounded data produce approximate results and compares them with actual astronomical measurements.


ARTICLE [PDF]]]> The mathematics of a total solar eclipse

Summary

This Royal Astronomical Society worksheet, “The Mathematics of a Total Solar Eclipse,” uses the geometry of solar eclipses to teach mathematical concepts to Years 10–11 students. It explains how the Moon can completely cover the Sun because their apparent sizes are similar, introducing similar triangles, proportionality, and scale calculations. 

Students first solve problems involving missing sides, then use real measurements of the Earth, Moon, and Sun to estimate the Earth–Sun distance, the diameters of the Sun and Earth, and the size of the Sun’s image in a pinhole camera. The worksheet also demonstrates how mathematical models based on rounded data produce approximate results and compares them with actual astronomical measurements.


ARTICLE [PDF]]]> <![CDATA[Rubiks cube and graph theory]]> https://mklab.gr/showthread.php?tid=1536 Sun, 09 Aug 2026 11:22:39 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1536 The demonstration highlights how graph theory models the Rubik's Cube's 43 quintillion positions as a Cayley graph to study transitions and shortest solution paths.

]]> The demonstration highlights how graph theory models the Rubik's Cube's 43 quintillion positions as a Cayley graph to study transitions and shortest solution paths.

]]>