https://mklab.gr/ Thu, 29 Oct 2026 06:15:14 +0000 MyBB https://mklab.gr/showthread.php?tid=2368 Thu, 29 Oct 2026 01:22:17 +0200 mklabgr]]> https://mklab.gr/showthread.php?tid=2368 Ptolemy’s Equation and Kin

The article by Katie Waddle explores a remarkable mathematical relationship connecting several areas of geometry and algebra through a common three-term equation: AB + CD = EF. The first example is Ptolemy’s theorem, which states that in a cyclic quadrilateral, the product of the lengths of the two diagonals equals the sum of the products of the lengths of the two pairs of opposite sides. The article shows that similar relationships appear in other mathematical settings, revealing unexpected connections between seemingly different geometric ideas.

The author examines three additional versions of this relationship: Casey’s theorem, concerning common tangents to circles; Penner’s lambda lengths, which arise in hyperbolic geometry; and Plücker’s identity, involving determinants of matrices and the geometry of higher-dimensional spaces. Although these results were developed in different historical periods and mathematical contexts, the article demonstrates that they can be transformed into one another using suitable changes of scale. This provides a unified perspective on results that might otherwise appear unrelated.

A central theme is that the same mathematical structure can emerge in different forms across distinct branches of mathematics. The article also explores connections with cross-ratios, projective geometry, and different models of the hyperbolic plane. By establishing links between classical geometry and modern algebraic concepts, Waddle illustrates how a familiar theorem can serve as a gateway to deeper mathematical ideas. The broader lesson is that recognizing common structures and relationships is one of the most powerful ways to discover connections and develop a unified understanding of mathematics.


ARTICLE]]> Ptolemy’s Equation and Kin

The article by Katie Waddle explores a remarkable mathematical relationship connecting several areas of geometry and algebra through a common three-term equation: AB + CD = EF. The first example is Ptolemy’s theorem, which states that in a cyclic quadrilateral, the product of the lengths of the two diagonals equals the sum of the products of the lengths of the two pairs of opposite sides. The article shows that similar relationships appear in other mathematical settings, revealing unexpected connections between seemingly different geometric ideas.

The author examines three additional versions of this relationship: Casey’s theorem, concerning common tangents to circles; Penner’s lambda lengths, which arise in hyperbolic geometry; and Plücker’s identity, involving determinants of matrices and the geometry of higher-dimensional spaces. Although these results were developed in different historical periods and mathematical contexts, the article demonstrates that they can be transformed into one another using suitable changes of scale. This provides a unified perspective on results that might otherwise appear unrelated.

A central theme is that the same mathematical structure can emerge in different forms across distinct branches of mathematics. The article also explores connections with cross-ratios, projective geometry, and different models of the hyperbolic plane. By establishing links between classical geometry and modern algebraic concepts, Waddle illustrates how a familiar theorem can serve as a gateway to deeper mathematical ideas. The broader lesson is that recognizing common structures and relationships is one of the most powerful ways to discover connections and develop a unified understanding of mathematics.


ARTICLE]]> https://mklab.gr/showthread.php?tid=2334 Thu, 22 Oct 2026 22:53:47 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2334 Source: Microsoft Research Blog | Publication date: October 21, 2026 |
Summary 

The Kannan–Lovász–Simonovits (KLS) conjecture, proposed in 1995, is a fundamental open problem in high-dimensional geometry, probability theory, and theoretical computer science. It concerns the geometry of high-dimensional probability distributions, specifically whether a simple, flat cut through a geometric object can identify its narrowest bottlenecks almost as effectively as the best possible cut of any shape. The conjecture predicts that the difference between these two approaches can be bounded by a universal constant independent of the number of dimensions. Resolving this question is important because the geometry of bottlenecks influences how efficiently random walks explore probability distributions, which is central to many sampling algorithms.


According to Microsoft Research, Xinzhi Zhang and his collaborator Zhao Song proved the conjecture by developing new mathematical ideas with assistance from AI agents. Their approach involved exploring more than 100 possible research directions, refining promising mathematical arguments, and discovering a recursive method that repeatedly improved the relevant bounds. The crucial breakthrough was proving that the errors introduced during these repeated refinements remain bounded by a universal constant, eliminating the remaining dependence on dimensionality. Earlier work had achieved bounds that still grew, albeit very slowly, with the number of dimensions. The final proof, posted on October 4, 2026, removed this dependence entirely.


The article highlights the importance of human–AI collaboration in mathematical research. Rather than solving the problem through a single, autonomous AI attempt, researchers directed multiple AI agents, assessed their intermediate results, discarded unproductive approaches, and developed the promising ideas into a rigorous proof. Independent researchers subsequently produced alternative proofs, and a formalization using Lean 4, a proof-assistant system, provided additional verification. These developments strengthen confidence in the result, although the article notes that further work is underway to simplify the proof and complete its formal verification. The resolution removes a longstanding theoretical obstacle and strengthens the foundations of research in convex geometry, probability, statistical computation, and randomized algorithms.


Key Takeaways

• A 31-year-old problem resolved: The KLS conjecture, formulated in 1995, has reportedly been proved, addressing a central challenge in high-dimensional geometry.
  
• A dimension-independent bound: The result establishes that the relationship between the best flat cut and the optimal cut can be bounded by a universal constant, regardless of dimensionality.
  
• AI assisted mathematical discovery: AI agents explored numerous potential approaches and supported calculations, derivations, manuscript preparation, and internal proof checks.
  
• Human mathematical judgment remained essential: Researchers guided the exploration, selected promising ideas, identified the key recursive strategy, and assembled the final argument.
  
• Independent verification matters: Alternative proofs and Lean 4 formalization provide complementary evidence supporting the result.
  
• Potential computational impact: The proof strengthens theoretical foundations for high-dimensional sampling, probability, optimization, and randomized algorithms.
  
• A model for future research: The work illustrates how AI can contribute to difficult mathematical problems when combined with structured exploration, expert guidance, and rigorous verification.
  

Overall conclusion: The reported resolution of the KLS conjecture is significant both for mathematics itself and for the developing role of AI in mathematical research. Its most important lesson is that AI-assisted discovery can be productive when researchers retain responsibility for mathematical reasoning, proof construction, and verification.

ARTICLE]]> Source: Microsoft Research Blog | Publication date: October 21, 2026 |
Summary 

The Kannan–Lovász–Simonovits (KLS) conjecture, proposed in 1995, is a fundamental open problem in high-dimensional geometry, probability theory, and theoretical computer science. It concerns the geometry of high-dimensional probability distributions, specifically whether a simple, flat cut through a geometric object can identify its narrowest bottlenecks almost as effectively as the best possible cut of any shape. The conjecture predicts that the difference between these two approaches can be bounded by a universal constant independent of the number of dimensions. Resolving this question is important because the geometry of bottlenecks influences how efficiently random walks explore probability distributions, which is central to many sampling algorithms.


According to Microsoft Research, Xinzhi Zhang and his collaborator Zhao Song proved the conjecture by developing new mathematical ideas with assistance from AI agents. Their approach involved exploring more than 100 possible research directions, refining promising mathematical arguments, and discovering a recursive method that repeatedly improved the relevant bounds. The crucial breakthrough was proving that the errors introduced during these repeated refinements remain bounded by a universal constant, eliminating the remaining dependence on dimensionality. Earlier work had achieved bounds that still grew, albeit very slowly, with the number of dimensions. The final proof, posted on October 4, 2026, removed this dependence entirely.


The article highlights the importance of human–AI collaboration in mathematical research. Rather than solving the problem through a single, autonomous AI attempt, researchers directed multiple AI agents, assessed their intermediate results, discarded unproductive approaches, and developed the promising ideas into a rigorous proof. Independent researchers subsequently produced alternative proofs, and a formalization using Lean 4, a proof-assistant system, provided additional verification. These developments strengthen confidence in the result, although the article notes that further work is underway to simplify the proof and complete its formal verification. The resolution removes a longstanding theoretical obstacle and strengthens the foundations of research in convex geometry, probability, statistical computation, and randomized algorithms.


Key Takeaways

• A 31-year-old problem resolved: The KLS conjecture, formulated in 1995, has reportedly been proved, addressing a central challenge in high-dimensional geometry.
  
• A dimension-independent bound: The result establishes that the relationship between the best flat cut and the optimal cut can be bounded by a universal constant, regardless of dimensionality.
  
• AI assisted mathematical discovery: AI agents explored numerous potential approaches and supported calculations, derivations, manuscript preparation, and internal proof checks.
  
• Human mathematical judgment remained essential: Researchers guided the exploration, selected promising ideas, identified the key recursive strategy, and assembled the final argument.
  
• Independent verification matters: Alternative proofs and Lean 4 formalization provide complementary evidence supporting the result.
  
• Potential computational impact: The proof strengthens theoretical foundations for high-dimensional sampling, probability, optimization, and randomized algorithms.
  
• A model for future research: The work illustrates how AI can contribute to difficult mathematical problems when combined with structured exploration, expert guidance, and rigorous verification.
  

Overall conclusion: The reported resolution of the KLS conjecture is significant both for mathematics itself and for the developing role of AI in mathematical research. Its most important lesson is that AI-assisted discovery can be productive when researchers retain responsibility for mathematical reasoning, proof construction, and verification.

ARTICLE]]> https://mklab.gr/showthread.php?tid=2317 Thu, 01 Oct 2026 20:50:24 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2317 Betti numbers are important invariants in algebraic topology that describe the structure of a topological space by measuring the number of independent “holes” of different dimensions. The $n$-th Betti number, denoted $b_n$, is the rank of the $n$-th homology group $H_n(X)$. Informally, $b_0$ counts connected components, $b_1$ counts one-dimensional or circular holes, and $b_2$ counts two-dimensional cavities. For example, a torus has $b_0=1$, $b_1=2$, and $b_2=1$, corresponding to one connected component, two independent circular holes, and one enclosed cavity.

Formally, the Betti number is defined by $b_k(X)=\operatorname{rank} H_k(X)$, where $H_k(X)=\ker\delta_k/\operatorname{Im}\delta_{k+1}$ is the $k$-th homology group. Equivalently, it can be viewed as the dimension of $H_k(X;\mathbb{Q})$ as a vector space over the rational numbers. Importantly, Betti numbers measure the free part of homology and therefore do not detect finite torsion. For instance, the projective plane has $H_1\cong\mathbb{Z}_2$, so its first Betti number is $b_1=0$, even though its first homology group is nontrivial. 

The Betti numbers can be collected into the Poincaré polynomial $P_X(x)=b_0+b_1x+b_2x^2+\cdots$, which provides a compact description of a space's topology. They are related to other fundamental invariants: for a finite CW complex, the Euler characteristic is $\chi(X)=\sum_i(-1)^i b_i$, while under suitable conditions on an $n$-dimensional manifold, Poincaré duality gives $b_k=b_{n-k}$. Betti numbers also appear in differential geometry through de Rham and Hodge theory, and they have applications beyond pure mathematics, including computer science, digital image analysis, and topological data analysis.

ARTICLE]]> Betti numbers are important invariants in algebraic topology that describe the structure of a topological space by measuring the number of independent “holes” of different dimensions. The $n$-th Betti number, denoted $b_n$, is the rank of the $n$-th homology group $H_n(X)$. Informally, $b_0$ counts connected components, $b_1$ counts one-dimensional or circular holes, and $b_2$ counts two-dimensional cavities. For example, a torus has $b_0=1$, $b_1=2$, and $b_2=1$, corresponding to one connected component, two independent circular holes, and one enclosed cavity.

Formally, the Betti number is defined by $b_k(X)=\operatorname{rank} H_k(X)$, where $H_k(X)=\ker\delta_k/\operatorname{Im}\delta_{k+1}$ is the $k$-th homology group. Equivalently, it can be viewed as the dimension of $H_k(X;\mathbb{Q})$ as a vector space over the rational numbers. Importantly, Betti numbers measure the free part of homology and therefore do not detect finite torsion. For instance, the projective plane has $H_1\cong\mathbb{Z}_2$, so its first Betti number is $b_1=0$, even though its first homology group is nontrivial. 

The Betti numbers can be collected into the Poincaré polynomial $P_X(x)=b_0+b_1x+b_2x^2+\cdots$, which provides a compact description of a space's topology. They are related to other fundamental invariants: for a finite CW complex, the Euler characteristic is $\chi(X)=\sum_i(-1)^i b_i$, while under suitable conditions on an $n$-dimensional manifold, Poincaré duality gives $b_k=b_{n-k}$. Betti numbers also appear in differential geometry through de Rham and Hodge theory, and they have applications beyond pure mathematics, including computer science, digital image analysis, and topological data analysis.

ARTICLE]]> https://mklab.gr/showthread.php?tid=2199 Fri, 25 Sep 2026 21:29:24 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2199 space, shape, size, distance, position, and relationships between figures. It is one of the oldest areas of mathematics, originating from practical problems involving surveying, construction, astronomy, and measurement. Classical Euclidean geometry studies fundamental objects such as points, lines, planes, angles, curves, surfaces, and solids, while concepts such as congruence, similarity, symmetry, and dimension provide ways to compare and classify geometric objects.

The foundations of geometry were developed systematically by the ancient Greeks, particularly through Euclid's Elements. Euclid organized geometry using definitions, axioms, theorems, and proofs, establishing the axiomatic method that remains central to mathematics. Important classical results include the Pythagorean theorem, $a^2+b^2=c^2$, relationships involving circles and triangles, and methods for calculating areas and volumes. Later mathematicians such as Archimedes developed sophisticated approaches to areas, volumes, curves, and $\pi$. 

From the 17th century onward, geometry became increasingly connected with algebra and analysis. Analytic geometry, developed by Descartes and Fermat, represents geometric objects through coordinates and equations—for example, a circle can be written as $(x-h)^2+(y-k)^2=r^2$. The 19th century produced an even more profound transformation with non-Euclidean geometry, differential geometry, projective geometry, and the study of curvature. Gauss's work showed that some geometric properties can be understood intrinsically, without reference to the surrounding Euclidean space.

Modern geometry is therefore much broader than the geometry of school textbooks. It includes differential geometry, algebraic geometry, topology, computational geometry, discrete geometry, projective geometry, affine geometry, convex geometry, and geometric group theory. Geometry now provides essential mathematical tools for physics, architecture, computer graphics, biology, medicine, engineering, and other branches of mathematics. A striking example is general relativity, where spacetime is described using geometric ideas, while algebraic geometry has also played an important role in major number-theoretic results such as the proof of Fermat's Last Theorem.

Key Takeaways

• Geometry studies space and form: shape, size, distance, position, angle, area, and volume.
  
• Euclidean geometry is based on axioms and logical deduction.
  
• The Pythagorean theorem is expressed as $a^2+b^2=c^2$.
  
• Analytic geometry connects geometry with algebra through coordinates and equations.
  
• A circle can be represented by $(x-h)^2+(y-k)^2=r^2$.
  
• Non-Euclidean geometry demonstrates that geometries different from Euclidean geometry can be mathematically consistent.
  
• Differential geometry studies curves, surfaces, curvature, and higher-dimensional spaces.
  
• Topology focuses on properties that remain unchanged under continuous deformation.
  
• Algebraic geometry connects geometric objects with algebraic equations.
  
• Computational and discrete geometry are important in computer science, algorithms, graphics, and optimization.
  
• Geometry has applications ranging from architecture and art to physics, medicine, engineering, and mathematics itself.
  
• The modern concept of a geometric space can extend far beyond ordinary three-dimensional physical space.
  

Source: Wikipedia — Geometry / PDF]]> space, shape, size, distance, position, and relationships between figures. It is one of the oldest areas of mathematics, originating from practical problems involving surveying, construction, astronomy, and measurement. Classical Euclidean geometry studies fundamental objects such as points, lines, planes, angles, curves, surfaces, and solids, while concepts such as congruence, similarity, symmetry, and dimension provide ways to compare and classify geometric objects.

The foundations of geometry were developed systematically by the ancient Greeks, particularly through Euclid's Elements. Euclid organized geometry using definitions, axioms, theorems, and proofs, establishing the axiomatic method that remains central to mathematics. Important classical results include the Pythagorean theorem, $a^2+b^2=c^2$, relationships involving circles and triangles, and methods for calculating areas and volumes. Later mathematicians such as Archimedes developed sophisticated approaches to areas, volumes, curves, and $\pi$. 

From the 17th century onward, geometry became increasingly connected with algebra and analysis. Analytic geometry, developed by Descartes and Fermat, represents geometric objects through coordinates and equations—for example, a circle can be written as $(x-h)^2+(y-k)^2=r^2$. The 19th century produced an even more profound transformation with non-Euclidean geometry, differential geometry, projective geometry, and the study of curvature. Gauss's work showed that some geometric properties can be understood intrinsically, without reference to the surrounding Euclidean space.

Modern geometry is therefore much broader than the geometry of school textbooks. It includes differential geometry, algebraic geometry, topology, computational geometry, discrete geometry, projective geometry, affine geometry, convex geometry, and geometric group theory. Geometry now provides essential mathematical tools for physics, architecture, computer graphics, biology, medicine, engineering, and other branches of mathematics. A striking example is general relativity, where spacetime is described using geometric ideas, while algebraic geometry has also played an important role in major number-theoretic results such as the proof of Fermat's Last Theorem.

Key Takeaways

• Geometry studies space and form: shape, size, distance, position, angle, area, and volume.
  
• Euclidean geometry is based on axioms and logical deduction.
  
• The Pythagorean theorem is expressed as $a^2+b^2=c^2$.
  
• Analytic geometry connects geometry with algebra through coordinates and equations.
  
• A circle can be represented by $(x-h)^2+(y-k)^2=r^2$.
  
• Non-Euclidean geometry demonstrates that geometries different from Euclidean geometry can be mathematically consistent.
  
• Differential geometry studies curves, surfaces, curvature, and higher-dimensional spaces.
  
• Topology focuses on properties that remain unchanged under continuous deformation.
  
• Algebraic geometry connects geometric objects with algebraic equations.
  
• Computational and discrete geometry are important in computer science, algorithms, graphics, and optimization.
  
• Geometry has applications ranging from architecture and art to physics, medicine, engineering, and mathematics itself.
  
• The modern concept of a geometric space can extend far beyond ordinary three-dimensional physical space.
  

Source: Wikipedia — Geometry / PDF]]> https://mklab.gr/showthread.php?tid=2188 Fri, 25 Sep 2026 20:44:34 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2188 Moira Dillon proposes that the foundations of human geometric thinking may be much older and more evolutionarily widespread than traditionally assumed. Rather than originating from a uniquely human, innate “geometry module,” our basic geometric abilities may have developed from cognitive systems used for navigation and spatial reasoning. Research involving humans and animals—including rats, chickens, and fish—shows that non-human animals can represent distance, direction, location, and spatial relationships without formal knowledge of geometry.

The proposal, called the “Wanderers Hypothesis for Geometry,” challenges theories that place Euclidean concepts such as parallelism and perpendicularity directly into the human mind from birth. According to Dillon's analysis, navigation provides a more fundamental form of geometry: organisms construct approximate representations of space that allow them to move efficiently and plan routes. This geometry does not reproduce Euclidean mathematics exactly, but it captures enough spatial structure to support practical reasoning about position, distance, direction, and shape.

The distinctive human contribution, according to Dillon, may therefore be language rather than a specialized mathematical faculty. Animals can use spatial representations to navigate, while humans can describe, manipulate, and recombine those representations symbolically. Language effectively allows humans to “wander” mentally without physically moving, transforming navigation-based spatial cognition into more abstract reasoning. This could help explain how humans progress from practical spatial experience to formal concepts such as geometric figures, relations, proofs, and mathematical problem-solving.

The analysis brings together evidence from developmental psychology, comparative cognition, cross-cultural studies of geometry learning, animal navigation, and artificial intelligence—including systems such as Google DeepMind's AlphaGeometry. It suggests an evolutionary pathway from embodied spatial behavior to increasingly abstract mathematical thought: navigation provides primitive geometric representations, human cognition reorganizes them, and language enables symbolic manipulation. The paper, “The cognitive origins of geometry,” was published in Trends in Cognitive Sciences on April 6, 2026, and is a peer-reviewed publication.

Key takeaways

• Geometry may have evolutionary roots: basic spatial reasoning appears to be shared with other animals rather than being exclusively human.
  
• Navigation may precede formal geometry: organisms can reason about distance, direction, and spatial relationships without knowing mathematical definitions.
  
• The Wanderers Hypothesis proposes that geometric cognition evolved from navigation-like mental processes.
  
• Euclidean geometry may be an abstraction: practical spatial cognition approximates mathematical geometry rather than implementing it exactly.
  
• Language may be the crucial human advantage: language lets humans manipulate spatial representations without physically navigating.
  
• From movement to mathematics: an evolutionary chain can be envisioned as
  navigation→spatial representation→language→abstract geometry.\text{navigation} \rightarrow \text{spatial representation} \rightarrow \text{language} \rightarrow \text{abstract geometry}.
  
• Educational implication: learning geometry might be connected to intuitive experiences involving movement, maps, shapes, distance, and spatial exploration—not only symbolic exercises.
  
• AI connection: the analysis also raises questions about whether systems such as AlphaGeometry acquire geometric reasoning through mechanisms analogous to human spatial abstraction.
  
• Central conceptual shift: instead of asking “Why are humans born knowing geometry?”, the research asks how evolutionary spatial cognition could have been transformed into formal mathematical reasoning.
  

ARTICLE]]> Moira Dillon proposes that the foundations of human geometric thinking may be much older and more evolutionarily widespread than traditionally assumed. Rather than originating from a uniquely human, innate “geometry module,” our basic geometric abilities may have developed from cognitive systems used for navigation and spatial reasoning. Research involving humans and animals—including rats, chickens, and fish—shows that non-human animals can represent distance, direction, location, and spatial relationships without formal knowledge of geometry.

The proposal, called the “Wanderers Hypothesis for Geometry,” challenges theories that place Euclidean concepts such as parallelism and perpendicularity directly into the human mind from birth. According to Dillon's analysis, navigation provides a more fundamental form of geometry: organisms construct approximate representations of space that allow them to move efficiently and plan routes. This geometry does not reproduce Euclidean mathematics exactly, but it captures enough spatial structure to support practical reasoning about position, distance, direction, and shape.

The distinctive human contribution, according to Dillon, may therefore be language rather than a specialized mathematical faculty. Animals can use spatial representations to navigate, while humans can describe, manipulate, and recombine those representations symbolically. Language effectively allows humans to “wander” mentally without physically moving, transforming navigation-based spatial cognition into more abstract reasoning. This could help explain how humans progress from practical spatial experience to formal concepts such as geometric figures, relations, proofs, and mathematical problem-solving.

The analysis brings together evidence from developmental psychology, comparative cognition, cross-cultural studies of geometry learning, animal navigation, and artificial intelligence—including systems such as Google DeepMind's AlphaGeometry. It suggests an evolutionary pathway from embodied spatial behavior to increasingly abstract mathematical thought: navigation provides primitive geometric representations, human cognition reorganizes them, and language enables symbolic manipulation. The paper, “The cognitive origins of geometry,” was published in Trends in Cognitive Sciences on April 6, 2026, and is a peer-reviewed publication.

Key takeaways

• Geometry may have evolutionary roots: basic spatial reasoning appears to be shared with other animals rather than being exclusively human.
  
• Navigation may precede formal geometry: organisms can reason about distance, direction, and spatial relationships without knowing mathematical definitions.
  
• The Wanderers Hypothesis proposes that geometric cognition evolved from navigation-like mental processes.
  
• Euclidean geometry may be an abstraction: practical spatial cognition approximates mathematical geometry rather than implementing it exactly.
  
• Language may be the crucial human advantage: language lets humans manipulate spatial representations without physically navigating.
  
• From movement to mathematics: an evolutionary chain can be envisioned as
  navigation→spatial representation→language→abstract geometry.\text{navigation} \rightarrow \text{spatial representation} \rightarrow \text{language} \rightarrow \text{abstract geometry}.
  
• Educational implication: learning geometry might be connected to intuitive experiences involving movement, maps, shapes, distance, and spatial exploration—not only symbolic exercises.
  
• AI connection: the analysis also raises questions about whether systems such as AlphaGeometry acquire geometric reasoning through mechanisms analogous to human spatial abstraction.
  
• Central conceptual shift: instead of asking “Why are humans born knowing geometry?”, the research asks how evolutionary spatial cognition could have been transformed into formal mathematical reasoning.
  

ARTICLE]]> https://mklab.gr/showthread.php?tid=2183 Fri, 25 Sep 2026 20:22:21 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2183 compact Bonnet pair. The result resolves a problem that had remained open for decades in classical differential geometry. The researchers constructed two globally different, closed torus-shaped surfaces that nevertheless have exactly the same intrinsic metric and the same mean curvature.

The underlying question goes back to the French mathematician Pierre Ossian Bonnet. A traditional principle suggested that if one knows the metric of a surface—essentially, all information about distances measured along it—and its mean curvature at every point, then the surface should be uniquely determined. In mathematical terms, the metric describes the intrinsic geometry, while the mean curvature describes how the surface bends in three-dimensional space. The new result demonstrates that these data can, in certain compact cases, fail to determine a unique global shape. 

The distinction between local and global geometry is central. The two tori have the same intrinsic distances and the same curvature information, so measurements made locally cannot distinguish them. Yet their global structures are different: they represent two distinct embeddings of essentially the same geometric data. Previously, counterexamples were known for non-compact surfaces, but for compact surfaces the situation was much more restrictive. In particular, for tori mathematicians knew that a given metric and mean curvature could correspond to at most two surfaces—but nobody had actually exhibited such a pair.

The work, by Alexander I. Bobenko, Tim Hoffmann and Alexander O. Sageman-Furnas, was published as Compact Bonnet Pairs: Isometric Tori with the Same Curvatures in Publications Mathématiques de l'IHÉS (2025). The result was subsequently recognized with the 2026 Frontiers of Science Award. Conceptually, the discovery is important because it shows that even very detailed local geometric information—metric plus curvature—may not encode the complete global geometry of a closed surface.

Key mathematical ideas

• Isometric surfaces: two surfaces have the same intrinsic geometry if their metrics agree; schematically, $g_1=g_2$.
  
• Mean curvature: for principal curvatures $k_1$ and $k_2$, the mean curvature is $H=\frac{k_1+k_2}{2}$.
  
• Bonnet-type uniqueness: the classical expectation was that $(g,H)$ should determine the surface, subject to appropriate conditions.
  
• Bonnet pair: two distinct surfaces sharing the same metric and mean curvature.
  
• Compact Bonnet pair: the new example consists of two closed tori, rather than surfaces extending infinitely or having boundaries.
  
• The key phenomenon is that local geometric data do not always determine global shape.
  
• For tori, the theory allowed at most two possibilities for a fixed metric and mean curvature; the breakthrough was finding an explicit pair realizing this possibility.
  
• The result is a striking example of the difference between intrinsic geometry and extrinsic geometry: two surfaces can look globally different while sharing important local geometric measurements.
  
• Field: differential geometry, especially the geometry of surfaces, with connections to topology and geometric analysis.
  

ARTICLE]]> compact Bonnet pair. The result resolves a problem that had remained open for decades in classical differential geometry. The researchers constructed two globally different, closed torus-shaped surfaces that nevertheless have exactly the same intrinsic metric and the same mean curvature.

The underlying question goes back to the French mathematician Pierre Ossian Bonnet. A traditional principle suggested that if one knows the metric of a surface—essentially, all information about distances measured along it—and its mean curvature at every point, then the surface should be uniquely determined. In mathematical terms, the metric describes the intrinsic geometry, while the mean curvature describes how the surface bends in three-dimensional space. The new result demonstrates that these data can, in certain compact cases, fail to determine a unique global shape. 

The distinction between local and global geometry is central. The two tori have the same intrinsic distances and the same curvature information, so measurements made locally cannot distinguish them. Yet their global structures are different: they represent two distinct embeddings of essentially the same geometric data. Previously, counterexamples were known for non-compact surfaces, but for compact surfaces the situation was much more restrictive. In particular, for tori mathematicians knew that a given metric and mean curvature could correspond to at most two surfaces—but nobody had actually exhibited such a pair.

The work, by Alexander I. Bobenko, Tim Hoffmann and Alexander O. Sageman-Furnas, was published as Compact Bonnet Pairs: Isometric Tori with the Same Curvatures in Publications Mathématiques de l'IHÉS (2025). The result was subsequently recognized with the 2026 Frontiers of Science Award. Conceptually, the discovery is important because it shows that even very detailed local geometric information—metric plus curvature—may not encode the complete global geometry of a closed surface.

Key mathematical ideas

• Isometric surfaces: two surfaces have the same intrinsic geometry if their metrics agree; schematically, $g_1=g_2$.
  
• Mean curvature: for principal curvatures $k_1$ and $k_2$, the mean curvature is $H=\frac{k_1+k_2}{2}$.
  
• Bonnet-type uniqueness: the classical expectation was that $(g,H)$ should determine the surface, subject to appropriate conditions.
  
• Bonnet pair: two distinct surfaces sharing the same metric and mean curvature.
  
• Compact Bonnet pair: the new example consists of two closed tori, rather than surfaces extending infinitely or having boundaries.
  
• The key phenomenon is that local geometric data do not always determine global shape.
  
• For tori, the theory allowed at most two possibilities for a fixed metric and mean curvature; the breakthrough was finding an explicit pair realizing this possibility.
  
• The result is a striking example of the difference between intrinsic geometry and extrinsic geometry: two surfaces can look globally different while sharing important local geometric measurements.
  
• Field: differential geometry, especially the geometry of surfaces, with connections to topology and geometric analysis.
  

ARTICLE]]> https://mklab.gr/showthread.php?tid=2178 Fri, 25 Sep 2026 19:53:36 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2178 The Freie Universität Berlin study “Beauty in/of Mathematics: Tessellations and Their Formulas” by Heinrich Begehr and Dajiang Wang explores how the visual beauty of tiling patterns is connected to sophisticated mathematical analysis. A tessellation covers a surface with geometric shapes without gaps or overlaps, but the researchers show that such patterns can also provide a practical framework for solving mathematical problems. Their work combines complex analysis, partial differential equations, and geometric function theory.

A central idea is the parqueting-reflection principle, in which a geometric region is repeatedly reflected across its edges to produce an entire tessellation. This is closely related to the classical reflection methods associated with Hermann Amandus Schwarz. The resulting symmetry is not merely decorative: it can be exploited to extend mathematical functions from one region to the surrounding reflected regions. In this way, geometric symmetry becomes an analytical tool.

One important application is the construction of kernel functions used in boundary-value problems. In particular, the method can produce Green, Neumann, and Schwarz kernels, which are fundamental in solving equations arising in mathematical physics and engineering. For example, a boundary-value problem such as the Dirichlet problem seeks a function $u$ satisfying an equation such as $\Delta u=0$ inside a domain together with prescribed boundary values $u|_{\partial\Omega}=f$. Tessellation and reflection can provide explicit representations for such solutions.

The research also extends these ideas beyond ordinary Euclidean geometry into hyperbolic geometry. The authors discuss Schweikart triangles, special hyperbolic triangles that can tile a circular disc and generate striking visual patterns. Such constructions connect mathematical aesthetics with computer graphics, architecture, mathematical physics, and visualization. The broader message is that mathematical beauty can reflect genuine structural properties: symmetry, repetition, geometry, and analytical efficiency can reinforce one another.

Key takeaways

• Tessellation is more than visual decoration: it can become a method for solving mathematical problems.
  
• The parqueting-reflection principle repeatedly reflects a geometric region to construct a complete tiling.
  
• Reflection symmetry can help construct solutions to boundary-value problems for differential equations.
  
• Important analytical objects such as Green, Neumann, and Schwarz kernels can be derived using these geometric constructions.
  
• A typical harmonic problem is $\Delta u=0$, together with boundary data such as $u|_{\partial\Omega}=f$.
  
• The work connects geometry, complex analysis, PDEs, and geometric function theory.
  
• The method works in both Euclidean and hyperbolic geometry.
  
• Schweikart triangles provide particularly striking examples of hyperbolic tessellation.
  
• Hyperbolic tessellations can be visualized inside a disk, producing infinitely intricate patterns.
  
• The mathematics has potential relevance to mathematical physics, engineering, architecture, and computer graphics.
  
• The research reinforces the idea that symmetry and aesthetics can correspond to deep mathematical structure, rather than beauty being merely subjective.
  
• The study was published in Applicable Analysis in 2025.
  

Freie Universität Berlin — “When Mathematics Meets Aesthetics”]]> The Freie Universität Berlin study “Beauty in/of Mathematics: Tessellations and Their Formulas” by Heinrich Begehr and Dajiang Wang explores how the visual beauty of tiling patterns is connected to sophisticated mathematical analysis. A tessellation covers a surface with geometric shapes without gaps or overlaps, but the researchers show that such patterns can also provide a practical framework for solving mathematical problems. Their work combines complex analysis, partial differential equations, and geometric function theory.

A central idea is the parqueting-reflection principle, in which a geometric region is repeatedly reflected across its edges to produce an entire tessellation. This is closely related to the classical reflection methods associated with Hermann Amandus Schwarz. The resulting symmetry is not merely decorative: it can be exploited to extend mathematical functions from one region to the surrounding reflected regions. In this way, geometric symmetry becomes an analytical tool.

One important application is the construction of kernel functions used in boundary-value problems. In particular, the method can produce Green, Neumann, and Schwarz kernels, which are fundamental in solving equations arising in mathematical physics and engineering. For example, a boundary-value problem such as the Dirichlet problem seeks a function $u$ satisfying an equation such as $\Delta u=0$ inside a domain together with prescribed boundary values $u|_{\partial\Omega}=f$. Tessellation and reflection can provide explicit representations for such solutions.

The research also extends these ideas beyond ordinary Euclidean geometry into hyperbolic geometry. The authors discuss Schweikart triangles, special hyperbolic triangles that can tile a circular disc and generate striking visual patterns. Such constructions connect mathematical aesthetics with computer graphics, architecture, mathematical physics, and visualization. The broader message is that mathematical beauty can reflect genuine structural properties: symmetry, repetition, geometry, and analytical efficiency can reinforce one another.

Key takeaways

• Tessellation is more than visual decoration: it can become a method for solving mathematical problems.
  
• The parqueting-reflection principle repeatedly reflects a geometric region to construct a complete tiling.
  
• Reflection symmetry can help construct solutions to boundary-value problems for differential equations.
  
• Important analytical objects such as Green, Neumann, and Schwarz kernels can be derived using these geometric constructions.
  
• A typical harmonic problem is $\Delta u=0$, together with boundary data such as $u|_{\partial\Omega}=f$.
  
• The work connects geometry, complex analysis, PDEs, and geometric function theory.
  
• The method works in both Euclidean and hyperbolic geometry.
  
• Schweikart triangles provide particularly striking examples of hyperbolic tessellation.
  
• Hyperbolic tessellations can be visualized inside a disk, producing infinitely intricate patterns.
  
• The mathematics has potential relevance to mathematical physics, engineering, architecture, and computer graphics.
  
• The research reinforces the idea that symmetry and aesthetics can correspond to deep mathematical structure, rather than beauty being merely subjective.
  
• The study was published in Applicable Analysis in 2025.
  

Freie Universität Berlin — “When Mathematics Meets Aesthetics”]]> https://mklab.gr/showthread.php?tid=2169 Fri, 25 Sep 2026 16:44:19 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2169 “An aperiodic monotile” by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss addresses a famous problem in tiling theory: whether there exists a single tile that can cover the entire plane, but only in non-periodic ways. Such a tile is called an aperiodic monotile, or informally an Einstein—from the German ein Stein, “one stone.” The paper presents a solution using a family of polygonal tiles, with the best-known representative being the “hat”. The current version is v3, revised in July 2024.

The central construction begins with the hat tile, a polygon assembled from eight kites. The authors show that hats can combine into larger structures called metatiles. These metatiles themselves obey substitution rules: a large-scale copy can be decomposed into smaller copies according to a fixed hierarchical pattern. Because these substitution structures can tile the entire plane, the authors establish that the hat itself can tile the plane. In symbolic form, the basic idea is a substitution hierarchy such as $T \rightarrow {T_1,T_2,\ldots,T_n}$, where increasingly large configurations are recursively constructed from smaller ones.

The much harder part is proving aperiodicity. A tiling is periodic if there exists a nonzero translation vector $v$ such that translating the entire tiling by $v$ leaves it unchanged. The authors demonstrate that the hat cannot admit such a periodic arrangement. Their argument combines geometric ideas with incommensurability: the geometric parameters of generic members of the tile family prevent the structures from repeating through a common translation period. They also provide a separate computer-assisted combinatorial proof showing that every hat tiling must possess a hierarchical structure. This hierarchy forces increasingly large-scale organization and therefore rules out periodicity.

An especially important result is that the hat is not an isolated curiosity. The authors construct a continuum of combinatorially equivalent aperiodic polygons. In other words, there is a continuous family of geometrically different shapes that share the same essential combinatorial structure and retain the aperiodic property for generic parameter choices. The work therefore does more than identify one exotic shape: it develops a framework connecting combinatorics, geometry, substitution systems, and tiling theory. The result resolves the long-standing monotile problem for topological-disk tiles and provides new mathematical tools for understanding how local geometric constraints can force global non-periodic order.

Key takeaways

• The problem: Find one tile that tiles the plane but never periodically.
  
• The solution: The authors exhibit the Hat, an aperiodic monotile.
  
• Tiling: The Hat can cover the plane without gaps or overlaps.
  
• Aperiodicity: No Hat tiling can possess a translational period $v\neq0$.
  
• Hierarchical structure: Hat tiles organize into larger metatiles, which recursively organize into still larger structures.
  
• Substitution: The construction can be represented schematically by recursive rules such as $T\mapsto{T_1,\ldots,T_n}$.
  
• Geometric argument: Incommensurability prevents the relevant structures from repeating periodically.
  
• Computer-assisted proof: A finite combinatorial analysis verifies constraints that force hierarchical, and hence aperiodic, tilings.
  
• Broader result: There is a continuum of geometrically distinct aperiodic polygons, not merely one special tile.
  
• Mathematical significance: The paper connects discrete geometry, combinatorics, tilings, substitution systems, and computational proof.
  
• Educational connection: It provides an excellent example of how simple local rules can impose highly nontrivial global behavior—a useful theme for teaching symmetry, recursion, transformations, geometry, and mathematical proof.
  

Read the full paper on arXiv]]> “An aperiodic monotile” by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss addresses a famous problem in tiling theory: whether there exists a single tile that can cover the entire plane, but only in non-periodic ways. Such a tile is called an aperiodic monotile, or informally an Einstein—from the German ein Stein, “one stone.” The paper presents a solution using a family of polygonal tiles, with the best-known representative being the “hat”. The current version is v3, revised in July 2024.

The central construction begins with the hat tile, a polygon assembled from eight kites. The authors show that hats can combine into larger structures called metatiles. These metatiles themselves obey substitution rules: a large-scale copy can be decomposed into smaller copies according to a fixed hierarchical pattern. Because these substitution structures can tile the entire plane, the authors establish that the hat itself can tile the plane. In symbolic form, the basic idea is a substitution hierarchy such as $T \rightarrow {T_1,T_2,\ldots,T_n}$, where increasingly large configurations are recursively constructed from smaller ones.

The much harder part is proving aperiodicity. A tiling is periodic if there exists a nonzero translation vector $v$ such that translating the entire tiling by $v$ leaves it unchanged. The authors demonstrate that the hat cannot admit such a periodic arrangement. Their argument combines geometric ideas with incommensurability: the geometric parameters of generic members of the tile family prevent the structures from repeating through a common translation period. They also provide a separate computer-assisted combinatorial proof showing that every hat tiling must possess a hierarchical structure. This hierarchy forces increasingly large-scale organization and therefore rules out periodicity.

An especially important result is that the hat is not an isolated curiosity. The authors construct a continuum of combinatorially equivalent aperiodic polygons. In other words, there is a continuous family of geometrically different shapes that share the same essential combinatorial structure and retain the aperiodic property for generic parameter choices. The work therefore does more than identify one exotic shape: it develops a framework connecting combinatorics, geometry, substitution systems, and tiling theory. The result resolves the long-standing monotile problem for topological-disk tiles and provides new mathematical tools for understanding how local geometric constraints can force global non-periodic order.

Key takeaways

• The problem: Find one tile that tiles the plane but never periodically.
  
• The solution: The authors exhibit the Hat, an aperiodic monotile.
  
• Tiling: The Hat can cover the plane without gaps or overlaps.
  
• Aperiodicity: No Hat tiling can possess a translational period $v\neq0$.
  
• Hierarchical structure: Hat tiles organize into larger metatiles, which recursively organize into still larger structures.
  
• Substitution: The construction can be represented schematically by recursive rules such as $T\mapsto{T_1,\ldots,T_n}$.
  
• Geometric argument: Incommensurability prevents the relevant structures from repeating periodically.
  
• Computer-assisted proof: A finite combinatorial analysis verifies constraints that force hierarchical, and hence aperiodic, tilings.
  
• Broader result: There is a continuum of geometrically distinct aperiodic polygons, not merely one special tile.
  
• Mathematical significance: The paper connects discrete geometry, combinatorics, tilings, substitution systems, and computational proof.
  
• Educational connection: It provides an excellent example of how simple local rules can impose highly nontrivial global behavior—a useful theme for teaching symmetry, recursion, transformations, geometry, and mathematical proof.
  

Read the full paper on arXiv]]> https://mklab.gr/showthread.php?tid=2168 Fri, 25 Sep 2026 16:39:18 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2168 “A Strongly Aperiodic Monotile in Three Dimensions” by Ioannis Tsiokos, submitted to arXiv in September 2026, presents a remarkable solution to a long-standing tiling problem in three-dimensional space. The author constructs a single three-dimensional shape, called Chair44 or $Q$, that can tile all of $\mathbb{R}^3$ using congruent copies, but cannot produce any tiling with an infinite-order symmetry. In other words, the shape necessarily produces non-periodic arrangements. This addresses a particularly demanding version of the three-dimensional Einstein problem: finding one shape that forces aperiodicity purely through its geometry.

The tile itself is geometrically quite interesting. It begins with a seven-cube chair: a $2\times2\times2$ cube with one corner cube removed. Its 24 exposed unit-square panels are modified with very small square-pyramid protrusions and indentations. These features act like a physical encoding of matching rules: neighboring tiles can only fit together in certain configurations. The resulting object is a rational polyhedral $3$-ball, meaning that it is topologically equivalent to a solid ball and its vertices have rational coordinates. Importantly, the construction does not impose external matching rules—the restrictions arise entirely from the shape itself.

The central mechanism is a hierarchical substitution structure. Eight appropriately rotated copies of the chair form a chair twice as large, and this process can be repeated indefinitely. Thus each individual tile belongs to a uniquely determined sequence of increasingly large “supertiles” with scales $2,4,8,\ldots,2^n$. If a translation were a symmetry of the tiling, it would have to remain a symmetry after passing to every larger scale. But the hierarchy forces the translation vector to be repeatedly halved; consequently the only possible translation is $0$. Formally, the authors establish $\operatorname{Per}(T)={0}$ and $|\operatorname{Sym}(T)|\leq24$ for every tiling $T$.

A particularly notable aspect is the extensive computer-assisted and formally verified proof. The authors enumerate possible configurations exactly rather than relying on numerical approximations. For example, $2{,}388$ candidate contact poses are reduced to exactly $44$ legal contacts, while other stages involve thousands of possible configurations and hundreds of thousands of collision checks. These computations were independently replayed in two Python implementations, while the finite statements and logical assembly were checked in Lean 4, using exact integers and rational arithmetic rather than floating-point calculations. The paper is explicitly presented as a proof submission pending refereeing, so its result should currently be regarded as a new preprint rather than an established peer-reviewed theorem.

Key takeaways

• Main result: A single three-dimensional polyhedral tile $Q$ is shown to tile $\mathbb{R}^3$ while forcing aperiodicity.
  
• The tile is called Chair44 (R44) and is based on a seven-cube chair.
  
• Its small geometric bumps and dents encode matching constraints through shape alone.
  
• Every tiling has no nonzero translational period:
  Per⁡(T)={0}.\operatorname{Per}(T)=\{0\}.
  
• Every tiling has a finite symmetry group:
  ∣Sym⁡(T)∣≤24.|\operatorname{Sym}(T)|\leq24.
  
• Every tiling is homochiral: all copies have the same handedness.
  
• Each tile belongs to a unique hierarchy of supertiles with scales
  20,21,22,….2^0,2^1,2^2,\ldots.
  
• The proof's key idea is recognizability + hierarchical substitution + period halving.
  
• The finite contact structure contains exactly 44 legal contacts after eliminating $2{,}388$ candidates.
  
• The computational part uses exact arithmetic, avoiding floating-point uncertainty.
  
• The finite enumerations and logical proof assembly are formally checked in Lean 4, modulo explicitly identified compiler hooks.
  
• The paper connects modern aperiodic tilings, combinatorics, discrete geometry, substitution systems, and formal proof.
  
• The work is currently a September 2026 arXiv proof submission, not yet a peer-reviewed publication. 
  

Read the full paper on arXiv]]> “A Strongly Aperiodic Monotile in Three Dimensions” by Ioannis Tsiokos, submitted to arXiv in September 2026, presents a remarkable solution to a long-standing tiling problem in three-dimensional space. The author constructs a single three-dimensional shape, called Chair44 or $Q$, that can tile all of $\mathbb{R}^3$ using congruent copies, but cannot produce any tiling with an infinite-order symmetry. In other words, the shape necessarily produces non-periodic arrangements. This addresses a particularly demanding version of the three-dimensional Einstein problem: finding one shape that forces aperiodicity purely through its geometry.

The tile itself is geometrically quite interesting. It begins with a seven-cube chair: a $2\times2\times2$ cube with one corner cube removed. Its 24 exposed unit-square panels are modified with very small square-pyramid protrusions and indentations. These features act like a physical encoding of matching rules: neighboring tiles can only fit together in certain configurations. The resulting object is a rational polyhedral $3$-ball, meaning that it is topologically equivalent to a solid ball and its vertices have rational coordinates. Importantly, the construction does not impose external matching rules—the restrictions arise entirely from the shape itself.

The central mechanism is a hierarchical substitution structure. Eight appropriately rotated copies of the chair form a chair twice as large, and this process can be repeated indefinitely. Thus each individual tile belongs to a uniquely determined sequence of increasingly large “supertiles” with scales $2,4,8,\ldots,2^n$. If a translation were a symmetry of the tiling, it would have to remain a symmetry after passing to every larger scale. But the hierarchy forces the translation vector to be repeatedly halved; consequently the only possible translation is $0$. Formally, the authors establish $\operatorname{Per}(T)={0}$ and $|\operatorname{Sym}(T)|\leq24$ for every tiling $T$.

A particularly notable aspect is the extensive computer-assisted and formally verified proof. The authors enumerate possible configurations exactly rather than relying on numerical approximations. For example, $2{,}388$ candidate contact poses are reduced to exactly $44$ legal contacts, while other stages involve thousands of possible configurations and hundreds of thousands of collision checks. These computations were independently replayed in two Python implementations, while the finite statements and logical assembly were checked in Lean 4, using exact integers and rational arithmetic rather than floating-point calculations. The paper is explicitly presented as a proof submission pending refereeing, so its result should currently be regarded as a new preprint rather than an established peer-reviewed theorem.

Key takeaways

• Main result: A single three-dimensional polyhedral tile $Q$ is shown to tile $\mathbb{R}^3$ while forcing aperiodicity.
  
• The tile is called Chair44 (R44) and is based on a seven-cube chair.
  
• Its small geometric bumps and dents encode matching constraints through shape alone.
  
• Every tiling has no nonzero translational period:
  Per⁡(T)={0}.\operatorname{Per}(T)=\{0\}.
  
• Every tiling has a finite symmetry group:
  ∣Sym⁡(T)∣≤24.|\operatorname{Sym}(T)|\leq24.
  
• Every tiling is homochiral: all copies have the same handedness.
  
• Each tile belongs to a unique hierarchy of supertiles with scales
  20,21,22,….2^0,2^1,2^2,\ldots.
  
• The proof's key idea is recognizability + hierarchical substitution + period halving.
  
• The finite contact structure contains exactly 44 legal contacts after eliminating $2{,}388$ candidates.
  
• The computational part uses exact arithmetic, avoiding floating-point uncertainty.
  
• The finite enumerations and logical proof assembly are formally checked in Lean 4, modulo explicitly identified compiler hooks.
  
• The paper connects modern aperiodic tilings, combinatorics, discrete geometry, substitution systems, and formal proof.
  
• The work is currently a September 2026 arXiv proof submission, not yet a peer-reviewed publication. 
  

Read the full paper on arXiv]]> https://mklab.gr/showthread.php?tid=2076 Wed, 23 Sep 2026 17:46:09 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2076 Dhruv Ranganathan, a Cambridge professor of algebraic geometry who won the 2026 London Mathematical Society Whitehead Prize. His research combines algebraic geometry, moduli spaces, and log geometry. The article explains how algebraic varieties—geometric shapes described by polynomial equations—can be studied through numerical and counting problems. 

For example, the bitangent problem asks how many lines can touch a curve at two points; such counts can reveal important information about the underlying geometric object. Moduli spaces take this idea further by treating all possible configurations of an object themselves as a geometric space. 

The central metaphor of Ranganathan's work is “shattering” a complicated geometric object into simpler pieces. In log geometry, one can study what happens when changing the coefficients of polynomial equations causes a variety to break into simpler components; tropical geometry then provides tools for understanding how those pieces can be reassembled. 

Ranganathan has applied these ideas to major theories such as Gromov–Witten and Donaldson–Thomas theory, helping demonstrate connections between approaches that initially look very different. The article presents this as a powerful strategy for turning difficult problems about complicated geometric spaces into more manageable problems about their simpler pieces.

ARTICLE]]> Dhruv Ranganathan, a Cambridge professor of algebraic geometry who won the 2026 London Mathematical Society Whitehead Prize. His research combines algebraic geometry, moduli spaces, and log geometry. The article explains how algebraic varieties—geometric shapes described by polynomial equations—can be studied through numerical and counting problems. 

For example, the bitangent problem asks how many lines can touch a curve at two points; such counts can reveal important information about the underlying geometric object. Moduli spaces take this idea further by treating all possible configurations of an object themselves as a geometric space. 

The central metaphor of Ranganathan's work is “shattering” a complicated geometric object into simpler pieces. In log geometry, one can study what happens when changing the coefficients of polynomial equations causes a variety to break into simpler components; tropical geometry then provides tools for understanding how those pieces can be reassembled. 

Ranganathan has applied these ideas to major theories such as Gromov–Witten and Donaldson–Thomas theory, helping demonstrate connections between approaches that initially look very different. The article presents this as a powerful strategy for turning difficult problems about complicated geometric spaces into more manageable problems about their simpler pieces.

ARTICLE]]> https://mklab.gr/showthread.php?tid=2072 Wed, 23 Sep 2026 16:07:10 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2072 “The Proof Mattered More Than the Prize,” tells the story of how Grigori Perelman solved the Poincaré conjecture, one of mathematics’ most famous problems. In 2002, Perelman posted a paper on arXiv containing ideas based on Richard Hamilton’s Ricci flow, followed by two further papers in 2003. Together, this work provided the crucial advances needed to resolve the Poincaré conjecture and, more broadly, the geometrization conjecture. The article explains the underlying topological question in accessible terms: essentially, whether a three-dimensional space with the right kind of “loop-shrinking” property must have the same fundamental structure as a three-dimensional sphere.

The central message is that the mathematical proof itself mattered more than the prizes attached to it. Perelman eventually declined both the Fields Medal and the $1 million Millennium Prize, attracting enormous public attention, but the article argues that these awards are secondary to the actual mathematical achievement. What ultimately established the result was not recognition, money, or fame, but the proof, the ideas behind it, and the mathematical community's ability to verify those ideas. The story therefore illustrates a broader principle about mathematics: recognition can celebrate a discovery, but it does not create its truth—the proof and the rigorous reasoning are what make the result meaningful. 

ARTICLE [PDF]]]> “The Proof Mattered More Than the Prize,” tells the story of how Grigori Perelman solved the Poincaré conjecture, one of mathematics’ most famous problems. In 2002, Perelman posted a paper on arXiv containing ideas based on Richard Hamilton’s Ricci flow, followed by two further papers in 2003. Together, this work provided the crucial advances needed to resolve the Poincaré conjecture and, more broadly, the geometrization conjecture. The article explains the underlying topological question in accessible terms: essentially, whether a three-dimensional space with the right kind of “loop-shrinking” property must have the same fundamental structure as a three-dimensional sphere.

The central message is that the mathematical proof itself mattered more than the prizes attached to it. Perelman eventually declined both the Fields Medal and the $1 million Millennium Prize, attracting enormous public attention, but the article argues that these awards are secondary to the actual mathematical achievement. What ultimately established the result was not recognition, money, or fame, but the proof, the ideas behind it, and the mathematical community's ability to verify those ideas. The story therefore illustrates a broader principle about mathematics: recognition can celebrate a discovery, but it does not create its truth—the proof and the rigorous reasoning are what make the result meaningful. 

ARTICLE [PDF]]]> https://mklab.gr/showthread.php?tid=2031 Thu, 17 Sep 2026 22:12:40 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2031 Heilbronn Triangle Problem, a classic problem in discrete geometry. The goal is to place nn points inside a region of area 1 so that the smallest triangle determined by any three points is as large as possible. In other words, we want to distribute the points as evenly as possible while avoiding three points becoming nearly collinear. The page examines three settings: points restricted to a unit square, points inside a unit-area triangle, and points inside an arbitrary convex region whose area is 1. 

The site provides the best-known values of A(n)A(n) for n=3n=3 through 3535, together with exact coordinates for the corresponding point configurations, symmetry information, proofs where available, and an online verifier. A key feature is that only some values are marked as proven optimal; many others are merely the best configurations currently known. 

The numerical values decrease rapidly as nn increases—for example, in the square the value falls from 0.50.5 for n=3n=3 to about 0.004100.00410 for n=35n=35. Thus, the page serves both as an introduction to the Heilbronn problem and as a catalogue of computational and mathematically verified progress on an optimization problem that remains difficult for larger numbers of points.


ARTICLE]]> Heilbronn Triangle Problem, a classic problem in discrete geometry. The goal is to place nn points inside a region of area 1 so that the smallest triangle determined by any three points is as large as possible. In other words, we want to distribute the points as evenly as possible while avoiding three points becoming nearly collinear. The page examines three settings: points restricted to a unit square, points inside a unit-area triangle, and points inside an arbitrary convex region whose area is 1. 

The site provides the best-known values of A(n)A(n) for n=3n=3 through 3535, together with exact coordinates for the corresponding point configurations, symmetry information, proofs where available, and an online verifier. A key feature is that only some values are marked as proven optimal; many others are merely the best configurations currently known. 

The numerical values decrease rapidly as nn increases—for example, in the square the value falls from 0.50.5 for n=3n=3 to about 0.004100.00410 for n=35n=35. Thus, the page serves both as an introduction to the Heilbronn problem and as a catalogue of computational and mathematically verified progress on an optimization problem that remains difficult for larger numbers of points.


ARTICLE]]> https://mklab.gr/showthread.php?tid=2008 Tue, 15 Sep 2026 22:34:13 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2008 “Topological structure of optimal flows on the Girl’s surface” by Maria Loseva and Alexandr Prishlyak studies dynamical systems on the Girl’s surface, one of the two known one-triple-point immersions of the real projective plane in three-dimensional space, the other being Boy’s surface. 

The authors first describe the surface using a cellular decomposition, representing the projective plane as a disk whose opposite boundary points are identified. They then investigate flows—motion generated by vector fields—on this singular surface. In particular, they classify the possible flows having a single fixed point and no separatrices, showing that there are exactly three distinct structures of this type. 

The central part of the paper concerns Morse–Smale flows, which are especially well-behaved dynamical systems whose trajectories and fixed points have a stable topological structure. The authors prove that Morse–Smale flows are precisely the structurally stable flows on both Boy’s and Girl’s surfaces. 

They then determine all possible optimal Morse–Smale flows on the Girl’s surface, classify the corresponding flows on the immersed projective plane, and describe their isotopy (continuous deformation) classes. The resulting classification provides a systematic way of encoding these flows through separatrix diagrams and topological graphs, connecting the geometry of the unusual Girl’s surface with broader ideas from dynamical systems, topology, and Morse theory. 

ARTICLE [PDF]]]> “Topological structure of optimal flows on the Girl’s surface” by Maria Loseva and Alexandr Prishlyak studies dynamical systems on the Girl’s surface, one of the two known one-triple-point immersions of the real projective plane in three-dimensional space, the other being Boy’s surface. 

The authors first describe the surface using a cellular decomposition, representing the projective plane as a disk whose opposite boundary points are identified. They then investigate flows—motion generated by vector fields—on this singular surface. In particular, they classify the possible flows having a single fixed point and no separatrices, showing that there are exactly three distinct structures of this type. 

The central part of the paper concerns Morse–Smale flows, which are especially well-behaved dynamical systems whose trajectories and fixed points have a stable topological structure. The authors prove that Morse–Smale flows are precisely the structurally stable flows on both Boy’s and Girl’s surfaces. 

They then determine all possible optimal Morse–Smale flows on the Girl’s surface, classify the corresponding flows on the immersed projective plane, and describe their isotopy (continuous deformation) classes. The resulting classification provides a systematic way of encoding these flows through separatrix diagrams and topological graphs, connecting the geometry of the unusual Girl’s surface with broader ideas from dynamical systems, topology, and Morse theory. 

ARTICLE [PDF]]]> https://mklab.gr/showthread.php?tid=1992 Tue, 15 Sep 2026 19:35:33 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1992 “The geometric theorems of Almada Negreiros” by Pedro J. Freitas examines the unusual intersection of mathematics and art in the work of Portuguese artist Almada Negreiros (1893–1970). Negreiros believed that a universal geometric system—his “Canon”—underlay visual art across cultures and historical periods. His artistic constructions made extensive use of geometric proportions, the golden ratio, divisions of the circle, and geometric shapes such as pentagrams. 

Remarkably, many of his drawings were not merely artistic compositions: they encoded geometric constructions and mathematical relationships. For example, some constructions produce rectangles with exact proportions involving the golden ratio, while others approximate divisions of a circle into 9, 10, or 14 equal parts. 

A particularly interesting aspect is the tension between exact mathematics and visual approximation. By the Gauss–Wantzel theorem, a circle cannot be divided exactly into 9 or 14 parts using only straightedge and compass, yet Negreiros produced remarkably accurate visual constructions; his approximation for a ninth of a circle has an error of only about 0.001% in one construction. 

He was more concerned with visual harmony and symbolic meaning than with rigorous mathematical distinction between exact and approximate results. The article argues that his work is therefore significant both artistically and mathematically: it demonstrates how geometric intuition can inspire original constructions while also revealing a distinctive philosophy in which geometry becomes a universal language connecting mathematics, art, history, and culture. 


ARTICLE [PDF]]]> “The geometric theorems of Almada Negreiros” by Pedro J. Freitas examines the unusual intersection of mathematics and art in the work of Portuguese artist Almada Negreiros (1893–1970). Negreiros believed that a universal geometric system—his “Canon”—underlay visual art across cultures and historical periods. His artistic constructions made extensive use of geometric proportions, the golden ratio, divisions of the circle, and geometric shapes such as pentagrams. 

Remarkably, many of his drawings were not merely artistic compositions: they encoded geometric constructions and mathematical relationships. For example, some constructions produce rectangles with exact proportions involving the golden ratio, while others approximate divisions of a circle into 9, 10, or 14 equal parts. 

A particularly interesting aspect is the tension between exact mathematics and visual approximation. By the Gauss–Wantzel theorem, a circle cannot be divided exactly into 9 or 14 parts using only straightedge and compass, yet Negreiros produced remarkably accurate visual constructions; his approximation for a ninth of a circle has an error of only about 0.001% in one construction. 

He was more concerned with visual harmony and symbolic meaning than with rigorous mathematical distinction between exact and approximate results. The article argues that his work is therefore significant both artistically and mathematically: it demonstrates how geometric intuition can inspire original constructions while also revealing a distinctive philosophy in which geometry becomes a universal language connecting mathematics, art, history, and culture. 


ARTICLE [PDF]]]> https://mklab.gr/showthread.php?tid=1926 Thu, 10 Sep 2026 00:54:58 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1926 Spherical trigonometry studies the relationships between the sides and angles of triangles drawn on the surface of a sphere. Unlike ordinary plane triangles, the sides of a spherical triangle are arcs of great circles, which are the geodesics of a sphere. This geometry is important in astronomy, geodesy, navigation, and calculations involving positions on Earth. A fundamental difference from Euclidean geometry is that the angles $A,B,C$ of a spherical triangle satisfy $A+B+C>\pi$. For a sphere of radius $R$, side lengths are usually expressed as angular quantities by dividing their arc lengths by $R$.

The central formulas are spherical versions of the familiar sine and cosine laws. The spherical law of cosines is $\cos a=\cos b\cos c+\sin b\sin c\cos A$, with analogous formulas obtained by cyclically permuting $a,b,c$. The spherical law of sines is $\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}$. For triangles that are very small compared with the sphere's radius, these formulas approach the ordinary Euclidean sine and cosine laws. Spherical trigonometry also includes Napier's analogies, Delambre's analogies, half-angle formulas, and special rules for right spherical triangles.

One of the most striking consequences of spherical geometry is the connection between area and angle sum. If a spherical triangle has angles $A,B,C$, its spherical excess is $E=A+B+C-\pi$. On a unit sphere, $E$ is exactly the area of the triangle. More generally, on a sphere of radius $R$, the area is $\text{Area}=R^2(A+B+C-\pi)=R^2E$. Thus, unlike in Euclidean geometry, the angle sum of a spherical triangle directly determines its area.

Key takeaways:

• Spherical triangles are formed by arcs of great circles.
  
• Their angles satisfy $A+B+C>\pi$, unlike Euclidean triangles where the sum is $\pi$.
  
• The spherical sine law and spherical cosine law are the main tools for solving spherical triangles.
  
• The spherical excess $E=A+B+C-\pi$ measures how much the angle sum exceeds $\pi$.
  
• The area of a spherical triangle is $\text{Area}=R^2E$.
  
• Spherical trigonometry has important applications in astronomy, navigation, geodesy, and calculations on the Earth's surface.
  

ARTICLE / ARTICLE [PDF]]]> Spherical trigonometry studies the relationships between the sides and angles of triangles drawn on the surface of a sphere. Unlike ordinary plane triangles, the sides of a spherical triangle are arcs of great circles, which are the geodesics of a sphere. This geometry is important in astronomy, geodesy, navigation, and calculations involving positions on Earth. A fundamental difference from Euclidean geometry is that the angles $A,B,C$ of a spherical triangle satisfy $A+B+C>\pi$. For a sphere of radius $R$, side lengths are usually expressed as angular quantities by dividing their arc lengths by $R$.

The central formulas are spherical versions of the familiar sine and cosine laws. The spherical law of cosines is $\cos a=\cos b\cos c+\sin b\sin c\cos A$, with analogous formulas obtained by cyclically permuting $a,b,c$. The spherical law of sines is $\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}$. For triangles that are very small compared with the sphere's radius, these formulas approach the ordinary Euclidean sine and cosine laws. Spherical trigonometry also includes Napier's analogies, Delambre's analogies, half-angle formulas, and special rules for right spherical triangles.

One of the most striking consequences of spherical geometry is the connection between area and angle sum. If a spherical triangle has angles $A,B,C$, its spherical excess is $E=A+B+C-\pi$. On a unit sphere, $E$ is exactly the area of the triangle. More generally, on a sphere of radius $R$, the area is $\text{Area}=R^2(A+B+C-\pi)=R^2E$. Thus, unlike in Euclidean geometry, the angle sum of a spherical triangle directly determines its area.

Key takeaways:

• Spherical triangles are formed by arcs of great circles.
  
• Their angles satisfy $A+B+C>\pi$, unlike Euclidean triangles where the sum is $\pi$.
  
• The spherical sine law and spherical cosine law are the main tools for solving spherical triangles.
  
• The spherical excess $E=A+B+C-\pi$ measures how much the angle sum exceeds $\pi$.
  
• The area of a spherical triangle is $\text{Area}=R^2E$.
  
• Spherical trigonometry has important applications in astronomy, navigation, geodesy, and calculations on the Earth's surface.
  

ARTICLE / ARTICLE [PDF]]]>