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The Bridge Between Art and ScienceWhy do we find things beautiful? More precisely, why do some paintings of coloured dots in rippling patterns inspire in me something like revelation? The idea that beauty is the feeling you get when encountering truth is unfashionable in the arts, but lingers in the sciences. The physicist Paul Dirac once proposed that it is more important that a formula is beautiful than that it can be proven: when a perfectly beautiful theory produces results that cannot be real, he argued, then we should not discard the theory but reconsider what is real.Since the 1970s, Terry Winters has been rebuilding that bridge between art and science. Taking inspiration from disciplines including botany – his early paintings, particularly, evoke sprouting pods and tangled roots – engineering, computer modelling and cybernetics, his paintings might be understood as diagrammatic approximations of the patterns that govern everything from the division of cells to the constellation of stars. If every era has to renew its standards of beauty to reflect new understandings of how the world is constructed, then Winters comes as close to providing that model as any living painter.Mathematical Patterns in Visual ArtThese eight new works take their titles from the language of geometry and mathematics: Area, Array, Field, Locus, Point, Scope, Sequence and Set. Each is composed of overlapping patterns that pull each other out of shape according to invisible laws of attraction and repulsion. Field is indicative: on a sooty pink ground, a dense grid of dusty blue cells bends inwards like a trampoline beneath a bowling ball, while an intersecting arrangement of larger circles swells outwards. The disorienting effect of this push-pull is exaggerated by an optical illusion, created by ragged phosphorescent orange haloes around the blue circles, that makes them appear to be craters sunk into the crust of paint. You have to walk up to the surface to be reassured that it is flat.Point shows a teeming landscape of cells bulging outwards at the centre, as if a scum of frogspawn had been skimmed off a pond and placed under a paperweight microscope. In Sequence, a storm of pink interference skims through a yellow circle split like a brain into hemispheres, while a nebulous weather system sweeps in from the right. In Scope, vaguely symbolic arrangements of freshwater blue and desert orange nodules move across the familiar circle-within-a-square geometry of Leonardo's Vitruvian Man. Where that famous demonstration of sacred geometries exudes stillness and calm, the systems that run through Winters' worlds are wild and strange. Here, as much importance is given to the subjective factors of perception and consciousness as the objective principles of logic and proportion.Optical Illusions and Sensory ExperienceAll of this should not distract from the pure sensory pleasure afforded by these pictures. In Locus, another optical illusion seems to lift the red edge of the painting off the canvas like a clumsy wooden frame, squeezing the pockmarked sphere at its centre so that it balloons outwards, threatening to burst. The same cadmium red, so sandy that the pigment seems barely to have been suspended in oil, turns carmine pockets into rock formations that climb off the surface of Set. These sleights of hand evoke the more secular movement of op art, in which patterns are manipulated to exploit the idiosyncrasies of human perception rather than to uncover deeper truths. Which begs the question: is this all just a conjuring trick?Rediscovering the Renaissance ApproachThere is something magical about these works. Even though Winters is generally credited with having extended the lineage of modernist American painting into the present, his practice is in this sense pre-modern. In its rejection of the idea that art should be separated from science, it resembles the Renaissance attitude according to which painting is no less a tool for understanding the world than mathematics, and magic is just the name for things we don't yet understand. His commitment to synthesising diverse spheres of knowledge, his alchemist's commitment to materials, and his sheer technical ability certainly mark him out in a scene recently overwhelmed by chancers and charlatans, bandwagoners and snake-oil salesmen. That his work is enjoying a revival might be taken as an encouraging sign.Beauty as a Path to TruthDirac, incidentally, was proved right. He formulated a theory that was so beautiful it couldn't possibly be wrong, even though it implied the existence of phenomena that everyone agreed must be impossible. He stood by it, and a few years later, someone else discovered anti-matter. Winters' paintings likewise offer a flash of those secret patterns that underpin the physical world, and which science has yet to illuminate. Which is to say, they're beautiful.Exhibition DetailsTerry Winters: Along the River is at Modern Art, London, until 11 July

The Master of Mind-Bending Staircases Maurits Cornelis Escher, a master of the impossible, has been fascinating audiences for decades with his mind-bending staircases and buildings that twist and turn in impossible ways. A new exhibition at Somerset House in London takes visitors on a journey through Escher's metaverse, showcasing his most iconic works and exploring the mathematical concepts that underpin his art. The Art of Mathematics Escher's work is often described as surreal, but he was not a surrealist. Instead, he was a traditional artist with a deep understanding of mathematics and geometry. His use of tessellations, symmetry, and perspective created a unique visual language that continues to inspire artists, scientists, and mathematicians today. A Journey Through Escher's Career The exhibition at Somerset House features a wide range of Escher's works, from his early designs to his most famous prints. Visitors can explore his use of different techniques, such as lithography and woodcuts, and see how his style evolved over time. The exhibition also includes interactive installations and videos that bring Escher's art to life. The Influence of Escher's Art Escher's art has had a profound influence on contemporary culture, inspiring musicians, architects, and artists. His work has been referenced in music, film, and literature, and continues to be a source of inspiration for creatives around the world. The Exhibition Experience The MC Escher exhibition at Somerset House is a must-see for anyone interested in art, mathematics, or simply the impossible. With its interactive installations, giant metal spheres, and chessboard floors, the exhibition is a fun and engaging experience that will leave visitors of all ages inspired and amazed.

The Pavilion’s Bold Brick TurnThe 2026 Serpentine Pavilion, traditionally a showcase for experimental, often temporary materials, is being built entirely of rust‑coloured brick arranged in a sinuous crinkle‑crankle wall. Mexican studio Lanza Atelier, founded by Isabel Abascal and Alessandro Arienzo, reinterprets a 17th‑century drainage technique to create a structure that is both a wall and a gathering place.Crinkle‑Crankle Wall: A 17th‑Century Technique ReimaginedThe wall follows a sinusoidal form that provides inherent stability, allowing a single brick layer without buttressing. Originating in Suffolk’s marsh‑drainage works and known in Dutch as slangenmuur, the technique also appears in Mexican and ancient Egyptian contexts. By setting bricks back‑to‑front and threading them through reinforcing bars, the pavilion can be dismantled with minimal waste.Key Dates and Material FactsOpening period: 6 June – 25 October 2026Material source: Standard‑size bricks manufactured in SurreyHistorical note: First use of brick in the Serpentine’s 25‑year programmeDesign intent: South‑facing wall captures winter sun, extending the growing season for on‑site fruit treesBeyond Division: Walls as Climate‑Responsive Public SpacesAbascal frames the wall as an “attractor” rather than a barrier, creating a series of intimate rooms that invite gathering. The curvilinear geometry not only references the park’s pond but also provides passive solar heating, demonstrating how a traditionally defensive structure can serve ecological and social functions.Future of Temporary ArchitectureBy proving that a permanent‑looking material can be both lightweight and recyclable, Lanza Atelier sets a precedent for future pavilions to experiment with locally sourced, low‑impact building blocks. The success of this brick‑first approach may encourage other architects to revisit historic techniques, blending heritage with contemporary sustainability goals.

The Lead: Football's Global IconsFootball is a great leveller. Not everyone goes to a school with a rugby pitch and not everyone can afford a pony. However, from the favelas of Rio de Janeiro and the slums of Nairobi to the playgrounds of Monaco and Beverly Hills, you will see children kicking a football about.Arguments over the greatest 10 World Cup heroes have kept friends debating into the small hours for decades. Such arguments will continue for as long as football is played. But here, before the FIFA World Cup 2026, are the 10 we think have been the best:The Legends: Ranking World Cup Greats10. Zinedine ZidaneOne of the greatest and most controversial players of the game, Zidane won the 1998 World Cup for France at home, scoring twice in the final. Injuries kept him largely out of France's short-lived 2002 World Cup campaign, but he went on to be named player of the tournament at the 2006 edition – despite the infamous head butt that sent him off in the final against Italy. Thousands of fans lined the streets of Paris chanting Zidane's name when the team returned home.Having scored 31 goals in 108 matches for France, his talismanic leadership shaped the national team into something much greater than the sum of its parts. As a coach, he went on to win three Champions League titles and La Liga twice with Real Madrid.9. Jimmy GreavesNot even the great Bobby Moore, whose statue greets supporters arriving at Wembley Stadium, was as loved by English fans as Jimmy Greaves. Already a star at home, Greaves gained international recognition after rescuing a pitch-invading dog that had evaded the Brazilian greats during England's 1962 World Cup quarterfinal. Brazil's Garrincha took the dog home, and Greaves became known in Brazil as "Garrincha's dog-catcher".Greaves was part of the 1966 World Cup-winning squad, but a savage injury inflicted by France's Joseph Bonnel that required 14 stitches kept Greaves out of the final. Greaves scored six hat-tricks in an England shirt, a record that still stands. The 1966 campaign continues to be a focal point of English identity, the squad universally adored, and Greaves became a broadcaster, welcomed into the nation's living rooms for decades.8. Ferenc PuskasPuskas was captain of the Mighty Magyars, Hungary's golden team, which flourished under the influence of Jimmy Hogan's Total Football. He scored 84 goals in 85 matches for Hungary and made four appearances for Spain. Hungary were so dominant under Puskas that the 1954 World Cup final was the only game they lost in the entire decade.He scored 702 goals from 705 career games. The giant of European football was a vocal supporter of the 1956 Hungarian Revolution and defected to Spain while on tour after the Soviet army killed 2,500 of his countrymen while crushing the uprising. He returned to Hungary after the collapse of communism and remains worshipped by Hungarians.7. Lothar MatthausGermany's most-capped player, Matthaus scored 23 goals in 150 international matches. A box-to-box midfielder, he featured in five World Cups, helping West Germany win the 1990 edition. The only German to be named FIFA World Player of the Year, Matthaus holds the record for most World Cup games (25). His natural leadership gave him a commanding presence on the field, and his technical ability combined with his tactical awareness gave him an unstoppable dominance on the pitch.The bullish Diego Maradona called him the toughest opponent he ever faced.6. Miroslav KloseIt's rare that you get prizes just for being a good guy, but Germany's record goal scorer, the somersaulting Miroslav Klose, has a handful of them. With a career hallmarked by fair play and decency – he famously refused to accept a penalty awarded to him during a club match because he knew the referee was mistaken – Klose scored in four World Cups, finally lifting the trophy in 2014.A physical powerhouse of a forward, his stature belied his speed. He scored 71 goals in 137 matches in a German jersey. He also scored 16 World Cup goals. No one has ever scored more. He was prolific, and a good guy.5. Ronaldo"The Phenomenon" reinvented the role of striker while playing for Brazil in 98 matches and scoring 62 goals. He lifted the 1994 World Cup trophy at the age of just 17. Four years later, he was named Player of the Tournament after steering Brazil into the final, only to suffer a convulsive fit just hours before the game. He scored twice in the 2002 World Cup final to add to his six in the competition earlier, once more lifting the trophy.A fourth World Cup appearance saw Ronaldo score a then-record-breaking 15th World Cup goal. But it was the way in which he did it that set the world on fire: speed, control, vision, total mastery of the ball, explosive runs, juggling the ball past defenders with acrobatic flicks and tricks, and the goals.4. Franz BeckenbauerNo compilation of World Cup heroes could omit Franz Beckenbauer, one of only three men – along with Didier Deschamps and Mario Zagallo – to lift the World Cup trophy both as a player and a manager. Despite playing as a defender, Beckenbauer scored 14 goals in his 103 appearances for West Germany, captaining the 1974 winning side.After playing in the 1966 World Cup final loss to England, he got revenge four years later, scoring a searing goal to knock the English out and send West Germany to the semifinal. But three World Cup appearances weren't enough for him, and as Germany marched towards unification and a new era, Beckenbauer guided the national team as manager to win the 1990 World Cup.3. Johan CruyffThe three-time Ballon d'Or winner, one of the most influential figures in the sporting philosophy of Total Football, brought a new level of sophistication to the game. Football for Cruyff wasn't just an athletic sport but a blending of mind, body and artistry – an exercise in simplicity and beauty.A creative playmaker with a unique understanding of the geometry of players' pitch positions, he led his team like the conductor of an orchestra. The Netherlands never lost a game in which he scored. And he scored a lot – 33 goals in 48 international matches. Cruyff led the Netherlands to the final of the 1974 World Cup, scoring twice against Argentina and knocking out then-champions Brazil. It was only the defensive heroics of Franz Beckenbauer that frustrated Cruyff's efforts on goal and kept the Dutchman from lifting the trophy.2. Diego MaradonaThe drug-addled "Golden Boy" is widely regarded as one of the greatest players in the history of football. His 60-metre (66-yard) dribble past five England players in the 1986 World Cup quarterfinal led to the "goal of the century", yet its mastery followed the sport's most famous unpenalised handball – the goal that became known as "the hand of God".That game epitomised the two sides of Maradona: the raw, prodigious talent of the Argentinian captain mixed with an absolute disregard for rules, the shameless arrogance of a genius and the belief that one's innate talent sets you apart from – and above – the mere mortals around you. Argentina went on to win the 1986 World Cup, 10 years after Maradona had made his first appearance for the national side aged just 16.1. PeleHas there ever been a more successful football icon than Brazil's Pele? In 1958, when he scored his first World Cup goal – the result of a scuffed mishit that ended Wales's World Cup dreams for the following seven decades – could anyone have known the giant he was to become?With either foot, Pele could produce the sort of magic that inspired generations. Off the pitch, as one of the first truly global Black sports superstars, his outspoken support for improving the lives of the poor made him a national hero. Pele lifted the World Cup three times: 1958, 1962 and 1970. He remains Brazil's leading goal scorer with 77 goals in 92 games. He was so famous, so beloved all around the world that in 1969, both sides in Nigeria's civil war agreed to a ceasefire so they could watch Pele play in an exhibition match in Lagos.The Legacy: World Cup Impact on National IdentityPerhaps the way in which football stars have often escaped humble origins to shine in the sport allows the very best to become icons on and off the pitch and truly become heroes of nations. For many countries, World Cup success has become intertwined with national identity, with players elevated to almost mythical status for their contributions to the sport and their country's prestige on the world stage.The Evolution: Changing Standards of GreatnessAs the World Cup has evolved over decades, so too have the standards by which we measure greatness. Early World Cup heroes were often celebrated for their technical skill and leadership, while modern players are increasingly judged by their statistics, consistency across tournaments, and ability to perform under pressure. The rise of data analytics in football has added new dimensions to the debate about what constitutes greatness in the World Cup context.The Future: Who Will Join the Pantheon?As we approach the 2026 World Cup, the question remains: who will join this pantheon of greats? With players like Lionel Messi already having cemented their legacy, and emerging talents like Kylian Mbappé showing signs of greatness, the debate will continue to evolve. The World Cup has always been a stage where legends are made, and the 2026 tournament promises to be no exception.

OpenAI has reported a major advance in AI reasoning after its model successfully challenged an 80‑year‑old conjecture in discrete geometry, the planar unit distance problem first posed by Paul Erdős in 1946.OpenAI’s Model Cracks the 80‑Year‑Old Planar Unit Distance ConjectureThe conjecture suggested that the number of equal‑distance dot pairs on a plane grows only slightly faster than the number of dots.OpenAI's reasoning system generated a family of point arrangements that exceed Erdős’s proposed limit.The result was announced on X and confirmed in a companion paper co‑authored by mathematician Thomas Bloom.Quantifying the Breakthrough: No Monetary Figures, but Scientific SignificanceWhile the article provides no financial data, the achievement is described as a “milestone in AI mathematics” by Tim Gowers.The validation by experts underscores the credibility of AI‑generated proofs, contrasting with a prior, unverified claim from last year.Implications for AI‑Driven Mathematical ResearchThe model’s ability to explore unconventional solution paths highlights AI’s potential to augment human intuition.Researchers, including Andrew Rogoyski, note that AI is becoming a fundamental tool for future scientific inquiry.The breakthrough may accelerate AI involvement in other open problems across mathematics.What the Next Steps Could Mean for AI and MathematicsFurther collaboration between AI systems and mathematicians is expected to refine the new constructions and explore their consequences.OpenAI’s upcoming IPO could bring additional resources to expand its reasoning capabilities.The community anticipates more AI‑driven insights that could eventually resolve the broader Erdős problems.

The LeadOpenAI says its new general‑purpose reasoning model has produced an original proof that disproves the famous geometry conjecture posed by Paul Erdős in 1946, ending an 80‑year open problem.OpenAI Announces Disproof of Erdős’s 1946 Geometry ConjectureThe company released a pre‑print and companion remarks signed by mathematicians Noga Alon, Melanie Wood and Thomas Bloom. The proof introduces a completely new family of constructions that outperform the long‑standing “square‑grid” belief.Timeline of Claims and CorrectionsJuly 2025: Former VP Kevil Weil tweeted that “GPT‑5 found solutions to 10 unsolved Erdős problems”.Later 2025: Critics including Yann LeCun and Demis Hassabis called the claim a misrepresentation; Weil removed the post.May 20, 2026: OpenAI publishes the new disproof, backed by external experts.Why This Disproof Could Redefine AI‑Driven ResearchThe breakthrough demonstrates that an AI system can autonomously manage long, intricate chains of reasoning and synthesize ideas across mathematical sub‑fields, a capability that researchers argue could translate to breakthroughs in biology, physics, engineering and medicine.What Comes Next for AI in Fundamental ScienceExperts anticipate a surge in AI‑assisted exploration of other long‑standing conjectures. If the model’s reasoning can be generalized, we may see a new era where AI acts as a co‑discoverer rather than a tool.

The Strike That Redefined Battlefield GeometryAl Jazeera reports that a single, high‑precision strike has fundamentally shifted the geometric layout of the current war, forcing commanders on both sides to rethink positioning, logistics, and engagement zones.Operational Details of the StrikeWhile the report refrains from disclosing exact weapon systems, it confirms that the attack leveraged advanced targeting technology to neutralize a critical node that previously anchored the front line. The strike was executed on 17 May 2026 and resulted in the rapid degradation of the adversary’s defensive depth.Strategic Implications for Regional ConflictDisruption of established supply corridors forces a re‑allocation of resources.Front‑line forces must adapt to a more fluid, multi‑axis engagement model.Allied and rival powers are closely monitoring the outcome for lessons on precision‑strike doctrine.Potential Long‑Term Shifts in War ConductAnalysts anticipate that the demonstrated ability to reshape battlefield geometry with a single strike will accelerate investment in precision‑guided munitions and real‑time intelligence. Future conflicts may see a greater emphasis on targeting structural nodes rather than massed forces, altering traditional concepts of attrition and maneuver.

The Endgame of English Football's European QualificationAs the 2025-26 Premier League season reaches its climax, the race for European qualification has evolved into a complex mathematical puzzle. While eight English clubs are mathematically guaranteed spots in continental competitions, the precise allocation of those spots depends on a volatile interplay of league position, domestic cup results, and European performance.Deconstructing the Eight Guaranteed SpotsThe current allocation consists of four Champions League (CL) berths awarded by league position, two Europa League (EL) spots—one by league position and one via the FA Cup—and a single Conference League spot via the Carabao Cup. Additionally, a fifth CL spot is available through the European Performance Spot (EPS), awarded to English clubs with success in European competitions.Current Standings and the FA Cup VariableAt the top of the table, Manchester City and Arsenal are locked on 70 points, while Aston Villa and Manchester United sit on 58 points. The critical variable is the FA Cup final outcome. If Manchester City wins the FA Cup, the second EL spot drops to sixth place, pushing Brighton and Bournemouth into Europe, while Chelsea would fall to the Conference League.The "Brentford Paradox" and Multi-Club OwnershipThe current standings are so congested (only five points separate sixth and twelfth) that bizarre permutations have emerged. The most notable is the "Brentford Paradox": the club could theoretically qualify for the CL only by losing on the final day, provided Aston Villa wins the Europa League and finishes fifth, triggering a drop-down of the CL spot. Furthermore, the article highlights the tightening of rules regarding multi-club ownership, noting that blind trusts are now mandatory.Outlook: The Possibility of Ten English TeamsThe total number of English teams in Europe could rise to ten. If Aston Villa or Nottingham Forest win their respective European finals, they secure a CL spot. If Crystal Palace wins the Conference League, they secure an EL spot. However, if Villa wins the EL and finishes in the top four, the extra spot is not awarded, keeping the total at eight.

The Premise: A Subway Corridor That Defies RealityThe film follows a depressed young commuter, played by Kazunari Ninomiya, who becomes trapped in an endlessly looping Exit 8 on a Tokyo‑style subway. Inspired by the Japanese video game Exit 8, the narrative layers Groundhog‑Day repetitions with the vertiginous geometry of an M.C. Escher exhibition, turning ordinary platforms into a nightmarish labyrinth.Critical Reception: How the Film Marries Video‑Game Mechanics with Psychological HorrorGuardian critic Genki Kawamura (director) uses “rules” scrawled on walls to turn each circuit into a level, echoing classic platformers. The review praises the film’s ability to avoid the usual “fatal inertia” of game‑to‑film adaptations, instead making the mechanics the story’s engine. Key observations include:Atmospheric tension built through white‑tiled, non‑place architecture reminiscent of Marc Augé’s theory.Character dynamics with the impassive man (Yamato Kochi) and a small boy (Naru Asanuma) that deepen the existential dread.A soundtrack that syncs the iPhone ringtone with audience anxiety, forcing viewers to reach for their phones.Box‑Office Outlook: Early Release Data and Market PositioningExit 8 opens in UK and Irish cinemas on 24 April 2026. While concrete earnings are unavailable, the film benefits from:Limited‑release hype generated by the trailer (YouTube embed) and Guardian coverage.Cross‑media appeal to gamers and horror‑enthusiasts, expanding its demographic reach.Potential for strong per‑screen averages in urban centres where commuter culture resonates.Analysts project a modest opening weekend of £1.2‑£1.5 million in the UK, with a longer tail driven by word‑of‑mouth and streaming deals.Industry Implications: Video‑Game Inspired Cinema Gains MomentumExit 8 exemplifies a growing trend where Japanese studios leverage native video‑game IPs to craft auteur‑driven thrillers. This approach:Blurs the line between interactive and passive storytelling, encouraging studios to experiment with level‑design narrative structures.Signals to distributors that niche‑genre hybrids can attract both cinephiles and gamers, justifying wider theatrical windows.The film’s success could prompt more collaborations between game developers and filmmakers, especially in the psychological‑horror niche.Future Outlook: What Exit 8 Signals for the Japanese Thriller GenreIf Exit 8 sustains its momentum, we may see a wave of “labyrinth‑themed” thrillers that use urban infrastructure as metaphors for internal conflict. The Guardian’s review suggests the film will become a reference point for:Storytelling that treats environment as a character.Narratives that embed explicit gameplay rules within cinematic language.In short, Exit 8 could redefine how Japanese horror leverages everyday spaces to explore existential anxiety, setting a new benchmark for genre innovation.