* . *
  • About
  • Advertise
  • Privacy & Policy
  • Contact
Sunday, June 1, 2025
Earth-News
  • Home
  • Business
  • Entertainment

    Unveiling the Enigmatic: First Looks at Destruction and Puck in ‘The Sandman

    Jackie Chan Reveals This Family Member ‘Never Watched’ The Whole Of Any Of His Movies – Yahoo

    Jackie Chan Reveals This Family Member ‘Never Watched’ The Whole Of Any Of His Movies – Yahoo

    Mavs CEO holds firm on new arena, entertainment district in Dallas – Dallas News

    Mavs CEO Stands Strong on Vision for New Arena and Entertainment District in Dallas

    Entertainment: On Your Marks, Get Set, Beer Run! – Urban Milwaukee

    Get Ready to Race: The Ultimate Beer Run Experience Awaits!

    Rachel Guttman Launches Entertainment Law Firm Gutt Law, PLLC [Exclusive] – MusicRow.com

    Rachel Guttman Unveils Exciting New Entertainment Law Firm: Gutt Law, PLLC!

    HYBE Cashes In: Offloads Final Stake in K-Pop Rival SM Entertainment for $177 Million!

  • General
  • Health
  • News

    Cracking the Code: Why China’s Economic Challenges Aren’t Shaking Markets, Unlike America’s” – Bloomberg

    Trump’s Narrow Window to Spread the Truth About Harris

    Trump’s Narrow Window to Spread the Truth About Harris

    Israel-Gaza war live updates: Hamas leader Ismail Haniyeh assassinated in Iran, group says

    Israel-Gaza war live updates: Hamas leader Ismail Haniyeh assassinated in Iran, group says

    PAP Boss to Niger Delta Youths, Stay Away from the Protest

    PAP Boss to Niger Delta Youths, Stay Away from the Protest

    Court Restricts Protests In Lagos To Freedom, Peace Park

    Court Restricts Protests In Lagos To Freedom, Peace Park

    Fans React to Jazz Jennings’ Inspiring Weight Loss Journey

    Fans React to Jazz Jennings’ Inspiring Weight Loss Journey

    Trending Tags

    • Trump Inauguration
    • United Stated
    • White House
    • Market Stories
    • Election Results
  • Science
  • Sports
  • Technology
    Bajeed Pattan Joins Forbes Technology Council as Innovation Leader – PRWeb

    Bajeed Pattan Takes the Helm as Innovation Leader at Forbes Technology Council!

    Lafayette Regional Technology Council – Tech Leadership That’s Homegrown and Future-Focused – Discover Lafayette

    Lafayette Regional Technology Council – Tech Leadership That’s Homegrown and Future-Focused – Discover Lafayette

    Drone technology demo in Cambria County showcases future of lifesaving medical deliveries – local21news.com

    Revolutionizing Healthcare: Drone Technology Takes Flight for Lifesaving Medical Deliveries in Cambria County

    Revolutionary Harvesting Technology Promises to Slash CAR-T Manufacturing Costs!

    Stop the Machines: The Rise of Anti-Technology Extremism – International Centre for Counter-Terrorism – ICCT

    Unplugged: The Surge of Anti-Technology Extremism

    Finland to head EU’s quantum defense technology project – Latest news from Azerbaijan

    Finland Takes the Lead in Pioneering EU’s Quantum Defense Technology Initiative!

    Trending Tags

    • Nintendo Switch
    • CES 2017
    • Playstation 4 Pro
    • Mark Zuckerberg
No Result
View All Result
  • Home
  • Business
  • Entertainment

    Unveiling the Enigmatic: First Looks at Destruction and Puck in ‘The Sandman

    Jackie Chan Reveals This Family Member ‘Never Watched’ The Whole Of Any Of His Movies – Yahoo

    Jackie Chan Reveals This Family Member ‘Never Watched’ The Whole Of Any Of His Movies – Yahoo

    Mavs CEO holds firm on new arena, entertainment district in Dallas – Dallas News

    Mavs CEO Stands Strong on Vision for New Arena and Entertainment District in Dallas

    Entertainment: On Your Marks, Get Set, Beer Run! – Urban Milwaukee

    Get Ready to Race: The Ultimate Beer Run Experience Awaits!

    Rachel Guttman Launches Entertainment Law Firm Gutt Law, PLLC [Exclusive] – MusicRow.com

    Rachel Guttman Unveils Exciting New Entertainment Law Firm: Gutt Law, PLLC!

    HYBE Cashes In: Offloads Final Stake in K-Pop Rival SM Entertainment for $177 Million!

  • General
  • Health
  • News

    Cracking the Code: Why China’s Economic Challenges Aren’t Shaking Markets, Unlike America’s” – Bloomberg

    Trump’s Narrow Window to Spread the Truth About Harris

    Trump’s Narrow Window to Spread the Truth About Harris

    Israel-Gaza war live updates: Hamas leader Ismail Haniyeh assassinated in Iran, group says

    Israel-Gaza war live updates: Hamas leader Ismail Haniyeh assassinated in Iran, group says

    PAP Boss to Niger Delta Youths, Stay Away from the Protest

    PAP Boss to Niger Delta Youths, Stay Away from the Protest

    Court Restricts Protests In Lagos To Freedom, Peace Park

    Court Restricts Protests In Lagos To Freedom, Peace Park

    Fans React to Jazz Jennings’ Inspiring Weight Loss Journey

    Fans React to Jazz Jennings’ Inspiring Weight Loss Journey

    Trending Tags

    • Trump Inauguration
    • United Stated
    • White House
    • Market Stories
    • Election Results
  • Science
  • Sports
  • Technology
    Bajeed Pattan Joins Forbes Technology Council as Innovation Leader – PRWeb

    Bajeed Pattan Takes the Helm as Innovation Leader at Forbes Technology Council!

    Lafayette Regional Technology Council – Tech Leadership That’s Homegrown and Future-Focused – Discover Lafayette

    Lafayette Regional Technology Council – Tech Leadership That’s Homegrown and Future-Focused – Discover Lafayette

    Drone technology demo in Cambria County showcases future of lifesaving medical deliveries – local21news.com

    Revolutionizing Healthcare: Drone Technology Takes Flight for Lifesaving Medical Deliveries in Cambria County

    Revolutionary Harvesting Technology Promises to Slash CAR-T Manufacturing Costs!

    Stop the Machines: The Rise of Anti-Technology Extremism – International Centre for Counter-Terrorism – ICCT

    Unplugged: The Surge of Anti-Technology Extremism

    Finland to head EU’s quantum defense technology project – Latest news from Azerbaijan

    Finland Takes the Lead in Pioneering EU’s Quantum Defense Technology Initiative!

    Trending Tags

    • Nintendo Switch
    • CES 2017
    • Playstation 4 Pro
    • Mark Zuckerberg
No Result
View All Result
Earth-News
No Result
View All Result
Home Science

Strangely Curved Shapes Break 50-Year-Old Geometry Conjecture

May 15, 2024
in Science
Strangely Curved Shapes Break 50-Year-Old Geometry Conjecture
Share on FacebookShare on Twitter

In an old Indian parable, six blind men each touch a different part of an elephant. They disagree about what the elephant must look like: Is it smooth or rough? Is it like a snake (so thinks the man touching the trunk) or a fan (as the man touching the ear proposes)? If the blind men had combined their insights, they might have been able to give a correct account of the nature of the elephant. Instead, they end up fighting.

For decades, topologists have hoped to avoid falling into a similar trap. They thought they could characterize mathematical shapes by synthesizing numerous local measurements. But newly discovered, paradoxically curved spaces show that this isn’t always possible. “Things can be much more wild than what we thought,” said Elia Bruè of Bocconi University in Italy, who worked with two other mathematicians to demonstrate this.

Topologists stretch and compress the shapes they study. An infinitely thin rubber band, from a topological perspective, is equivalent to a circle, because you can easily deform it into a circular shape. Topologists tend to characterize shapes according to their global properties: Do they have holes, like a doughnut? Do they go on forever, like an infinite plane, or are they “compact” like the surface of a sphere? Do their “straight” lines go on indefinitely — making them what mathematicians call “complete” — or are there dead ends?

But as with the elephant in the parable, it can be hard to directly perceive the global nature of topological shapes. And so mathematicians want to understand their relationship to local geometric properties, like curvature. What can you say about a shape’s global topology, given information about how it curves at every point?

In 1968, John Milnor, a renowned mathematician then at Princeton University, conjectured that an average sense of a complete shape’s curvature was enough to tell us that it couldn’t have infinitely many holes. For the next 50 years, many results supported his claim. “You were tempted to believe it was true, because it was true in so many realistic cases,” said Jeff Cheeger of the Courant Institute of Mathematical Sciences at New York University. “And how in God’s name could you construct a counterexample to it?”

In this area of mathematics, said Vitali Kapovitch of the University of Toronto, “the Milnor conjecture was probably the biggest open problem.”

And so in 2020, Bruè and two colleagues set out to prove it. They ended up finding a counterexample instead — and built an entirely new kind of topological shape in the process. “It’s fantastic work,” Cheeger said. “A landmark.”

A Holey Grail of Topology

To understand Milnor’s conjecture, it helps to first consider how topologists and geometers think about curvature.

Both study manifolds — spaces that look flat when you zoom in on them. A tiny ant on the surface of a sphere, doughnut or other two-dimensional manifold will perceive its immediate neighborhood to be no different from a two-dimensional plane. But if the ant moves a little bit in any direction, it might notice that the space begins to shift, or curve. The idea of a locally flat manifold generalizes easily into higher dimensions. But curvature is tougher to define.

Take, for example, the simplest case: a one-dimensional object such as a circle. Surprisingly, these one-dimensional spaces cannot, in a mathematical sense, be intrinsically curved. A one-dimensional geometer walking along a circle, unable to perceive more than one dimension, would think she was traveling in a straight line — and would be surprised to find herself retracing her steps.

But if you embed a circle in a two-dimensional plane, it’s apparent that it has constant, positive extrinsic curvature. (The relevant distinction here is between intrinsic and extrinsic curvature: what you can see if you’re stuck inside the space, versus what you can see from outside it.)

Smaller circles bend more quickly as you move around them, and therefore have higher extrinsic curvature; bigger circles have lower curvature. (A straight line, in this sense, is like an infinitely big circle. Its curvature is zero, indicating that it’s completely flat.) We can also apply this definition to more complicated shapes that have changing curvature, by considering how big a circle you would need to match the shape at any given point. In this way, curvature is a local property: Every point on a manifold has an associated curvature.

For a surface — a two-dimensional manifold — there are many ways to place circles so that they match the surface’s curves. At a given point, you can measure curvature in any direction by placing an appropriately sized circle in that direction. But, surprisingly, it’s possible to define the surface’s curvature at that point with just one number. If you find the directions that give you the biggest and smallest curvature values, and multiply those values together, you get a number called the Gaussian curvature. This number summarizes information about how the surface bends in a useful way. Even more surprisingly, the Gaussian curvature turns out to be an intrinsic property: It doesn’t depend on any higher-dimensional background space the surface might be placed into. In this sense, paradoxically, cylinders are not intrinsically curved, though spheres are.

>>> Read full article>>>
Copyright for syndicated content belongs to the linked Source : Quanta Magazine – https://www.quantamagazine.org/strangely-curved-shapes-break-50-year-old-geometry-conjecture-20240514/

Tags: curvedscienceStrangely
Previous Post

How to explore Promenade Plantée in Paris, France

Next Post

How to watch Stage 11 of the Giro as sprint beckons

John Hancock Multimanager Lifestyle Moderate Portfolio Q1 2025 Commentary (JALMX) – Seeking Alpha

John Hancock Multimanager Lifestyle Moderate Portfolio Q1 2025 Commentary (JALMX) – Seeking Alpha

June 1, 2025
Editorial: The world promised by AI isn’t necessarily a better one – Pittsburgh Post-Gazette

Editorial: The world promised by AI isn’t necessarily a better one – Pittsburgh Post-Gazette

June 1, 2025
Little Rock economy growing faster than other similarly-sized cities, study shows – thv11.com

Little Rock’s Economy Outpaces Peers: A Promising Growth Story!

June 1, 2025

Unveiling the Enigmatic: First Looks at Destruction and Puck in ‘The Sandman

June 1, 2025
Why Gen Z Will Demand Crypto-Enabled Health Systems – Forbes

How Gen Z is Shaping the Future of Crypto-Enabled Health Systems

June 1, 2025
State Sen. Skoufis declares war on en­ter­tainment ticketing practices in closing days of session – Spectrum News

State Senator Skoufis Takes a Stand Against Unfair Ticketing Practices in Final Session Push!

June 1, 2025
Bajeed Pattan Joins Forbes Technology Council as Innovation Leader – PRWeb

Bajeed Pattan Takes the Helm as Innovation Leader at Forbes Technology Council!

June 1, 2025
With college sports in limbo and key issues coming to a head, the spotlight is on the SEC: ‘It’s going to get heated’ – Yahoo Sports

SEC Showdown: Tensions Rise as College Sports Face Uncertain Future

June 1, 2025
Upgrades coming to ecological park in Sunbury – WNEP

Upgrades coming to ecological park in Sunbury – WNEP

June 1, 2025
This will ‘devastate scientific research’: 16 states sue the Trump administration over National Science Foundation directive – CNN

This will ‘devastate scientific research’: 16 states sue the Trump administration over National Science Foundation directive – CNN

June 1, 2025

Categories

Archives

June 2025
MTWTFSS
 1
2345678
9101112131415
16171819202122
23242526272829
30 
« May    
Earth-News.info

The Earth News is an independent English-language daily published Website from all around the World News

Browse by Category

  • Business (20,132)
  • Ecology (655)
  • Economy (671)
  • Entertainment (21,577)
  • General (15,254)
  • Health (9,713)
  • Lifestyle (673)
  • News (22,149)
  • People (671)
  • Politics (679)
  • Science (15,891)
  • Sports (21,175)
  • Technology (15,658)
  • World (659)

Recent News

John Hancock Multimanager Lifestyle Moderate Portfolio Q1 2025 Commentary (JALMX) – Seeking Alpha

John Hancock Multimanager Lifestyle Moderate Portfolio Q1 2025 Commentary (JALMX) – Seeking Alpha

June 1, 2025
Editorial: The world promised by AI isn’t necessarily a better one – Pittsburgh Post-Gazette

Editorial: The world promised by AI isn’t necessarily a better one – Pittsburgh Post-Gazette

June 1, 2025
  • About
  • Advertise
  • Privacy & Policy
  • Contact

© 2023 earth-news.info

No Result
View All Result

© 2023 earth-news.info

No Result
View All Result

© 2023 earth-news.info

Go to mobile version