• Email Us: [email protected]
  • Contact Us: +1 718 874 1545
  • Skip to main content
  • Skip to primary sidebar

Medical Market Report

  • Home
  • All Reports
  • About Us
  • Contact Us

Too Many Balls Will Cause A Sausage Catastrophe, And Now There’s Finally Proof

January 25, 2024 by Deborah Bloomfield

Big news, everyone: the sausage catastrophe has been reproduced with tiny balls.

Look. It’s a little-known fact in mathematics, but the more difficult your theorem is to prove, the sillier its name has to be. Take, for example, the fact that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres – or to use its common name, the hairy ball theorem. Or, hey, need an algorithm to determine whether a set of sections provides a basis for the Mordell–Weil abelian group of a given elliptic surface? Why not try the Cox-Zucker machine! And let’s not even start on all the theorems and concepts named after Tits.

Advertisement

We tell you this only to make it clear that despite sounding like the kind of weird punk band your parents used to listen to in 1976, Sphere Packing and the Sausage Catastrophe are actually a big deal in math circles – and a recent paper confirming a long-suspected result on the problem is genuinely impressive and important.

Indeed, “the theory of infinite packings of convex bodies, in particular, lattice packings of spheres is a fundamental and classical topic in mathematics,” wrote mathematicians Martin Henk and Jörg Wills, who were not involved in this new result, back in 2020. 

“[It] plays a role in various branches of mathematics [such] as number theory, group theory, geometry of numbers, algebra, and [it] has numerous applications to coding theory, cryptography, crystallography and more,” they noted.

So, what exactly is this problem that has such wide applicability? Well, in a word: balls.

Advertisement

More specifically, it’s this: what is the most efficient way to pack balls? It sounds like an easy question – all the hardest ones do – but it has evaded a definite answer for centuries. Even the solutions we do have seem weirdly overcomplicated: for instance, we’ve known since 2005, and been pretty sure since 1611, that the best way in three dimensions is to use the “cannonball” method – yes, like how pirates used to do – but only if you have an infinite number of them. Which, let’s face it, even the saltiest of seadogs did not.

Unfortunately, that’s the kind of nonsense you end up with when you let mathematicians tackle a simple problem – which is why this new result came from a group of physicists, instead. 

“In reality, however, all packings are inherently finite, which means that their extension is limited in space,” the researchers write (classic non-mathematical starting point there). “This raises the question of what is the most efficient way to pack equally-sized spheres in either a container with a predefined shape, or inside a flexible container like the smallest convex hull that encloses the spheres.”

And unlike Kepler’s clusters of cannonballs, the answer is totally unintuitive: it’s a sausage. At least, at first.

Advertisement

“One of my students observed a linear packing, but it was quite puzzling,” Hanumantha Rao Vutukuri, assistant professor in experimental physics at the University of Twente, told New Scientist. His team hadn’t even been thinking about this Keplerian puzzler at the time; they were experimenting with putting spherical nanoscale particles inside microscopic containers known as vesicles.

“We thought that there was some fluke, so he repeated it a couple of times and every time he observed similar results,” Vutukuri said. “I was wondering, ‘why is this happening?’”

Now, was it unexpected? Yes. But was it completely unpredictable? No – and in fact, it was predicted, by mathematician Fejes Tóth all the way back in 1975. Proving it, however, was a different story – and the team ran up against some challenges pretty early on in their experiments.

“The vesicles kept rupturing with more than nine particles,” said Marjolein Dijkstra, Professor of Soft Condensed Matter at Utrecht University, in a statement. “This prevented us from testing how the stacking of particles would change if we added more than nine.”

Advertisement

Luckily, we have something Kepler never had: computers. Faced with these burst vesicles, the researchers moved instead to computer simulations of the problem, investigating the most efficient way to pack now up to 150 balls.

And it’s a good thing they went so high. Had they stopped at 55, they would only have obtained half an answer – because at 56, something catastrophic happened.

Literally. It’s called the “sausage catastrophe.” It’s when “a sudden transition in the packing density occurs from a linear to a cluster arrangement, where the coordinates of the particles extend in all three dimensions,” the paper explains – in other words, all of a sudden, it becomes more efficient to make a smush of balls, like Kepler did, than a line.

That, ultimately, is why we should care about this weird problem. Not just because it’s fun and provides closure to Kepler’s pirate friends, but because it shows that experiments really can have an important place in math.

Advertisement

“Our systematic investigation of these clusters with various shapes has allowed us to directly prove, with a practical approach inspired by the physics of colloids, the existence of previously unidentified clusters that exhibit a packing efficiency that is superior to the sausage configuration,” the team writes. “Nevertheless, it remains to be determined whether mathematical proofs can be developed for the packing of these clusters and for the entire Fejes Tóth conjecture.” 

“Finite sphere packing is still an open and intriguing problem,” they conclude. “We believe that our work can serve as a catalyst for further research in this direction.”

The paper is published in the journal Nature Communications.

Deborah Bloomfield
Deborah Bloomfield

Related posts:

  1. Tennis-Scrappy Sakkari survives gruelling three-setter to beat Andreescu
  2. Cricket-NZ players reach Dubai after ‘specific, credible threat’ derailed Pakistan tour
  3. Vatican trial prosecutors concede case gaps, willing to investigate more
  4. The Scottish Mummy That Turned Out To Be Made Of Three People

Source Link: Too Many Balls Will Cause A Sausage Catastrophe, And Now There's Finally Proof

Filed Under: News

Primary Sidebar

  • Bright Northern Lights Across America Expected This Week As 3 Coronal Mass Ejections Fly Towards Earth
  • Brain Implant Enables Paralyzed Man To Feel And Use Objects Using Someone Else’s Hands
  • “This Is A Really Big Deal”: Brain Training Significantly Improves Key Neurochemical Levels In World First
  • “Wholly Unexpected”: First-Ever Fossil Paranthropus Hand Raises Questions About Earliest Tool Makers’ Identity
  • For Centuries, Nobody Knew Why Swiss Cheese Has Holes. Then, The Mystery Was Solved.
  • Scientists Studied The Infamous “Chicago Rat Hole” And They Have Some Bad News
  • Massive 166-Million-Year-Old Sauropod Footprints Become The Longest Dinosaur Trackway In Europe
  • Do Spiders Dream? “After Watching Hundreds Of Spiders, There Is No Doubt In My Mind”
  • IFLScience Meets: ESA Astronaut Rosemary Coogan On Astronaut Training And The Future Of Space Exploration
  • What’s So Weird About The Methuselah Star, The Oldest We’ve Found In The Universe?
  • Why Does Red Wine Give Me A Headache? Many Scientists Blame It On The Grape Skins
  • Manta Rays Dive Way Deeper Than We Thought – Up To 1.2 Kilometers – To Explore The Seas
  • Prof Brian Cox Explains What He Finds “Remarkable” About Interstellar Object 3I/ATLAS Story
  • Pioneering “Pregnancy Test” Could Identify Hormones In Skeletons Over 1,000 Years Old
  • The First Neolithic Self-Portrait? Stony Human Face Emerges In 12,000-Year-Old Ruins At Karahan Tepe
  • Women Are Diagnosed With ADHD 5 Years Later Than Men, Even With Worse Symptoms
  • What Is Cryptozoology? We Explore The History And Mystery Of This Controversial Field
  • The Universe’s “Red Sky Paradox” Just Got Darker: Most Stars Might Never Host Observers
  • Uranus And Neptune May Not Be “Ice Giants” But The Solar System’s First “Rocky Giants”
  • COVID-19 Can Alter Sperm And Affect Brain Development In Offspring, Causing Anxious Behavior
  • Business
  • Health
  • News
  • Science
  • Technology
  • +1 718 874 1545
  • +91 78878 22626
  • [email protected]
Office Address
Prudour Pvt. Ltd. 420 Lexington Avenue Suite 300 New York City, NY 10170.

Powered by Prudour Network

Copyrights © 2025 · Medical Market Report. All Rights Reserved.

Go to mobile version