Posts Tagged ‘fourth dimension’
“Topology is precisely the mathematical discipline that allows the passage from local to global”*…
Jordana Cepelewicz on two new topographical results that bring some order to the confoundingly difficult study of four-dimensional shapes…
The central objects of study in topology are spaces called manifolds, which look flat when you zoom in on them. The surface of a sphere, for instance, is a two-dimensional manifold. Topologists understand such two-dimensional manifolds very well. And they have developed tools that let them make sense of three-dimensional manifolds and those with five or more dimensions.
But in four dimensions, “everything goes a bit crazy,” said Sam Hughes, a postdoctoral researcher at the University of Oxford. Tools stop working; exotic behavior emerges. As Tom Mrowka of the Massachusetts Institute of Technology explained, “There’s just enough room to have interesting phenomena, but not so much room that they fall apart.”
In the early 1990s, Mrowka and Peter Kronheimer of Harvard University were studying how two-dimensional surfaces can be embedded within four-dimensional manifolds. They developed new techniques to characterize these surfaces, allowing them to gain crucial insights into the otherwise inaccessible structure of four-dimensional manifolds. Their findings suggested that the members of a broad class of surfaces all slice through their parent manifold in a relatively simple way, leaving a fundamental property unchanged. But nobody could prove this was always true.
In February, together with Daniel Ruberman of Brandeis University, Hughes constructed a sequence of counterexamples — “crazy” two-dimensional surfaces that dissect their parent manifolds in ways that mathematicians had believed to be impossible. The counterexamples show that four-dimensional manifolds are even more remarkably diverse than mathematicians in earlier decades had realized. “It’s really a beautiful paper,” Mrowka said. “I just keep looking at it. There’s lots of delicious little things there.”
Late last year, Ruberman helped organize a conference that created a new list of the most significant open problems in low-dimensional topology. In preparing for it, he looked at a previous list of important unsolved topological problems from 1997. It included a question that Kronheimer had posed based on his work with Mrowka. “It was in there, and I think it was a little bit forgotten,” Ruberman said. Now he thought he could answer it…
Read on for the details: “Mathematicians Marvel at ‘Crazy’ Cuts Through Four Dimensions,” from @jordanacep in @QuantaMagazine.
* Rene Thom
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As we savor surprising shapes, we might send carefully-modeled birthday greetings to William Bowie; he was born on this date in 1872. A geodetic engineer who joined the United States Coast and Geodetic Survey in 1895, he investigated isostasy (a principle that dense crustal rocks to tend cause topographic depressions and light crustal rocks cause topographic elevations).
Bowie was the first President of the American Geophysical Union from 1920 to 1922 and served as president a second time from 1929 to 1932. The William Bowie Medal, the highest honor of the AGU, is named in his honor.
“Once we introduce the possibility of applying the quantum principle to the universe, we are forced to consider parallel universes”*…
In the Antarctic, things happen at a glacial pace. Just ask Peter Gorham. For a month at a time, he and his colleagues would watch a giant balloon carrying a collection of antennas float high above the ice, scanning over a million square kilometres of the frozen landscape for evidence of high-energy particles arriving from space.
When the experiment returned to the ground after its first flight, it had nothing to show for itself, bar the odd flash of background noise. It was the same story after the second flight more than a year later.
While the balloon was in the sky for the third time, the researchers decided to go over the past data again, particularly those signals dismissed as noise. It was lucky they did. Examined more carefully, one signal seemed to be the signature of a high-energy particle. But it wasn’t what they were looking for. Moreover, it seemed impossible. Rather than bearing down from above, this particle was exploding out of the ground.
That strange finding was made in 2016. Since then, all sorts of suggestions rooted in known physics have been put forward to account for the perplexing signal, and all have been ruled out. What’s left is shocking in its implications. Explaining this signal requires the existence of a topsy-turvy universe created in the same big bang as our own and existing in parallel with it. In this mirror world, positive is negative, left is right and time runs backwards. It is perhaps the most mind-melting idea ever to have emerged from the Antarctic ice – but it might just be true…
Strange particles observed by an experiment in Antarctica could be evidence of an alternative reality where everything is upside down: “We may have spotted a parallel universe going backwards in time.”
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As we consider our alternatives, we might recall that it was on this date in 1912, in his “Manuscript on the Special Theory of Relativity,” that Einstein first identified the fourth dimension as time… or so it is widely accepted. Some physicists believe that Einstein was making a subtler– and much more complicated– suggestion, “x4 = ict”: that the fourth dimension, not “physical” like the other three, but emergent (in a way “understandable” as time) as the fourth dimension expands from the other three at the rate of “c.”
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