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A New Mathematical Proof Shows How Some Spaces Can't Always Be Divided

The mathematics of leftovers.
​Image: Dmitry Dzhus/Flickr

​Consider ​Zeno's arrow. There it is, suspended just inches from my face, bearing down at almost the speed of light. The arrow halves the distance between its tip and the spot right between my eyes in just a few yoctoseconds, and then that half is even more quickly divided again. The halving continues as the arrow divides the distances between itself and my face again and again. … And again and again and again and again and again and again and again and again.

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Where does the halving stop and the face-splitting begin? Ignoring the IRL restriction of the Planck length, why should these divisions ever stop? And if they never stop, does that mean the arrow never reaches its destination? How does anything ever reach its destination? The arrow and everything else must be motionless.

That's some dorm room philosophizing for sure, but Zeno's paradox offers a way into the triangulation of spaces. Simply, can a space be divided up an infinite number of times? In other words, what would prevent that endless division from happening? In two and three dimensions, the mathematical answer is that nothing would prevent an endless division of space. Quantum physics would eventually prevent a literal divisioning, but as an abstract notion, we might just keep cutting. Forever.

What about other spaces? The fourth and fifth dimensions, the billionth dimension, and beyond. (Actually, it's already been disproved for the fourth, but not for anything higher.) Does the same basic idea hold true absolutely? The idea that, yes, it does is called the triangulation conjecture. It's never been proven, just intuitively assumed (as above), yet a mathematician at the University of California recently offered ​a proof of his own—showing that the conjecture isn't true.

In other words, some spaces can't be cut. There exists some final, uncuttable slice of extradimensional reality (a manifold, properly) that Zeno's arrow might leap to finally meet its target. The catch in thinking about the whole thing is that it's the very tool mathematicians use to visualize higher dimensions, triangulation, that's being disproved. Triangulation offers a continuity between our intuitive, friendly four dimensions (space-plus-time) and those completely abstract higher realms.

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Image: Harvard.edu

What we're really talking about is what's known as an invariant. In computer science, this is the part of the algorithm or program that remains true no matter what. Given any possible input and every possible program execution, this one thing will remain true always. Turns out something like this applies to mathematical spaces/dimensions via an invariant known as the Euler characteristic.

At Quanta, Kevin Hartnett ​explains it as such:

To find the Euler characteristic of a two-dimensional surface, first divide it into any number of polygons. Now count the number of vertices, subtract the number of edges, then add the number of faces. The resulting integer will come out the same no matter how many polygons you use to triangulate the manifold. The Euler characteristic of a sphere is 2; that of a torus is 0. In two dimensions, any two manifolds with the same Euler characteristic are topologically equivalent.

The problem with this division, at least among some manifolds, is that the first geometric divisions made don't necessarily match up with the last ones, like trying to match up two halves of two different zippers. There is space leftover, particularly as more and more dimensions are added and the situation becomes more complex This meshing, between the beginning of one space or dimension and the final boundary before the next dimension, isn't guaranteed.

In the most general terms, the invariant question has to do with the transformation of some space into another space. If we can bend and twist one manifold into another sort of manifold, we can say that it's topologically invariant. If we have to break it apart—introduce some discontinuity—than the invariant fails, which is bad. Find a topological property of some manifold that is not invariant and we've shown that the space is not ​homeomorphic, which is where we start to find our screwy boundaries.

Triangulation is a homeomorphism, a continuity between spaces. If a topological region isn't homeomorphic, the problem of applying the same division to each space starts to become clear as the boundaries between spaces become ugly.

The UC mathematician behind the latest proof, Ciprian Manolescu, based his argument on a whole new sort of invariant using what's called Floer homology, which involves the translation of unimaginable spaces to our regular old three-dimensional world via a "3-manifold." It's sort of like how the world seems flat from our perspective, but it's really this spinning sphere. So, we might have some space in some dimension that's a nightmarish knot of infinity, but we can look at it (and come up with invariants for it) as if it were some manageable element of our intuitive reality (like, well, a knot).

So, Manolescu came up with a whole new invariant, which he calls just "beta." Using beta, he was able to show a necessary discontinuity in some spaces, e.g. where things just didn't line up. As such, these spaces were to shown to be indivisible in the most complete sense. There is always something left over.