It can be tempting to assume that your intuitions about three-dimensional space carry over to higher-dimensional realms. After all, adding another dimension simply creates a new direction to move around in. It doesn’t change the defining features of space: its endlessness and its uniformity.

But different dimensions have decidedly different personalities. In dimensions 8 and 24, it’s possible to pack balls together especially tightly. In other dimensions, there are “exotic” spheres that look irremediably crumpled. And dimension 3 is the only one that can contain knots — in any higher dimension, you can untangle a knot even while holding its ends fast.

Now, mathematicians have put the finishing touches on a story of dimensional weirdness that has been 65 years in the making. For many decades, researchers have wanted to know which dimensions can host particularly strange shapes — ones so twisted that they cannot be converted into a sphere through a simple procedure called surgery. The existence of these shapes, mathematicians have shown, is intimately intertwined with fundamental questions in topology about the relationships between spheres of different dimensions.

Over the years, mathematicians found that the twisted shapes exist in dimensions 2, 6, 14, 30 and 62. They also showed that such shapes could not possibly exist in any other dimension — save one. No one could determine the status of dimension 126.

Three mathematicians have now settled this final problem. In a paper posted online last December, Weinan Lin and Guozhen Wang of Fudan University in Shanghai, along with Zhouli Xu of the University of California, Los Angeles, proved that 126 is indeed one of the rare dimensions that can host these strangely twisted shapes.

It’s “a very long program, finally finished,” said Ulrike Tillmann of the University of Oxford.

The proof, which use