Back in 1989, the Canadian mathematician Alexander Dewdney published a letter in Scientific American, describing a discovery so world-changing I’m astonished how little attention it's received in the years since. Dewdney explained that the letter had been sent to him by a friend. It detailed how this friend had been travelling in a South American country – he wouldn't say which one – when he happened across a secretive money-making operation, deep in the forest. I don’t mean a counterfeiting operation: the gang discovered by Dewdney’s friend was literally creating wealth from nothing. They were exploiting a remarkable finding in geometry called the Banach-Tarski paradox, which states that, given any solid sphere in three-dimensional space, there is a way of cutting up the sphere so that you can reassemble it into two spheres, each identical to the first one.

No, that’s really what it says! The ingenious South Americans were cutting up gold balls, then reassembling them into a greater number of identical gold balls. And then cutting up those gold balls, and reassembling them, and so on.

I was reminded of this shocking finding (tell me again why I've bothered working for a living, all these years, when I could have been cutting up gold?) this week, when BoingBoing republished the charming video below. In it, the science writer Mario Tomatis demonstrates how to cut up a slab of chocolate, then reassemble it so that you get more chocolate than before. He’s exploiting a different mathematical anomaly, the so-called Curry Paradox – but with equally amazing results. Take a look:

Anyone with remaining doubts should scrutinise this neat animation of the Curry Paradox by Michael Bach, from which the images below are taken. The coloured tiles are identical in each image, but in the second version, the missing square has been filled in. I don’t think I really need to spell out the implications for the debt ceiling struggle, national deficits, austerity cutbacks, or global poverty. But put it this way: given the right cutting apparatus, we could do this with gold as easily as with chocolate.

In this configuration, there's a missing square... Photograph: www.michaelbach.de Photograph: www.michaelbach.de

...but in this version it's been filled in. Photograph: www.michaelbach.de Photograph: www.michaelbach.de

Now, there are always naysayers, professional pessimists who make it their business to mock or disdain the revolutionary thinking of others. For example, there are people who point out that the Banach-Tarski paradox applies only to massless mathematical spheres, not spheres of real matter. There are those who argue that Curry’s Paradox isn't really a paradox but an illusion, because it depends on the fact that neither of the hypotenuses of those triangles are straight lines. (One bows out and one bows in, which accounts for the “missing” space.) There are even those who think we shouldn’t take Dewdney’s South American anecdote seriously because it was printed in the April edition of Scientific American, and his friend’s name was given as Arlo Lipof.

But come on! Desperate times call for innovative thinking. Washington is paralysed. People are suffering. It’s time to think outside the box!

Or just cut the box up into pieces, then rearrange it into a bigger one.