Possible practical applications in pure mathematics?

The place we’ve rented in York is unfurnished so we’ve had to find things to sit on, lie on and eat from. Most of this stuff has come from second-hand furniture shops. In one place, I found a cool old trunk which I immediately decided should be our coffee table/toy box. Here it is:

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The sofa in this photo was free!

I love the trunk. It’s big, spacious inside and very strong and sturdy. The only problem is that it’s kind of grungy inside and not the sort of place you would really want to store children’s toys.

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So I bought this clean, bright piece of fabric with the idea of lining the inside of the trunk.

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But how do you turn a one-dimensional piece of fabric into a three-dimension box-shape? I asked my mathematician husband for ideas. He told me that what I wanted was a “conformal area-preserving bijection from the piece of fabric to the inside surface of the suitcase”. WTF! He then told me that he could probably prove that one existed but that he wouldn’t be able to find it for me. If ever there was an example of the uselessness of pure mathematics then this is surely it 😉

I decided to use my favourite problem-solving strategy – trial and error – and this is what I got:

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