'Maths Without Numbers' by Milo Beckman
'Before you go tell your loved ones that you read a book about math and learned that a square is a circle, keep in mind: Context matters. A square is a circle, in topology. A square is most certainly not a circle in art or architecture, or in everyday conversation, or even in geometry, and if you try to ride a bike with square tires you won't get far.' pp.7-8.
'Like a line:
(Illustration - pen drawing of a line, a 'C' and an almost closed circle.)
A line can be bent almost into a circle, but to finish the job we'd need to click the ends together--not allowed. No matter how you manipulate a line, you'll always have those two special points on either end, where the shape just stops. You can't get rid of end-points. You can move them around and stretch them apart, but the two end-points are an unchanging feature of the shape.
For a similar reason, a figure-eight is a different shape too. There aren't any end-points, but there's still a special point in the middle where the lines cross, where there are four arms reaching out instead of the usual two at any other point. Stretch and squeeze all you want, you can't get rid of a crossing-point either. p.9.
'The circle (aka S-one) and the infinite line (named R-one) are the only manifolds in the first dimension. To avoid end-points, you either have to loop back around or just go on and on forever. And don't forget: Because all the shapes in topology are stretchy, this also covers any closed-loop shape and any goes-on-forever shape. It doesn't have to be literally a circle or a straight line.' p.16.
Math Without Numbers
The third dimension, dough-type manifolds, is pretty well understood at this point, though it took a hundred years and a million-dollar prize to get there, and we still don't have a totally neat and clean classification like the lower dimension. In dimensions five and up, topologists use a set of techniques called "surgery theory" to operate on manifolds and construct new ones.
That just leaves dimension four.
I wish I could tell you what's going on in dimension four. I'm not sure there's anyone who really knows. It's a weird boundary case: too many dimensions to do visually, but not enough to use sophisticated surgery tools. There are entire textbooks dedicated to what little we know about four-manifolds, and I couldn't make sense of anything past the opening pages. A professional topologist once told me she'd wanted to work on four-manifolds as an undergraduate but was advised to steer clear.' pp.23-24."Like a line" ... Yes. Take two that cross. Then the universes open up.
Milo Beckman (2021) Math Without Numbers. London, Penguin Books. Illustrated by M. Erazo.
Previously: 'surgery'
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