Elkement, who’s one of my longest blog-friends here, put forth this suggestion for an ‘E’ topic. It’s a good one. They’re author of the Theory and Practice of Trying to Combine Just Anything blog. Their blog has recently been exploring complex-valued numbers and how to represent rotations.
Consider a book. It’s a collection. It’s easy to see the ordered setting of words, maybe pictures, possibly numbers or even equations. The important thing is the ideas those all represent.
Set the book in a library. How can this change the book?
Perhaps the comparison to other books shows us something the original book neglected. Perhaps something in the original book we now realize was a brilliantly-presented insight. The way we appreciate the book may change.
What can’t change is the content of the original book. The words stay the same, in the same order. If it’s a physical book, the number of pages stays the same, as does the size of the page. The ideas expressed remain the same.
So now you understand embedding. It’s a broad concept, something that can have meaning for any mathematical structure. A structure here is a bunch of items and some things you can do with them. A group, for example, is a good structure to use with this sort of thing. So, for example, the integers and regular addition. This original structure’s embedded in another when everything in the original structure is in the new, and everything you can do with the original structure you can do in the new and get the same results. So, for example, the group you get by taking the integers and regular addition? That’s embedded in the group you get by taking the rational numbers and regular addition. 4 + 8 is 12 whether or not you consider 6.5 a topic fit for discussion. It’s an embedding that expands the set of elements, and that modifies the things you can do to match.
The group you get from the integers and addition is embedded in other things. For example, it’s embedded in the ring you get from the integers and regular addition and regular multiplication. 4 + 8 remains 12 whether or not you can multiply 4 by 8. This embedding doesn’t add any new elements, just new things you can do with them.
Once you have the name, you see embedding everywhere. When we first learn arithmetic we — I, anyway — learn it as adding whole numbers together. Then we embed that into whole numbers with addition and multiplication. And then the (nonnegative) rational numbers with addition and multiplication. At some point (I forget when) the negative numbers came in. So did the whole set of real numbers. Eventually the real numbers got embedded into the complex numbers. And the complex numbers got embedded into the quaternions, although we found real and complex numbers enough for most of our work. I imagine something similar goes on these days.
There’s never only one embedding possible. Consider, for example, two-dimensional geometry, the shapes of figures on a sheet of paper. It’s easy to put that in three dimensions, by setting the paper on the floor, and expand it by drawing in chalk on the wall. Or you can set the paper on the wall, and extend its figures by drawing in chalk on the floor. Or set the paper at an angle to the floor. What you use depends on what’s most convenient. And that can be driven by laziness. It’s easy to match, say, the point in two dimensions at coordinates (3, 4) with the point in three dimensions at coordinates (3, 4, 0), even though (0, 3, 4) or (4, 0, 3) are as valid.
Why embed something in another thing? For the same reasons we do any transformation in mathematics. One is that we figure to embed the thing we’re working on into something easier to deal with. A famous example of this is the Nash embedding theorem. It describes when certain manifolds can be embedded into something that looks like normal space. And that’s useful because it can turn nonlinear partial differential equations — the most insufferable equations — into something solvable.
Another good reason, though, is the one implicit in that early arithmetic education. We started with whole-numbers-with-addition. And then we added the new operation of multiplication. And then new elements, like fractions and negative numbers. If we follow this trail we get to some abstract, tricky structures like octonions. But by small steps in which we have great experience guiding us into new territories.
I hope to return in a week with a fresh A-to-Z essay. This week’s essay, and all the essays for the Little Mathematics A-to-Z, should be at this link. And all of this year’s essays, and all A-to-Z essays from past years, should be at this link. Thank you once more for reading.