This post should give you the feeling of bijecting between domains without knowing a lot of mathematics. Which is part of getting the intuitive feeling of mathematics with less work.
Besides automorphisms, there’s another interesting kind of bijection. I’ll try to give you the feeling of bijecting between different domains (a kind of analogy) without requiring much prior knowledge.
Like I said yesterday, a bijection is an invertible total mapping. It ≥ covers ↓ the target and ≤ injects ↑ one-to-one into the target. This is thinking of spaces as wholes—deductive thinking—rather than example-by-example thinking. (There’s a joke about an engineer and a mathematician who are friends and go to a talk about 47-dimensional geometry. The engineer after the talk tells the mathematician friend that it was hard to visualise 47 dimensions; how did you do it? The mathematician replies “Oh, it’s easy. I simply considered the problem in arbitrary
N dimensions and then set
N=47!” I used to be frustrated by this way of thinking but after X years, it finally makes sense and is better for some things.)
So, graphically, a bijection is surjective/covering/ ≥ / ↓
and injective/one-to-one (not one-to-many)/ ≤ / ↑
This amounts to a mathematical way of saying two things are “the same”, when of course there are a lot of ways in which that could be meant. The equation
x=x is the least interesting one so “sameness” has to be more broad than “literally the same”. The same like how? Bijection as a concept opens the door to ≥1 kinds of comparison.
That was a definition. Now on to the example which should give you the feeling of bijecting across domains and the feeling of payoff after you come up with an unintuitive bijection.
Let’s talk about an “ideal city” where the streets make a perfectly rectangular lattice. I’m standing at 53rd St & 140th Ave and I want to walk/bike/cab to 60th St & 147th Ave.
How many short ways can I take to get there?
The first abstraction I would do from real life to a drawing is to centre the data. A common theme in statistics and mathematically it’s like removing the origin. I can actually ignore everything except the 7×7 block between me and my destination to the northeast.
(By the way, by “short” paths I mean not circling around any more than necessary. Obviously I could take infinitely more and more circuitous routes to the point of circling the Earth 10 times before I get there. But I’m trying not to go out of my way here.)
Now the problem looks smaller. Just go from bottom left corner to top right corner.
I drew one shortest path in red and two others in black. To me it would be boring to go north, north, north, north, north, north, north, east, east, east, east, east, east, east. But if I want to count all the ways of making snakey red-like paths then I should bracket the possibilities by those two black ones.
When I try to draw or mentally imagine all the snakey paths, I lose track—looking for patterns (like permute, then anti-inner-permute, but also pro-inner-anti-inner-inner-permute…these are words I make up to myself) that I probably could see if I understood the fundamental theorem of combinatorics, but I’ve never been able to fully see the additive pattern.
But, I know a shortcut. This is where the bijection comes in.
Every one of these paths is isomorphic to a rearrangement of the letters
Every time I “flip” one of the corners in the picture—which is how I was creating new snakes in between the black brackets—that’s just like interchanging an
N and an
Of course! It’s so obvious in hindsight.
And now here’s the payoff. Rearrangements of strings of letters like
AAABBBCCCCD are already a solved problem.
I explained how to count combinatorial rearrangements of letters here. It’s 1026 words long.
The way to get the following formula is to  derive a trick for over-counting,  over-count and then  quotient using the same trick.
- since the rearrangements of
AAAAAAABBBBBBBare isomorphic to the rearrangements of
- and since the rearrangements of
NNNNNNNEEEEEEEare isomorphic to the short paths I could take through the city to my destination,
the correct answer to my original question—how many short ways to go 7 blocks east and 7 blocks north—is
I asked the Berkeley Calculator the answer to that one and it told me 3432. Kind of glad I didn’t count those out by hand.
So, the payoff came from (1) knowing some other solved problem and (2) bijecting my problem onto the one with the known solution method.
But does it work in New York? Even though NYC is kind of like a square lattice, there may be a huge building making some of the blocks not accessible.
Or maybe ∃ a “Central Park” where you can cut a diagonal path.
And things like “Broadway” that cut diagonally across the city.
And some dead ends in certain ranges of the ciudad. And places called The Flat Iron Building where roads meet in a sharp V.
So my clever discovery doesn’t quite work in a non-square world.
However now I maybe also gave you a microcosm of mathematical modelling. The barriers and the shortcuts could be added to a computer program that counts the paths. We could keep adjusting things and adding more bits of reality and make the computer calculate the difference. But the “basic insight”, I feel, is lacking there. After all I could have written a computer program to permute the letters
NNNNNNNEEEEEEE or even just literally model the paths in the first place. (At least with such a small problem.) But then there would be no Eureka moment. I think it’s in this sort of way that mathematicians mean their world is more beautiful than the real one.
As mathematical modellers we inherit deep basic insights—like the Poisson process and the Gaussian as two limits of a binary branching process—and try to construct a convoluted sculpture using those profound insights as the basis. For example maybe I could stitch together a bunch of square lattice pieces together. Maybe for instance two square lattices representing different boroughs and connected only by a single congested bridge. Since I solved the square lattice analytically, the computational extensions will be less mysterious to me if I use the understood pieces. Unless I can be smart enough to figure out how to count triangles & multi-block industrial buildings & shortcuts & construction roadblocks and find an equally excellent insight into how the various discrepancies change the number at the end of my computation (rather than just reading it off and having an answer but no wisdom), I’m left using the excellent insight as a starting point and doing some dirty computations from there—no wisdom at all, no map, just scrapping in the wilderness—a lot of firepower and no idea how to use it. I might as well be spraying a tree with a shotgun instead of cutting the V with an axe and letting its weight do the work.