mathematical aesthetics
this is a big issue: "elegance"
There are *different* aesthetics / components
*structure/patterns*,
and *underlying* patterns
*gets to the heart*
slickness/surprise: *unexpected* structure
Oh!
"ordinary" science (and math) is dominated by "follow known structure"
...which is weirdly at odds with some of the attraction of math
(and the answers tend to be "work hard/chug away")
[
Importance of aesthetics:
why do we care?
 ethics/value: what is worthwhile math?
what should we (personally) do / remember / teach / fund?

]
. enumeration
. specific generic objects boring
(but *statements about* generic objects *are* interesting!
e.g., generically nowhere differentiable)
. AIT says that many results will be *ugly*
(AIT itself is pretty though)
. computers like ugly proofs!
*surprise* is a good feature; *mystery* isn't
it's ok to be unexpected, but not *unexplained*:
in *retrospect*, you should be able to explain / understand
Mystery:
see "I hate elementary proofs"
surprise:
EG of mean time to get to other vertex on a cube:
pretty b/c:
 for most graphs, this question is ugly
(indeed, this is the largest natural graph for which this answer works)
...and it *doesn't* generalize to higher dimensions
 the structure/symmetry that makes it work is *surprising*
 connection to decay is cute if you're not expecting it
(though you actually should: this is a general principle)
so: there's a hidden sporadic symmetry,
and a nice general pattern
A more significant example is the brachistochrone:
a priori variational questions seem impossible:
the space of possible curves is immense!
The point is that a solution must satisfy specific constraints
which make it solveable.
(NB: tautochrone is easier, b/c not minimizing)
The fact that the answer has a nice form
suggests that it can be seen for a nice reason.
EG of 4color theorem
Compare with "embedability in plane/torus/etc.":
these also come down to enumerations (if you don't contain these...)
...but embed in proj plane has a long list, iirc.
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Value apart from aesthetics:
 applicability / usefulness
 difficulty
(again, w/o applicable or elegance, not so interesting:
can be 1000 pages of equations)
 "depth"
(somewhat discredited: there are elementary proofs of prime number, say)
...but does capture something beyond mearly historical:
a level of sophistication (further removed from naive concerns)
oh!
deepbutnotapplicable is "airy abstraction"
(v. interesting within field, not outside)
deepandcentral is a motherload:
v.important *and not accessible before*