
What Is the Most Surprising Result in Mathematics? - luu
http://math.stackexchange.com/questions/2949/which-one-result-in-maths-has-surprised-you-the-most
======
ColinWright
Here are some I like:

* Bi-color the edges of a (countably) infinite complete graph. There is a mono-chromatic (countably) infinite complete induced sub-graph.

* Between any two rationals there are uncountably many irrationals, and between any two irrationals there are only countably many rationals.

* There is an extension of Lesbesgue measure on R^2 that is isometry invariant, finitely additive, and defined for all subsets. (This can't be extended to R^3 - that's the real point of Banach-Tarksi)

* There is a 3D object with infinite surface area but finite volume. You can't paint the surface, but you can fill it with paint, then empty it out.

* The 3D version of the Jordan–Schönflies Theorem (an "obvious" strengthening of the Jordan Curve Theorem) is wrong.

And just for fun:

* 2=4 (See [http://www.solipsys.co.uk/new/TwoEqualsFour.html](http://www.solipsys.co.uk/new/TwoEqualsFour.html))

 _(Added in edit: I suspect this will shortly fall foul of PG 's auto-
detection of flame wars - it has many more comments than points)_

~~~
evilduck
Not a mathematician, but "uncountably many" sounds like a loaded phrase unique
to the field of mathematics. How does "uncountably many" differ from
"infinitely many"? Are we not sure that it goes on forever or do we know that
it's just an extremely large finite set?

~~~
ColinWright
It is a precise term with a precise meaning. You can look it up and found
many, many explanations online, some of which will be right, few of which will
be truly helpful.

Let me add to them.

If you can put a set into one-to-one correspondence with the counting number
(which are 1, 2, 3, and so on) then we call the set "countable" or "countably
infinite". Examples include (but are not limited to):

* The even counting numbers: 2, 4, 6, 8, 10, and so on;

* The integer squares: 0, 1, 4, 9, 16, 25, and so on;

* The primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, and so on;

* The rational numbers (fractions);

* Finite strings made from a finite collection of symbols;

... and more. You may choose to think about how matchings can be made between
each of these and the counting numbers, or each with the others - some
ingenuity required.

However, there are infinite collections that _cannot_ be put into one-to-one
correspondence with the counting numbers. Examples include:

* Infinite non-terminating strings of 0s and 1s;

* The reals;

* All sets (finite and infinite) of counting numbers;

... and more.

Such infinite collections, since they cannot be matched up with the counting
numbers, are called "Uncountably Infinite Sets".

To prove that such collections can never be matched up with the counting
numbers is a typical exercise for 1st year undergrads in mathematics, or for
precocious mathematics enthusiasts. I first met it when I was about 8 years
old and reading Martin Gardner. The usual proof is Cantor's "Diagonalisation
Argument"[0][1], and you can find a version of that here:

[http://www.solipsys.co.uk/new/CantorVisitsHilbertsHotel.html](http://www.solipsys.co.uk/new/CantorVisitsHilbertsHotel.html)

That includes a genuine proof of the existence of a set that is infinite, and
cannot be put in one-to-one correspondence with the counting numbers.

Still, many people find that unconvincing. I personally prefer Cantor's first
argument that there exist transcendental numbers, which is equivalent. Here it
is, adapted to show that any list of numbers is incomplete. You'll need to
concentrate:

Take any list of numbers: c_0, c_1, c_2, _etc._

Take any interval [a0,b0], and cross out numbers from your list until you find
one, say, c_i, in the interval (including an endpoint). Take some strict sub-
interval [a1,b1] within [a0,b0] such that a0<a1, b1<b0, and c_i is not in
[a1,b1]. Now continue crossing off numbers, and each time you find one in your
current interval, take a sub-interval to exclude it.

The left hand end-points, a0, a1, a2, _etc,_ form a strictly increasing
sequence of points. This sequence is bounded above, so it has an upper bound.
Consider the least upper bound.

This cannot be on your original list, because everything was, at some point,
excluded. Hence your original list cannot be complete.

And in case you think I can only ever construct one missing number, I can
actually construct infinitely many missing numbers. At each stage instead of
taking just one sub-interval, I take _two_ subintervals, each excluding the
number from your list.

I'm intending to write all this up soon, so if you're interested, I can let
you read a draft as a way of improving your understanding, and giving me an
attentive proof-reader.

[0] Not to be confused with the diagonal path in the argument that the
fractions are countable.

[1]
[http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument](http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument)

------
btilly
My answer would be that there are existence proofs for things that we cannot
produce examples of. (There are actually many such proofs, but at some point I
stopped being shocked.) And furthermore there exist existence proofs for
classes of things, at least some of the examples of which we can NEVER verify
without adding new axioms to our axiom system!

Let me give a concrete example. A minor of a graph is any graph that you can
get from the first by deleting points, deleting edges, or contracting points
and joining edges. See
[https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo...](https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theorem)
for a theorem which says that any family of graphs that is closed under taking
minors (ie if G is in the family, then every minor of G is as well) can be
characterized by a finite set of forbidden minors. Any graph without a minor
in the forbidden set, is in the family.

The best known example of such a family is planar graphs. Those are graphs
that you can draw in the plane without crossings. The forbidden minors are K_5
(5 points, all connected to each other) and K_3_3 (2 groups of 3 points, every
point in each group connects to the every point in the other). Any graph
without either of those as minors, is planar.

So, given any family of graphs that is closed under taking minors, we must
have a concrete set of forbidden minors. Furthermore it turns out that given a
finite set of forbidden minors, we can use them create a polynomial-time
algorithm for recognizing whether a given graph is in the family.

Here is where it gets strange.

Given a family of graphs, we do not have a procedure to produce the forbidden
minors. Such a procedure would be wonderful - it would let us do interesting
things like solve the Halting problem! That is, of course, an impossible thing
to do. This proves that there is, in fact, no such procedure. So we have
proved existence of finite, concrete lists, yet proved that in general it is
impossible to construct said lists.

It gets worse. There are many interesting families of graphs which are closed
under minors, many of which we do not have any decision procedure for at all!
(For example graphs embeddable in 3-dimensions without anywhere embedding a
knot.)

So we've proven the existence of finite answers to concrete mathematical
problems. Yet cannot produce those answers. In general we can NEVER produce
those answers. Yet we know that they exist.

Now for the philosophy question: in what sense do those unfindable,
unverifiable answers actually exist?

~~~
tel
This is closely related to proof by the probabilistic method, which shows up
quite often in pattern recognition literature and combinatorics (Erdös was
fond of it). This method says that instead of even proving the exact existence
of something (despite missing any examples) you simply prove that the
probability of its existence is positive (and thus cannot be zero).

At some level these are just proofs that lower-bound the size of the set of
instances, and it may be possible to convert any such argument in to one that
is more directly measuring the count of instances. In practice however they
tend to go through using bounds which are derived from measurable to arguments
on random variables, so the conversion is less obvious to me.

I wish my copy of DGL wasn't 2 hours away or I'd dig up one.

(Edit: Wikipedia to the rescue! The "second example" on this page goes through
using Markov's Bound on distributions
[http://en.wikipedia.org/wiki/Probabilistic_method](http://en.wikipedia.org/wiki/Probabilistic_method))

~~~
sanderjd
Forgive my mathematical ignorance, but you've piqued my curiosity and I
haven't had any luck in searching - what is DGL?

~~~
jfarmer
I think he means this textbook:
[http://www.szit.bme.hu/~gyorfi/pbook.pdf](http://www.szit.bme.hu/~gyorfi/pbook.pdf)

~~~
tel
Yep, sorry! (Also, did not know that was online. It is totally going on my
ipad now)

------
jperras
This is from the realm of mathematical physics, but Noether's Theorem
([http://en.wikipedia.org/wiki/Noether%27s_theorem](http://en.wikipedia.org/wiki/Noether%27s_theorem))
is quite possibly the most beautiful result that I can think of:

Briefly, "Any differentiable symmetry of the action of a physical system has a
corresponding conservation law".

The extension of this theorem in QED, The Ward-Takahashi Identity, is also
incredibly astounding, if a bit obtuse if you're not well versed in
renormalizable field theories.

------
ramblerman
"Closed as too localized"

I have an ever growing contempt for over-eager stack overflow editors.

~~~
josephagoss
I agree, every single interesting and fascinating question on stack exchange
sites seems to have been closed for some reason or another. This overzealous
closing of anything slightly outside the box is really tiresome.

~~~
insteadof
Anything outside the box belongs on another site. They have a format that
works for them and rules to follow. We're not being forced to use only Stack
Exchange sites for our non-HN posts.

------
mjn
Gödel's incompleteness theorems came as something of an unwelcome surprise to
those who, at the time, were hoping to put David Hilbert's program into
practice.

------
bitL
Material implication and the fact that logic based on it is considered OK.

E.g. "if the moon was made of cheese, then P=NP" is considered true. This
would have been funny if this kind of proofs wasn't used in proofs of basic
facts of set theory, e.g. the theorem about empty set being a subset of any
set.

I am glad there are people trying to address these issues with a new logic,
such as relevance logic, though almost every mathematician I know is fully
reconciled with the former type of logic without questioning it.

~~~
scotty79
p => q sort of means, if you assume p, you can get q by flawless reasoning. It
is actually true that when you assume someting false you can get from it to
any conclusion by correct reasoning. The path between cheese moon and P=NP
might be hard to imagine but it probably leads through 1=0 somewhere and you
have o properly define what you mean by moon, cheese, and made of.

------
hypersoar
In a class I took once, the professor presented a theorem he was going to
prove (I think it was "surface groups are subgroup separable"). He told us
"the story goes that when the great mathematician, Serre, heard about this
theorem, he was so surprised he dropped his fork".

Edit: fixed quotation marks

~~~
StavrosK
I was going to throw a fit about that unmatched quote, but the second one
canceled it out, phew.

~~~
civilian
Yeah but now you have an unmatched parenthesis! When will it end?!?!

~~~
StavrosK
Damn you!

~~~
Filligree
Here, let me help)

~~~
StavrosK
Phew, thanks, man.

------
bluecalm
And of course SO police closed this very interesting topic killing many
potential very interesting posts. That is despite people liking it, discussing
it and upvoting it. I am sorry for this a bit off-topic comment but I really
hope there will be more voices against this moronic policy.

~~~
rtpg
If you start making exceptions to rules then you're going to end up causing
even more strife. I'm willing to pay the price to not get "what's the best
programming language for X" style questions on SO (for example).

On a related note, there's a blog post on the SO blog concerning closed
questions, pretty interesting stuff.

~~~
billforsternz
The trouble is that the proportion of quality content that ends up getting
closed for arbitrary, capricious, and downright infuriating reasons (just read
the text of the "explanation" in this case) is enormous. Often when looking
for answers on Stack Overflow I find _all_ of the useful answers are closed.
This makes no sense and reflects the stubborn attitudes of the decision makers
there, nothing else.

~~~
rtpg
out of curiosity,do you have an example of useful answers that are closed?
I've found a lot of interesting answers that are closed, but not really
"useful" in the SO sense: something that answers a specific question.

~~~
billforsternz
Sorry, late reply; I suppose it comes down to the definition of "a specific
question". Often I will be looking for recommended tools/resources/approaches.
Look at something like [http://stackoverflow.com/questions/17489/best-
environment-fo...](http://stackoverflow.com/questions/17489/best-environment-
for-python-on-windows)

I like questions like this being closed, particularly if they draw out a ton
of useful information like this one. Maybe my pain could be eased a little if
they had some better boilerplate explanations. To me "Not constructive" in
this case is just misleading, wrong and rude.

~~~
billforsternz
s/like/don't like

------
raverbashing
For me:

\- The Gauss theorem still looks weird to me, that the shape doesn't matter,
all it matters is what's contained inside
([http://en.wikipedia.org/wiki/Divergence_theorem](http://en.wikipedia.org/wiki/Divergence_theorem))

\- The Carmichael function (which is a generalization of Euler's theorem which
is a generalization of Fermat's little theorem)
[http://en.wikipedia.org/wiki/Carmichael_function](http://en.wikipedia.org/wiki/Carmichael_function)

------
tzs
I was surprised by the Khinchin–Lévy constant. For almost any real number, if
you take the sequence of convergents of its continued fraction,
$\frac{p_1}{q_1}$, $\frac{p_2}{q_2}$, ..., this is true:

\\[\lim_{n \to \infty}{q_n}^{1/n}=e^{\pi^2/(12\ln2)}\\]

The number on the right is called the Khinchin–Lévy constant.

You can use MathJax to render the LaTeX in this comment with the bookmarklet
available here: [http://checkmyworking.com/misc/mathjax-
bookmarklet/](http://checkmyworking.com/misc/mathjax-bookmarklet/)

------
scoot
For me, the fact that 0.999... = 1, not least because the elementary proofs
are easy enough for even me to understand!

[http://en.wikipedia.org/wiki/0.999..](http://en.wikipedia.org/wiki/0.999..).

~~~
jgreen10
I always think that's just a bug in / the inelegance of the "..." notation.

------
lotharbot
My favorite surprising result in mathematics is that you can create chaos in
the mathematical sense [1] with a bi-infinite binary sequence and the bitshift
operation. Essentially, "multiplying by two" is chaotic in the right
circumstances.

The technical description can be found starting at page 567 of [0], though the
Google Books preview has some important pages missing.

[0] Stephen Wiggins - Introduction to Applied Nonlinear Dynamical Systems and
Chaos
[http://books.google.com/books?id=GYcOfuZDOKMC&lpg=PA565&ots=...](http://books.google.com/books?id=GYcOfuZDOKMC&lpg=PA565&ots=bF12jI5K8_&dq=bi-
infinite%20sequence%20chaos&pg=PA567#v=onepage&q=bi-
infinite%20sequence%20chaos&f=false)

[1] The key attributes of chaos are described in
[http://en.wikipedia.org/wiki/Chaos_theory#Chaotic_dynamics](http://en.wikipedia.org/wiki/Chaos_theory#Chaotic_dynamics)

~~~
dmvaldman
multiplying by two, and adding I presume. makes sense.

to give some related knowledge, all matrices can be decomposed into direct
sums of shifts and scales. this is the base of the jordan normal form [1].
infinite matrices can certainly encapsulate chaos.

[1]
[http://en.wikipedia.org/wiki/Jordan_normal_form](http://en.wikipedia.org/wiki/Jordan_normal_form)

~~~
lotharbot
There's actually no adding. _Just_ multiplying by two. That's what makes the
result so surprising.

The key is the metric on the bi-infinite sequence space. Differing in the
digit closest to the decimal point on either side gives a distance of 1/2,
differing one digit further out is 1/4, then 1/8, 1/16, and so on (the total
distance is the sum; the maximum distance between two sequences is 2 if they
differ in every digit.) So two sequences are "close" to one another if their
middle digits are all the same.

This gives you the key attributes of chaos:

(1) Sensitive Dependence on Initial Conditions. Two sequences can be identical
for BIG_NUM digits on each side of the decimal point, and then completely
different outside of that area. This means their initial distance apart is
roughly 1/2^BIG_NUM. But after BIG_NUM multiply-by-two operations, the
trajectories have diverged to a distance of around 2.

(2) Topological Mixing. A similar construction to above -- select the first N
digits of a sequence to make it fall into a particular neighborhood, and the
next M digits to make it fall into a different neighborhood after the
appropriate number of bitshifts. This can be done for any pair of
neighborhoods.

(3) Density of periodic orbits. Pick any point X in the space, and a distance
epsilon. Construct a periodic orbit that gets within epsilon of X simply by
taking the central digits of X (using log_2(epsilon) to select how many
digits) and repeating those digits.

There you have it -- chaos as a result of a single bitshift operation on bi-
infinite sequences, with no addition or other operations.

------
bjterry
I found this post to be an interesting description of Banach-Tarski, if anyone
is as confused as I was: [http://scientopia.org/blogs/goodmath/2012/01/06/the-
banach-t...](http://scientopia.org/blogs/goodmath/2012/01/06/the-banach-
tarski-non-paradox/)

------
gjm11
One of my favourites, which isn't on the first page there (I didn't look
beyond that) but contains _three_ surprises, is Goodstein's theorem. It has
the nice extra feature that it can be explained pretty simply, so here goes.

When you write a number down in (say) base 10, what you're really doing is
writing it as a sum of [small thing] x [power of 10]. So, e.g., 1234 means 1 x
10^3 + 2 x 10^2 + 3 x 10^1 + 4 x 10^0. "Small thing" means "non-negative
integer, less than the base". And of course you can do this in bases other
than 10.

[EDIT: previous paragraph had a stupid mistake in it; thanks, grannyg00se, for
catching it!]

But when you do this, the _exponents_ may be large numbers; in particular,
they may be larger than the base. So consider the following operation, given a
non-negative integer and a base: write the integer as a sum of [small] x
[power of base], and then do the same thing to the exponents, and to the
exponents in that, etc., until you've got rid of all the numbers >= the base.

OK. Now here's a process you can carry out. Take a number. Write it "strongly
in base 2" according to the procedure above. Now change all the 2s to 3s,
generally making the number much much bigger. Now subtract 1.

Now write it "strongly in base 3" again, change all the 3s to 4s, and subtract
1. And again: 4 -> 5, subtract 1. 5 -> 6, subtract 1.

If you try this on some not-too-large number, you'll notice that the base-
increasing operation typically increases the number _hugely_ , whereas of
course subtracting 1 decreases it _just a little bit_.

So here's _Goodstein 's theorem_: start with any number you like and carry out
this process; eventually the numbers will stop increasing and you'll end up at
0.

(Surprise #1: what a ridiculous idea! Obviously the numbers are increasing
really really fast; how can they end up at zero?)

It turns out that Goodstein's theorem is actually rather easy to prove, using
a bit of machinery from set theory. There's a generalization of the natural
numbers called the _ordinals_ , and doing it properly requires some theorems
about them, but here's the idea. Replace all the bases with "infinity". Assume
some simple rules for comparing expressions full of infinities (e.g., if a<b
then (anything finite) x infinity^a < infinity^b); then every time you
subtract 1 and re-express your number "strongly in base b", the corresponding
thing full of infinities decreases. And of course when you replace all the
bases with infinity, the "increase the base by 1" operation does nothing. And
now -- and this is the bit that actually needs some set theory -- it turns out
that _you can 't have an infinite decreasing sequence of these things-full-of-
infinities_ (they're "well-ordered"). Which means that eventually you have to
end up at zero.

(Surprise #2: this is, even when the details are filled in, a startlingly
short proof for such a surprising theorem.)

All that stuff involving infinities is a bit weird, though. Can't we find a
nice proof that works entirely in terms of ordinary finite integers? Once you
formalize what "works entirely in terms of ordinary finite integers" means, it
turns out that the answer is no! Goodstein's theorem is not provable from the
"Peano axioms" describing the natural numbers; you _need_ something stronger
to prove it. Such as, in this case, set theory.

(Surprise #3: that something so innocuous requires such powerful mathematical
machinery to resolve.)

~~~
DanielRibeiro
My favorite one is Russel's Paradox[1], which actually became the proof that
there is no such thing as the Set of all Sets (under traditional set
theory[2]).

Which was one of my first steps into Category Theory, as Set is a proper
category[3].

[1]
[https://en.wikipedia.org/wiki/Russell%27s_paradox](https://en.wikipedia.org/wiki/Russell%27s_paradox)

[2]
[https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory)

[3]
[http://en.wikipedia.org/wiki/Category_of_sets](http://en.wikipedia.org/wiki/Category_of_sets)

------
SeanDav
As a non mathematician the most surprising for me is that 1 is not a prime
number. It still feels like a kludge because it produces inconvenient results
to have 1 a prime - so let's just exclude it.

~~~
mjw
This isn't a surprising _result_ though, it's a surprising (to you)
_definition_.

You could redefine "prime" to include 1 if you wanted to without changing the
fundamental truth of important results about the natural numbers. It would
just mean you'd have to phrase many of them in a slightly more roundabout way,
e.g. with "suppose x is a prime number not equal to 1" needing to appear in
key theorems where before "suppose x is prime" sufficed. (Perhaps some other
theorems might be more concise, although I'd be surprised if there were a net
benefit -- the community has probably had this debate a bunch of times before
settling on the current consensus.)

Definitions are chosen to be convenient for the purposes of communication,
there's nothing inherently true or false about them. An argument for one
definition being better than another might involve demonstrating that
mathematical results people care about can be stated more clearly, concisely
and efficiently using your alternative definition. The recent tau vs pi
controversy providing an amusing example :)

~~~
mjw
As a programming analogy: imagine mathematics is a big codebase and "prime" is
a commonly-used function defined within that codebase.

You could refactor "prime" to allow 1, provided you refactor all the call
sites to work with that new definition too. Consensus seems to be that this
refactoring would add LOC and reduce code readability, probably leading to
someone submitting a patch soon after with a "prime-but-not-one" function and
using that everywhere.

------
mathattack
I know Im missing the point, but this jumps out at me....

"closed as too localized by t.b., Zev Chonoles Sep 5 '11 at 22:18

This question is unlikely to help any future visitors; it is only relevant to
a small geographic area, a specific moment in time, or an extraordinarily
narrow situation that is not generally applicable to the worldwide audience of
the internet. For help making this question more broadly applicable, visit the
help center. If this question can be reworded to fit the rules in the help
center, please edit the question."

------
cobookman
I do enjoy Euler's formula's use in Electrical Engineering/DSP:
[http://en.wikipedia.org/wiki/Euler%27s_formula](http://en.wikipedia.org/wiki/Euler%27s_formula)
One of the many applied uses would be the FFT, or calculating Electrical
Impedance.

There is also Green's Theorem:
[http://en.wikipedia.org/wiki/Green%27s_theorem](http://en.wikipedia.org/wiki/Green%27s_theorem)

------
ezequiel-garzon
The Prime Number Theorem! The average spacing between primes under N tends
to... ln(N).

By the way, Gauss conjectured it at the tender age of 14...

------
joe_the_user
I find Skolem's Paradox to be a remarkable thing.

Uncountable sets are the foundation for a large portion of modern math but the
sense in which they "really exist" is at least interesting.

[http://plato.stanford.edu/entries/paradox-
skolem/](http://plato.stanford.edu/entries/paradox-skolem/)

------
mhartl
The divergence of the harmonic series (sum_{n=1}^∞ 1/n) is one of the most
surprising elementary results.

------
kriro
Not really a result, more of a discovery but the Mandelbrot set still amazes
me to this day :)

Not really mathematics but close enough: The fact that a Rule 110 cellular
automaton is Turing complete really fascinates me as well.

~~~
mpyne
The Mandelbrot set was what introduced me to programming as an adolescent. :)

------
ctdonath
My 3 favorites:

2^(2^(2^...-1)-1)-1 seems prime for all nestings, but the numbers get enormous
fast

John Conway's "surreal numbers" create all math from an empty set

Fractals ('nuf said)

~~~
gizmo686
>2^(2^(2^...-1)-1)-1 seems prime for all nestings, but the numbers get
enormous fast

Can you explain this. I read your expression (with infinite nesting) as
x=2^(x-1), whose solutions are 1 and 2.

Looking at it as a series, I see: 2-1=1 2^(2-1)-1=1 2^(2^(2-1)-1)-1=1
2^(2^(2^...-1)-1)-1=1

Also, assuming that a sequence does have the properties you describe (always
prime, and gets large very fast), it seems like that sequence would either
have to be hard to compute, or unknown to all of the mathematicians working on
finding large primes.

~~~
crygin
The issue is that we have no proof that they are primes in general, and they
become large so quickly that verifying their primehood is (currently)
impossible.

~~~
anonymous
The issue gizmo and I see with the sequence is that it is all ones if the
first element is 2 - 1 = 1.

Are you supposed to start with 2^2 - 1 = 3; then 2^3 - 1 = 7; 2^7 - 1 = 127;
etc? Those do seem to be all prime.

~~~
ErsatzVerkehr
Mersenne primes whose exponents are themselves Mersenne primes.

------
revskill
To me, it's Euler formula: e^(iπ) + 1 = 0. It combines three most important
constants of mathematics into one formula. Sweet !

~~~
nemo1618
that...isn't true. The standard form of Euler's identity is e^(iπ) + 1 = 0.

Of course, all the cool kids nowadays use e^(iτ) = 1. ;)

~~~
revskill
Ohh, my mistake. Thank you for correction.

------
banachtarski
100% agreed on the bit about holomorphic functions. Once I forget the
derivation, the result surprises me all over again.

------
lisper
i (the square root of -1) raised to the i'th power is a real number (it's
e^(-pi/2))

------
RBerenguel
Missed sphere eversion on the first pages of this list...

~~~
gjm11
It's the third highest voted answer. "The fact that you can turn a sphere
inside out differentiably.", 52 points.

~~~
RBerenguel
Fourth, actually. Indeed, this eases my opinion :) Since I was reading
diagonally I think my eye moved away to Cauchy's Integral Formula, since I
have worked a little in the complex field. Thanks for pointing this out ;)

------
jgreen10
e^(i*tau)=1 is definitely my favourite, but I don't really understand what x^i
means.

~~~
prezjordan
Write out e^x as a power series, then just plug in ix instead of x :) Fun
stuff happens.

------
e3pi
Poised inert(0) being mathematically ameliorated with those processes constant
including Fourier's common denominator assuaging all cycles of all myriad
things, and of at most a single exceptionally egregious serious error, genteel
aplomb in action, while abruptly hazarding the whole ball of wax, London
gentleman's Reform Club wagerer closed contour circumnavigates eastward around
a steaming Victorian Whitechapel® worshipful churchwarden's lit'l
chimney("Caminetto") simmering essential singularity, where right of state
police Scotland Yard's criminal suspicion and under-cover investigation of
Philius Fogg's first err'd then favorite and NOT his discovered saving
redemption, rather obsequious servant/valet Passepartout's essential human
artifice required singularity: the International Date Line, is not only
imaginary(2):

2 _pi_ e^(pi/2 _e^(pi /2_e^(pi/2 _...und zu weiter...e^(pi /2_i)))

and within an epsilon neighborhood of NOT the author's, and NOT of his
contemporaneous Cauchy's and Fourier's Prime Meridian, but its primary
protagonist's Observatory Greenwich.

By Capt J Luc Picard's Bigger Theorem:

if an entire holomorphic process has an essential singularity at w, then
within any open set epsilon neighborhood containing lit'l rill w, takes on all
possible values, with at most a single exception, infinitely often- the
necessary pathological(1) attending entropic heat, ignorance, indifference and
state sanctioned `algebraically' indiscriminate `transcendent' stupidity among
all possible colliding events abruptly attenuates to less than epsilon by the
renormalization resonance of Philius' corporeal holotropism of the weighed
measure of all possible events and Fourier's common denominator being
identical with the simultaneity in the conservation of the contour integral's
representation of both signal and noise of all possible events' -all
coincident and of arbitrary duration- their unstable equilibrium catalyzes
over and precipitates the synch jaunt with nicely holomorphic Philius' lit'l
rill flywheel into actually winning the bet, Fogg fists repugnant Fix's
despicable state-sanctioned stupid and recklessly endangering souless
`algebraic' orthotropic tyranny into a bloody nose, and gets the real girl,
and really conformally reforms into the really really rich.

(0) `Lazy' -parsed, inert Inertia:

Inertia does not contain within itself the slightest suggestion of a regular
rising and falling of the contents of life; it offers itself at every movement
with the same freshness and efficiency; by its far reaching effects and by
reducing things to one and the same standard value, that is by leveling out
countless fluctuations, mutual alterations of distances and proximity of
oscillations and equilibrium, it levels out what would otherwise impose far-
reaching change upon the possibilities for the individual's activities and
experiences. Inertia is a real object as well as the integration of all
objects. However its pure notion may be, inertia appears as the warm spring of
life, flows into the schemata of all things, allowing them to blossom and
unfold their very essence, no matter how diverse or antagonistic.

Inertia tends towards a point at which, as a pure symbol, it completely
absorbs all exchange and measurement, all static constants and dynamic
processes.

It is the facilitator creating contact with one another. Know such unique
qualities of inertia, for the development of the rhythmical and specific
objective styles of life, because the incomparable depth of their opposition
illustrates the remarkable actuality of unique inertia being all their common
factor. Inertia is the only scalar-independent intrinsic reference and so the
ONLY absolute frame of reference from the absolute constant speed of light, as
well anything else's inertia, anything else's matter for that matter.

The perception which is in more agreement to natural law, is not only in
agreement with reality but is reality. Natural law perception of reality, but
so too also perceptive -is sentient- being reality. In their realm the
feelings that inertia excites possess a psychological similarity with this.

In so far as inertia becomes the absolutely commensurate expression and
equivalent of all values that arise to abstract heights way above the whole
broad diversity of objects; it becomes the center in which the most opposed,
the most estrange, and the most separable of things find their common
denominator.

(1) Le tour du monde en quatre-vingts jours was written during difficult
times, both for France and for Verne. It was during the Franco-Prussian War
(1870-1871) in which Verne was conscripted as a coastguard, the French army
losing every single campaign to the newly united German/Prussian Hohenzollern
ubermenchen blot und stahl; and Jules was having money difficulties (his
previous works were not paid royalties), his father had died recently, and he
had witnessed a shockingly horrible state-ordered public execution which had
markedly damaged him.

Ingnorant of those qualities transcendental, you will suffer those chaotic
consequences algebraic."

(2) "Imagination is the one weapon in the war against orthotrope reality". \--
Jules de Gaul, linux fortuna

