# Archive for May 18th, 2013

### Unintentional Haiku in the Princeton Companion to Mathematics

Posted by Angela Brett in Haiku Detector on May 18, 2013

I’ve had a copy of the Princeton Companion to Mathematics for a while, and intended to start a series called ‘forms and formulae’, where I’d write about some of the articles using poetic forms from the Princeton Encyclopaedia of Poetry and Poetics (addendum: I have since started a series called Forms and Formulae doing just that.) However, both books are huge and difficult to read on the bus, and the articles are long, so so far all I’ve managed to do in that vein is write a poem about platonic solids in a duel, and procrastinate my way out of writing about the entries whose names were alphabetically closest to Emmental. So I was excited to discover this morning that there is a pdf of the Princeton Companion to Mathematics available for free, apparently legally. Finally I can carry it around with me on my iPad and write poems about it whereever I want. But I don’t even need to do that, now; thanks to Haiku Detector, I can easily find the poems that are already in it. And boy are there some nice ones. Some were missed because Haiku Detector doesn’t know how to pronounce Greek letters and a lot of other mathematical notation, and the book sometimes hyphenates at the ends of lines so it looks like they’re good places for line breaks when they’re not. But these are the best ones I found. First off, some which don’t even sound like they’re about mathematics:

Watch your hand as it

reaches out gracefully to

pick up an object.The difference between

the two definitions of

a secret is huge.These ideas will

occupy us for the rest

of the article.This opens you up

to new influences and

opportunities.In our case, there are

two natural properties

that one should ask for.Suppose that households

are able to observe one

another’s outputs.Everything is now

a martingale and there can

be no arbitrage.The magician can

at once identify which

digit has been changed.This definition

has the advantage of great

flexibility.Let us briefly sketch

the argument, since it is

an instructive one.Moreover, it was

a thought that took many years

to be clarified.The blocks are the sets

of seed varieties used

on the seven farms.

If you didn’t know where it came from, this could be about anything, but it also sums up the appeal of mathematics:

But then again, who

can deny the power of

a glimpse at the truth?

And a more transparent statement about the nature of mathematics:

“All roads lead to Rome,”

and the mathematical

world is “connected.”

But I really love it when you can’t tell it’s about mathematics until the last line:

The answer turns out

to be that we should weaken

our hypotheses.It is important

to have a broad awareness

of mathematics.We will focus on

the most important special

case: vector bundles.Sometimes relations

are defined with reference to

two sets A and B.This remains as an

outstanding open problem

of mathematics.Church’s thesis is

therefore often known as the

Church–Turing thesis.How, though, can we be

sure that this process really

does converge to x?It turns out that both

choices are possible: one

automorphismWe shall now describe

the most important of these

extra assumptions.Several themes balance

in Hilbert’s career as a

mathematician.Indeed, the study

of such designs predates their

use in statistics.This turns out to be

a general fact, valid

for all manifolds.However, it is

a well-understood kind of

singularity.In particular,

we can define the notion

of winding numbers.This is exactly

the task undertaken in

proof complexity.

Questions mathematicians ask themselves:

How much better would

you do if you could compound

this interest monthly?Why are spherical

harmonics natural, and

why are they useful?What consequence should

this have for the dimension

of the Cantor set?Can we reduce this

computational problem

to a smaller one?How about checking

small numbers a, in order,

until one is found?For what values of

the edge-probability

p is this likely?Is every even

number greater than 4 the

sum of two odd primes?Can one make sense of

the notion of a random

continuous path?

Perhaps this is the answer:

In mathematical

research now, there’s a very

clear path of that kind.

This one sounds like some kind of ‘how many roads must a man walk down’ question:

How many walks of

length 2n are there that start

and end at 0?

And while this isn’t actually a haiku, I can imagine it being sung in response to that song, with ‘the number of such walks’ to the tune of ‘the answer my friend’:

The number of such

walks is clearly the same as

W (k − 1).

Mathematicians don’t always answer questions in ways that other people find useful:

If instead we were

to ask each person “How big

is your family?”In particular,

the average family size

becomes infiniteIt follows that at

some intermediate r

the answer changes.

Things only a mathematician would feel the need to state explicitly:

This is a sum of

exponentials — hence the phrase

“exponential sums.”What makes them boring

is that they do not surprise

us in any way.

Proof is left as an exercise for the reader; it probably takes several pages, but:

If you do know it,

then the problem becomes a

simple exercise.Once this relative

primality is noticed,

the proof is easy.All we have to do

is use one more term in the

Taylor expansion.Doing things this way

seems ungainly to us, but

it worked very well.It is not hard to

see that the two approaches

are equivalent.(Of course, one needs to

check that those two expressions

really are equal!)But this subtlety

is not too important in

most applications.

Some interesting statements:

For every person

P there exists a drink D

such that P likes D.That is exactly

what a sphere is: two disks (or

cups) glued together!Thus, recursion is

a bit like iteration

but thought of “backwards.”Nevertheless, it

turns out that there are games that

are not determined.(It can be shown that

there is exactly one map

with this property.)The remainders get

smaller each time but cannot

go below zero.There are other ways

to establish that numbers

are transcendental.(The term “Cartesian

plane” for R2 is therefore

anachronistic.)As usual, we

identify R2 with

the complex plane C.Note that a block of

size 1 simply consists of

an eigenvector.The upshot is that

we should always use a prime

number as our base.Among the other

important number fields are

the cyclotomic fields.Thus we obtain a

number that is less than the

quantity we seek.So we might define

the “points” of a ring R to

be its prime ideals.(For both halves, the pinched

equator is playing the

part of the point s.)Thus, we have deduced

that length-minimizing curves

are geodesics.For example, the

geodesics on the sphere

are the great circles.The generators

correspond to loops around each

of the two circles.The image of this

map will be a closed loop C

(which may cross itself).We consider what

happens to C if we add

a small ball to it.It is not hard to

show that the orbits form a

partition of X.There are many ways

of combining groups that I

have not mentioned here.I have thrown classes

of groups at you thick and fast

in this last section.To apply Newton’s

method, one iterates this

rational function.A quick overview

of physics will be useful

for the discussion.can get away with

not understanding quantum

mechanics at all.The quantum version

of Hamilton’s principle

is due to Feynman.These encapsulate

the idea of a proof

by contradiction.(A graph is simple

if it has neither loops nor

multiple edges.)It is really an

algorithm that inputs

n and outputs an.(An involution

is a permutation that

equals its inverse.)If the tree has 2

vertices, then its code is

the empty sequence.But the number of

possible orders of A,

B, and C is 6.Number theory is

one of the oldest branches

of mathematics.The percolation

and Ising models appear

to be quite different.First, Albert shouts out

a large integer n and

an integer u.

This one is interesting if you imagine it’s about lines of poetry:

Another affine

concept is that of two lines

being parallel.

A mathematical protest slogan:

equality if

and only if x and y

are proportional.

A title of the mathematician’s equivalent of a song about unrequited love:

5.1.5

Why Is It so Difficult

to Prove Lower Bounds?

A series of short films:

10 Differences in

Economic Life among

Similar People

And something said in a soothing tone after a litany during a maths/mass:

Now let us return

to polynomials with

n variables.

The probability of finding a good haiku in the end matter is low, but I think this one’s pretty neat, even if it only has the right syllable counts if you say the ‘and’ in 906 but not 753:

law of large numbers,

753,

906