4) Harmonic analysis

Quickies

pi goes on and on and on …
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed?

Chebychev said it and I’ll say it again:
There’s always a prime between n and 2n.

Man has pondered
Since time immemorial
Why 1 is the value
Of zero-factorial.

Three jolly sailors from Blaydon-on-Tyne
They went to sea in a bottle by Klein.
Since the sea was entirely inside the hull
The scenery seen was exceedingly dull.

— Frederick Windsor
from The Space Child’s Mother Goose, (c) 1958.

---

Cheers!

e to the x dx,
e to the y dy,
Sine x, cosine x,
Natural log of y,
Derivative on the left
Derivative on the right
Integrate, integrate,
Fight! Fight! Fight!

e to the x dx dy
Radical transcendental pi
Secant cosine tangent sine
3.14159
2.71828
Come on folks let’s integrate!

e to the i dx dy
e to one over y dy
Cosine secant log of pi
Disintegrate ‘em RPI !!!

Square root, tangent
Hyperbolic sine,
3.14159
e to the x dy dx
Sliderule, slipstick,
TECH TECH TECH!

And for the computer scientists:

Shift to the left, shift to the right!
Pop up, push down, byte, byte, byte!

---

Limericks

‘Tis a favorite project of mine
A new value of pi to assign.
I would fix it at 3
For it’s simpler, you see,
Than 3.14159…

If inside a circle a line
Hits the center and goes spine to spine
And the line’s length is d
the circumference will be
d times 3.14159…

If (1+ x) (real close to 1)
Is raised to the power of 1
Over x, you will find
Here’s the value defined:
2.718281…

In arctic and tropical climes,
The integers, addition, and times,
Taken (mod p) will yield
A full finite field,
As p ranges over the primes.

If n in a Taylor series
Goes 2 to 11 by threes
For n = 1
Convergence is done
‘Twixt 0 and 2, I believe.

Mathematics: of sciences, queen
Has more rules than I’ve ever seen.
There are no exceptions,
Just number deceptions.
On calculators, I am quite keen.

A mathematician confided
That the Moebius band is one-sided
And you’ll get quite a laugh
If you cut one in half
‘Cause it stays in one piece when divided.

A mathematician named Klein
Thought the Moebius band was divine
Said he: “If you glue
The edges of two
You’ll get a weird bottle like mine.”

A go-go lap dancer, a pip,
Was able to peel in a zip.
But she read science fiction
And died of constriction
Attempting a Moebius strip.

The Moebius strip is a pain,
When you cut it again and again,
But if you should wedge
A large disk round the edge
Then you just get a projective plane.

If you have a cross-cap on your sphere,
And you give it a circle-shaped tear,
Then just shake it about
And untangle it out
And a Moebius strip will appear!

A mathematician named Crottle
Poured water into a Klein bottle.
When asked, “Do you doubt
That some will run out?”
He replied, “No, I don’t. Quite a lot’ll.”

There was young maiden named List
Whose mouth had a funny half-twist.
She’d turned both her lips
Into Moebius strips…
‘Til she’s kissed you, you haven’t been kissed!

There was a young fellow named Fisk,
A swordsman, exceedingly brisk.
So fast was his action,
The Lorentz contraction
Reduced his rapier to a disc.

A conjecture both deep and profound
Is whether the circle is round;
In a paper by Erdos,
written in Kurdish,
A counterexample is found.

A challenge for many long ages
Had baffled the savants and sages.
Yet at last came the light:
Seems old Fermat was right–
To the margin add 200 pages.

A calc student upset as could be
That his antiderivative didn’t agree
With the one in the book
E’en aft one more look.
Oh! Seems he forgot to write the “+ C”.

A graduate student from Trinity
Computed the cube of infinity;
But it gave him the fidgets
To write down all those digits,
So he dropped math and took up divinity.

A mathematician called Bird,
Had students who thought him absurd.
There were cries of derision
When he said long division,
Meant one into one made a third.

A mathematician called Rumbold,
One day, quite by accident, stumbled
On the Meaning of Life,
Then went on, for his wife,
To find out why all her apple pies crumbled.

To a tightrope walker named Zekund
The a due to gravity beckoned.
His performance was great
At about 9.8
Meters per second per second.

Consider the pitiful plight
Of a runner who wasn’t too bright.
For he sprinted so fast,
That he vanished at last
By red-shifting himself out of sight.

In the near-light speed space-ship I’m in,
I went rocketting off from my twin;
But since I’ve been away
I’ve aged hardly a day
And just look at the state that he’s in!

There once was a prof, Dr. K.,
Who taught calculus everyday.
From dawn until noon,
Integrating to the moon.
To him, derivatives were okay.

There’s a leather-clad separatrix,
a vector-valued dominatrix
who divides a phase plane
into pleasure and pain
when she gets hold of more than one matrix.

— (c) Courtney Gibbons

Jack the mathematician
Had a very strange mission.
The problems he wrote,
He often would gloat,
Sent many a student a-wishin.’

— (c) Dillon Glover

There was a prof named Kowalski
Who taught all this calculus to me.
On the final — no pass;
I must retake the class.
Why, we should all be so lucky.

— (c) Eric Seifert

Despite all the might fine teachin,’
I can’t help but find myself thinkin’
That Calculus I
Will be much more fun
The second time o’takin’.

— (c) Matt Begeman

Along cam Sir Isaac Newton
Doin’ mathematical computin’.
One day he contrived
To anti-derive
When findin’ signed areas is suitin’.

There once was a user named Fred,
Who one day used grep, awk, and sed.
He parsed a huge text stream,
Used regexps to the extreme,
Now his file’s tail is its head.

function createLimmerick(){
var scanning=terriblySlick;
if(lines==5)
&&rhyme=="live"
do(laugh(); performNewTrick();)}

---

Limeriquations

Euler’s Equation:

Here are a few limericks about this one.

I used to think math was no fun,
‘Cause I couldn’t see how it was done.
Now Euler’s my hero,
For I now see why 0
Equals e ^{pi i} + 1.

e raised to the pi times i,
And plus 1 leaves you nought but a sigh.
This fact amazed Euler
That genius toiler,
And still gives us pause, bye the bye.

The Pythagorean Theorem:

A triangle’s sides a, b, c,
With a vertex of 90 degrees,
If that vertext be
‘Tween sides a and b,
The root a² plus b² is c.

— (c) Andrew Adams

Equation 1:

A Dozen, a Gross and a Score,
Plus three times the square root of four,
Divided by seven,
Plus five times eleven,
Equals nine squared and not a bit more.

— (c) John Saxon, textbook writer

Equation 2:

Integral v-squared dv
From 1 to the cube root of 3
Times the cosine
Of three pi over 9
Equals log of the cube root of e.

Equation 3:

One over point one-oh-two-three,
When raised to the second degree,
Divided by seven
Then minus eleven
Is approximately equal to e.

— (c) A. F. Cooper

Equation 4:

Th’integral from e-squared to e
Of 1 over v dot dv,
When raised to the prime
Between five and nine,
Is e to the i pi by 3.

— (c) M. M. Bishop

Equation 5:

The integral from naught to pi
Of sine-squared of 2 phi d-phi,
When doubled and then
Not altered again,
Is log (minus 1) over i.

— (c) M. M. Bishop

Equation 6:

To find Euler’s Gamma of three,
Integrate to infinity
From zero, dx
x-squared on exp(x),
Or three bang divided by three.

— (c) M. M. Bishop

Equation 7:

‘Cause phi-squared less phi, minus 1,
Is exactly equal to none,
The golden mean phi,
Which so pleases the eye,
Is half of root 5 add on one.

— (c) M. M. Bishop

Equation 8:

The square root of minus 2 pi
On th’square root of inverse sine phi;
All that need be done
Is let phiequal one:
It’s twice exp of i pi on i.

— (c) Andrew Adams

Figure 1:

If a circle through B, like so,
Has arc AD with center O,
The angle at B,
Wherever B be,
Is half of the angle at O.

— (c) M. M. Bishop

Figure 2:

A body with mass m kg
Feels a force of magnitude T.
When its weight t’wards the ground
Is added it’s found
To speed up at T on m, less g.

— (c) Andrew Adams

---

Piems

There is in mathematical circles (ha!) a great and lengthy tradition of making mnemonics to memorize the digits of pi:

pi = 3.14159 26535 89793 23846 26433 83279 502….

In a traditional pi mnemonic, such as the one above, the number of characters in each word is used to represent a digit of pi, with the ocassional 10-, 11-, 12, or 14-digit word denoting two digits of pi. Punctuation is ignored. A famous example is the following:

God! I need a drink –
Alcoholic, of course –
After all those lectures
Involving radical equations.

Below are several more examples, listed in order of increasing decimal precision.

3.1416

May I draw a circle?

— Anon, found in Dmitri Borgmann’s Language on Vacation, (c) 1965

Yes, I have a number.

— H. E. Licks, Recreations in Mathematics, (c) 1917

3.14159

Wow, I made a great discovery!

— Anon

3.14159 2

Now I need a verse recalling pi.

— Irene Fisher & Dunstan Hayden, Geometry, (c) 1965

3.14159 265

How I wish I could enumerate pi easily today.

— Anon

Yes, I know I shall recollect my number right.

— Wallace Lee, Math Miracles, (c) 1960

May I draw a round enclosure as circle known?

— Dmitri Borgmann, Language on Vacation, (c) 1965

3.14159 26535

May I have a large container of coffee, sugar and cream?

— Anon

3.14159 26536 (rounded)

But I must a while endeavour
To reckon right the ratios.

— Anon, Mathematical Gazette, vol. 10, (c) October 1921

3.14159 26535 8

Sir! I send a rhyme excelling
In sacred truth and rigid spelling.

— F. R. S., Nature, vol. 72, no. 1875, (c) October 1905

3.14159 26535 8

God! I need a drink –
Alcoholic, of course –
After all those lectures
Involving radical equations.

— Anonymous mathematicians everywhere

How I want a drink,
Alcoholic of course,
After the heavy chapters
Involving quantum equations.

— Sir James Jeans, (c) 1932

3.14159 26535 8979

Now I have a score notations
Of digits large and small,
teaching diameter’s precise relations,
And we can remember ‘tall.

— G. E. Gude, Scientific American Supplement, vol. 77, no. 1994, (c) March 1914

3.14159 26535 89793 2384

How I wish I could recollect pi.
Eureka! cried the great inventor.
Christmas pudding, Christmas pie
Is the problem’s very center.

— Anon, found in Alan D. Baddeley’s The Pyschology of Memory, (c) 1976

Now I sing a silly roundelay
Of radical roots, and utter “Lackaday!
Euclidean results imperfect are, my boy…
Mnemonic arts employ!”

— Willard R. Espy, An Almanac of Words at Play, (c) 1975

3.14159 26535 89793 23846 264

Now I know a spell unfailing,
An artful charm, for tasks availing,
Intricate results entailing.
Not in too exact a mood…
(Poetry is pretty good!)

— Anon, Nature, vol. 72, no. 1878, (c) October 1905.

How I want a drink,
Alcoholic of course,
After the heavy chapters
Involving quantum equations.
All of thy geometry,
Herr Planck, is fairly hard.

— Anonymous extension to Sir Jeans’ famous mnemonic above.

3.14159 26535 89793 23846 26433 8

For circumscribing a round enclosure or circle, every man
might remember ingenious numbers measuring one by one
diameter into circle or circle upon its own diameter…

— Anon, Mathematical Gazette, vol. 4, no. 65, (c) July 1907.

3.14159 26535 89793 23846 26433 83279

See, I have a rhyme assisting
My feeble brain, its tasks sometime resisting,
Efforts laborious can by its witchery
Grow easier, so hidden here are
The decimals all of circle’s periphery.

— L. R. Stokelbach, The Scientific American Supplement, no. 1994, (c) March 1914.

Now I — even I — would celebrate
In rhymes unapt, the great
Immortal Syracusan, rivaled nevermore
Who in his wondrous lore
Passed on before,
Left men his guidance how to circles mensurate.

— Adam C. Orr, Lierary Digest, vol. 32, no. 3, (c) January 1906.

Now I will a rhyme construct,
By chosen words the young instruct,
Cunningly devised endeavors
Con it and remember ever
Widths in circle here you see
Sketched out in strange obscurity.

— Anon, The Dark Horse, (c) 1951.

3.14159 26535 89793 23846 26433 83279 5

Sir: I wish I could recapture my memory about Sir
Jeans’ diabolic mnemonics! However, invention
now of any reliable easy phrase is beyond what shy
and fumbling aid my present intellect gives.

— Bill Powers, found in Willey Ley’s The Borders of Mathematics, (c) 1967.

May I have a month, professor,
To figure these, you brain assessor?
Calculate, student, calculate now!
As the figuring gets longer,
My friend, hope you get stronger
And no figures incorrect allow!

— Aaron L. Buchman, School Science and Mathematics, (c) 1953.

You I sing, O ratio undefined
By strict assay and lined,
Sequence limitless. Stunned regarding you,
We see eternity — alas — unwind
In random cast and rue,
Dejected out of measure, reckoning blind.

— John Freund, The Mathematics Teacher, vol. 62, (c) 1969.

May I tell a story purposing to render clear
the ratio circular perimeter-breaths, revealing
one of the problems most famous in modern days,
and the greatest man of science anciently known.

— C. J. Jackson, Mathematical Gazette, vol. 4, (c) 1907.

3835 digits of pi…

Of course, the all out winner, a massive mnemonic encoding 3835 digits of pi, is Michael Keith’s Cadaeic Cadenza, a fourteen chapter collection of digit-mindful paraphrasings of entire works of literature, including Edgar Allan Poe’s The Raven, Shakespeare’s Hamlet, Lewis Carroll’s Jabberwocky, and Carl Sandburg’s Grass, among others. Not only does he manage to preserve the story, structure, and rhyme schemes of these works, but Keith additionally encodes further pi mnemonics within the text itself. For example, the twelfth chapter includes an acrostic which itself encodes the digits of pi based on standard alphabetical number assignments. While absolutely useless in helping one recall the expansion of pi, it’s a breathtaking piece of mathematical and linguistic art.

---

4:45 AM

I keep writing things down
and then losing the pieces of paper
because individually the work isn’t any good
but collectively it’s enough to scare Clive Barker.

Now I study sequences and series
and spend much of my time
showing where things tend as
they approach infinity.

Infinity is strange.
There is no concept of
next to within its grasp
and yet it can be countable.

I keep studying math
because to me I have always
found myself immensely satisfied
when I come up with the right answer.

Now it seems the further I go
I find myself with less answers
and more questions and I only
hope that math will know.

I like the quiet of night
when the sleeping world
reaches its serendipitous pinnacle
and I can listen to no one talk.

In that delicate moment of the
restless yearning for rest I can
see answers on the insides of
my eyelids written in bright
flashes of neon light.

The few times that I
switch on the lamp to write
them down, they are no longer
there to be seen.

— (c) Brian Grimsley
4:45 A.M.
November 3, 1999

---

Any questions?

(Canon, “O du eselhafter Martin”, by Mozart)

The professor	Students 1, 2, and 3
Now then, are there any questions, Any problems, any questions? If there are none, then I am done, And I can bid you all good day. For there’s no reason I should stay here, Since I’ve said all I have to say here. If there are none then I am done, I wish you luck on the examination. And so my friends, I bid you all goodbye, I hope you liked this course as much as I. Goodbye, goodbye, Goodbye to one and all I say goodbye. Just one more thing — and do not laugh, I hope you’ll take the second half: Physics, Physics, Physics 11b.
For there’s no reason I should stay here, For I’ve said all I have to say here. If there are none then I am done, I wish you luck on the examination. And so my friends, I bid you all goodbye, I hope you liked this course as much as I. Goodbye, goodbye, Goodbye to one and all I say goodbye. Just one more thing — and do not laugh, I hope you’ll take the second half: Physics, Physics, Physics 11b. Now then, are there any questions, Any problems, any questions? If there are none, then I am done, And I can bid you all good day.	#1: Ha, he asks if there are any questions. Holy smoke have I got questions! I’ve got a ton, and every one, Would take him half a day to do. But I don’t really want to stay here Since he’s said all he has to say here But it’s agreed that I shall need Much more than luck on the examination. And so I think I’ll let him say goodbye I guess that he is as relieved as I Goodbye, goodbye, Thank God the course is over now, goodbye One thing he said that makes me laugh He hopes I’ll take the second half Ha ha, ha ha, ha ha, Don’t make me laugh.
Now then, are there any questions, Any problems, any questions? If there are none, then I am done, And I can bid you all good day. For there’s no reason I should stay here, Since I’ve said all I have to say here. If there are none then I am done, I wish you luck on the examination.	#2: Ha, he asks if there are any questions. Holy smoke have I got questions! I’ve got a ton, and every one, Would take him half a day to do.
And so my friends, I bid you all goodbye,	#2: Ha, he asks if there are any questions. #3: But I don’t really want to stay here
I hope you liked this course as much as I.	#2: Holy smokes have I got questions! #3: Since he’s said all he has to say here
Goodbye, goodbye,	#2: I’ve got a ton, and every one, #3: But it’s agreed that I shall need
Goodbye to one and all I say goodbye.	#2: Would take him half a day to do #3: Much more than luck on the examination.
Just one more thing — and do not laugh,	#1: Ha, he asks if there are any questions. #2: But I don’t really want to stay here #3: And so I think I’ll let him say goodbye
I hope you’ll take the second half:	#1: Holy smokes have I got questions! #2: Since he’s said all he has to say here #3: I guess that he is as relieved as I
Physics, Physics,	#1: I’ve got a ton, and every one, #2: But it’s agreed that I shall need #3: Goodbye, goodbye,
Physics 11b.	#1: Would take him half a day to do #2: Much more than luck on the examination. #3: Thank God the course is over now, goodbye
Now then, are there any questions,	#1: But I don’t really want to stay here #2: And so I think I’ll let him say goodbye #3: One thing he said that makes me laugh
Any problems, any questions?	#1: Since he’s said all he has to say here #2: I guess that he is as relieved as I #3: He hopes I’ll take the second half
If there are none, then I am done,	#1: But it’s agreed that I shall need #2: Goodbye, goodbye, #3: Ha ha, ha ha, ha ha, don’t make me laugh.
And I can bid you all good day.	#1: Much more than luck on the examination. #2: Thank God the course is over now, goodbye #3: Ha, he asks if there are any questions.
	#1: And so I think I’ll let him say goodbye #2: One thing he said that makes me laugh #3: Holy smokes have I got questions!#1: I guess that he is as relieved as I #2: He hopes I’ll take the second half #3: I’ve got a ton, and every one,#1: Goodbye, goodbye, #2: Ha ha, ha ha, ha ha, don’t make me laugh. #3: Would take him half a day to do

— (c) Tom Lehrer, 1951.

Download this song from Professor Walter Smith’s “Physical Revue” page.

---

The axes not analyzed

---

Brokenspacebar

myspacebarisbroken,
ohwhatshallIdo?
myspacebardoesn'tspace,
nospacewillitdo.
howcanIkeeptyping,
howcanIupdate?
ah!I'llremapmykeyboard
(whouses"f8"?)

— (c) Courtney Gibbons

---

The derivative song

(To the tune of “There’ll Be Some Changes Made.”)

Introductory Dialog, spoken by the Professor:
To cover that great theory would be my fondest hope,
Were it not that of this course it’s far beyond the scope.
This brings us to the question, though,
Of how much math you ought to know:
Most of it is inconsequential,
But the derivative is essential.
But you oughtn’t have trouble, ought you?
If you remember that song I taught you:

The Derivative Song

You take a function of x and you call it y,
Take any x-nought that you care to try,
You make a little change and call it delta x,
The corresponding change in y is what you find nex’,
And then you take the quotient and now carefully
Send delta x to zero, and I think you’ll see
That what the limit gives us, if our work all checks,
Is what we call dy/dx,
It’s just dy/dx.

— (c) Tom Lehrer,
American Mathematical Monthly,
81 (1974), p. 490.

Download this song from Professor Walter Smith’s “Physical Revue” page.

---

Did I miss anything?

Nothing. When we realized you weren’t here
we sat with our hands folded on our desks
in silence, for the full two hours.

Everything. I gave an exam worth
40 percent of the grade for this term
and assigned some reading due today
on which I’m about to hand out a quiz
worth 50 percent.

Nothing. None of the content of this course
has value or meaning.
Take as many days off as you like:
any activities we undertake as a class
I assure you will not matter either to you or me
and are without purpose.

Everything. A few minutes after we began last time
a shaft of light suddenly descended and an angel
or other heavenly being appeared
and revealed to us what each woman or man must do
to attain divine wisdom in this life and the hereafter.
This is the last time the class will meet
before we disperse to bring the good news to all people
on earth.

Nothing. When you are not present
how could something significant occur?

Everything. Contained in this classroom
is a microcosm of human experience
assembled for you to query and examine and ponder.
This is not the only place such an opportunity has been
gathered

but it was one place

And you weren’t here.

— by Tom Wayman
from Did I miss Anything? Selected Poems 1973-1993,
(c) 1993 Harbor Publishing

---

The drinking song (for mathematicians)

Chorus
We drink to mathematics, upon whose structure lies
The physics of both space and time, and why the butter flies.
We drink our coffee every day, and nightly quaff down beer:
The postulates and axioms come out as theorems here.

Rene from Chartres was a man who had it all worked out.
He said, “I think therefore I am — of this there is no doubt!
And by some lines in space you can all points therein describe,
So with that thought I think I shall another beer imbibe”

Chorus

Pierre de Fermat out of France, he was a gambling man.
He said “I read these books of mine to find out all I can.
“xⁿ plus yⁿ‘s not zⁿ if n does exceed two,
But the margins of this book cannot contain this wond’rous proof!”

Chorus

An apple fell and Newton saw it plummet to the ground.
He said “I wonder what doth make the Moon to go around?”
Gravity was born that day, and so was calculus,
With its fluxions and its fluents and its dots to torment us.

Chorus

Bernoulli: father, son, and brother, and an uncle too.
Daniel was a relative, a cousin once removed.
Math and physics gained a lot, for they fought endlessly,
And nothing breeds success so much as sibling rivalry!

Chorus

Euler did believe in God, just like his fellow man.
Diderot the atheist said “Prove God if you can!”
“Sir, (a + bn) over n is x so therefore God exists!”
And Diderot could not refute a proof as good as this.

Chorus

Joseph Louis Lagrange wrote a book full of mathematics.
In English it would be the Analytical Mechanics.
He said “Look high and low, you’ll find no pictures here!
So kind sir would you pour for me another pint of beer!”

Chorus

Marquis de Laplace did write some works on motions up in heaven.
He started work when but a lad of only ten and seven.
Napoleon said “I see no sign of God in this!”
Laplace replied “I have no need for that hypothesis.”

Chorus

Karl Friedrich Gauss had a choice at one year ‘ere his score,
To study mathematics, or words forevermore.
With compass and straightedge, he made his choice anon:
He went around and then inscribed a septendecagon.

Chorus

Cauchy and Bolzano thought that math was in a mess.
Separately they sought to make it much more rigorous!
Bolzano was Bohemian, and Cauchy came from France.
And with their work analysis continued its advance.

Chorus

Evariste Galois did write the theory of the group.
And with his knife did toast the King before the course of soup.
And at my age been laid to rest a dozen years or more.
So unto algebra I drink this next drink that I pour.

Chorus

Linear equations can be solved by anyone.
The cubics and quadratics and the quartics have been done.
Abel said “The quintics cannot likewise fall!
They cannot be resolved by means of radicals at all!”

Chorus

Riemann said that parallels need not exist in space.
Lobachevsky said that two or more could be in place.
The axioms of Euclid, which give us only one,
Cannot describe the space near massive bodies like the sun!

Chorus

Kronecker thought numbers should be things you can count on.
To root of 2 and rationals, he said “Vile things, begone!
Numbers should be whole! Let’s get rid of the reals!
And in their place we’ll substitute the theory of ideals!”

Chorus

Cantor showed that aleph₀ can count the integers.
And aleph₁ the reals, and aleph₂ the curves.
For aleph-three-or-more, we’ve got no use at all.
So let us sing of aleph₀ beer bottles on the wall.

Chorus

Banach, Tarski and Sierpinski were a bunch of Poles
Creators of pathologies, like gaskets full of holes!
“If you cut a sphere just so, from one you shall make two!”
(To comprehend this proof you need another pint of brew!)

Chorus

von Neumann was a man of math, from Hungary he came.
He did his work with automats and theories of the game.
He said “By minimax, you might as well coins flip!
“So you can play your game while I will have another sip!”

Chorus

The saga of mathematics has not yet in full been told.
The story, it goes on and on, it’s six millenia old.
And yet I fear this song, has gone on long enough.
So take my glass and fill it with some alcoholic stuff….

We drink to mathematics, upon whose structure lies
The physics of both space and time, and why the butter flies.
We drink our coffee every day, and nightly quaff down beer:
The postulates and axioms come out as theorems here.
Yes, the postulates and axioms come out as theorems here!

— (c) Jeff Suzuki, 2005

---

The drinking song (for philosophers)

Immanuel Kant was a real pissant
who was very rarely stable.
Heidegger, Heidegger was a boozy beggar
who could think you under the table.
David Hume could out consume
Wilhelm Friedrich Hegel,
And Wittgenstein was a beery swine
who was just as sloshed as Schlegel.

There’s nothing Nietzsche couldn’t teach ya
‘Bout the raisin’ of the wrist.
Socrates himself was permanently pissed.

John Stuart Mill, of his own free will,
After half a pint of shandy was particularly ill.
Plato, they say, could stick it away,
Half a crate of whiskey every day!
Aristotle, Aristotle was a bugger for the bottle,
And Hobbes was fond of his Dram.
And Rene Descartes was a drunken fart:
“I drink, therefore I am.”

Yes, Socrates himself is particularly missed;
A lovely little thinker, but a bugger when he’s pissed.

— (c) Monty Python’s Flying Circus

---

The electron in gold

There was an electron in gold
Who said, “Shall I do as I’m told?
Shall I snuggle down tight
With a brief flash of light
Or be Auger outside in the cold?”

Said the K-shell electron in gold,
“I’m thinking of leaving the fold
To be hit like a hammer
By an outgoing gamma.
In freedom I’ll live till I’m old.”

Said the K-shell electron in gold,
“I wonder if I might be bold
And make a slight shift
From this circular drift
And change this damned atom to platinum.”

— (c) Arthur H. Snell

If, like me, your physics needs a little help, the three stanzas refer to fluorescent yield, internal conversion, and electron capture, respectively.

---

Epigram
Nature and Nature’s laws lay hid in night.
God said, “Let Newton be!” and all was light.

— (c) Alexander Pope

It did not last: the Devil howling “Ho!
Let Einstein be!” restored the status quo.

— (c)  Sir John Collins Squire

---

Euclid drew a circle

Old Euclid drew a circle
On a sand-beach long ago.
He bounded and enclosed it
With angles thus and so.
His set of solemn greybeards
Nodded and argued much
Of arc and of circumference,
Diameter and such.
A silent child stood by them
From morning until noon
Because they drew such charming
Round pictures of the moon.

— (c) Vachel Lindsay

---

Failing my calculus

(To the tune of “Closer to Fine” by the Indigo Girls.)

I’m trying to get through this alive,
Maybe figure out these tangents and cosines,
And the best thing you’ve ever done for me
Is to help me take this course less seriously,
It’s only math, after all
yeah

Well, I kind of understood my high school algebra,
And in trig at least I pulled a solid “B”,
But Calculus — I don’t know why I’m in it,
Excuse me, but I think I’ve reached my limit,
As I approach despair.

Chorus:

These d‘s and these x‘s, they make me so nervous,
Who cares what the area under a curve is?
I don’t know the answer to these questions, slopes and curves and tangent lines,
And the less I see the point of the derivative,
I’m failing my Calculus.
I’m failing my Calculus.

Well, I went to see the teacher, with a sheepish grin,
And he looked at me like I was just a wart upon his chin,
I’ve tried so hard, but I can’t understand it,
I’m so right brained that I can write left handed,
I spent four hours prostrate, staring at the wall,
And I still don’t understand e.

Chorus

I stopped by the Web at three a.m.
To seek help from alt.algebra, and possibly from Ken,
But this is just a way that my brain won’t work,
So listen, if you’ll only do my homework,
I’ll help your favorite charity.

Chorus

— (c) Kenny Felder
North Carolina State University

---

Finite simple group of order 2

The path of love is never smooth
But mine’s continuous for you
You’re the upper bound in the chains of my heart
You’re my Axiom of Choice, you know it’s true

But lately our relation’s not so well-defined
And I just can’t function without you
I’ll prove my proposition and I’m sure you’ll find
We’re a finite simple group of order two

I’m losing my identity
I’m getting tensor every day
And without loss of generality
I will assume that you feel the same way

Since every time I see you, you just quotient out
The faithful image that I map into
But when we’re one-to-one you’ll see what I’m about
‘Cause we’re a finite simple group of order two

Our equivalence was stable,
A principal love bundle sitting deep inside
But then you drove a wedge between our two-forms
Now everything is so complexified

When we first met, we simply connected
My heart was open but too dense
Our system was already directed
To have a finite limit, in some sense

I’m living in the kernel of a rank-one map
From my domain, its image looks so blue,
‘Cause all I see are zeroes, it’s a cruel trap
But we’re a finite simple group of order two

I’m not the smoothest operator in my class,
But we’re a mirror pair, me and you,
So let’s apply forgetful functors to the past
And be a finite simple group, be a finite simple group,
Let’s be a finite simple group of order two
(Oughter: “Why not three?”)

I’ve proved my proposition now, as you can see,
So let’s both be associative and free
And by corollary, this shows you and I to be
Purely inseparable. Q. E. D.

— (c) The Klein Four Group

---

Foundations of arithmetic

One day when Mugg the Missing Link was prowling through the woods,
In search of wives and mammoth-meat and other useful goods,
Whom should he see, on pushing out from deep arboreal shade,
But Ogg, the Paleolithic Man, cross-legged in a glade.

This Ogg had made a neat array of pebbles on the ground,
In number they were twenty-one, the most that could be found,
And Ogg, with one red-hairy hand pressed to his bony brow,
Was staring at hese pebbles like a ruminating cow.

           o o o o o o o
           o o o o o o o
           o o o o o o o

Thought Mugg – for he was Primitive – I should be very dull
To lose this opportunity of busting in his skull;
My club weighs half a hundredweigh, he doesn’t wear a hat
(And here he wondered) Yes, but what the devil is he at?

For Ogg was touching pebbles and then prodding at his digits,
Until the weirdness of it all afflicted Mugg with fidgets:
“Invented any goodish wheels just recently?” he hollered,
And doubled up in merriments, his face raw-beefy coloured.

Ogg looked at him in pity, then he drummed upon his chest:
“I’ve don a Think!” he bellowed “Monkey Mugg. I’ve done a think!
And I would write it down, but no one’s yet invented ink.”

Mugg moved a little closer, and his eyes and mouth were round,
And stared in trepidation at those pebbles on the ground.
Ogg pointed with a nailed red-hairy sausage at the rows
And said, “Three people’s hand-plus-two is hand-plus-feet-plus-nose.”

oooo       oooo       oooo       oooo     oooo
     o   +      o   +      o   =      o o       + ooooO Oooo + A
o          o          o
o          o          o

“And this is hand-plus-two of people’s three-for-each-by-name,
So three times hand-plus-two and hand-plus-to time three’s the same!”
Mugg scratched his matted hairy head, not knowing what to say.
Said Ogg, “It’s all made clear by this rectangular array.”

                             o o o
                             o o o
           o o o o o o o     o o o
           o o o o o o o  =  o o o
           o o o o o o o     o o o
                             o o o
                             o o o

“Three rows of hand-plus-two and hand-plus-two short rows of three
are just the same according to which way you look, you see?
In brief, a tripe heptad is the same as seven trebles,
And may quite possibly be true of other things than pebbles.”

Mugg viewed it from all angles, then he gave a raucous belch
And trod on a Batrachian that perished with a squelch.
He growled, “I do not understand these arithmetic quirks,
But maybe we should try to discover if it works.”

So home they went to get their wives and drag them by the hair,
For Mugg had feet-plus-hand-plus-four, while Ogg had just a pair;
But what with all their screeching and their running every way,
At first they would not form a neat rectangular array.

So Ogg he then positioned each by holding of her down
While Mugg with mighty club in hand, just dinted in her crown;
And when they had them all in place, like pebbles, they could see
That three times hand-plus-two in wives was hand-plus-two times three!

      o<= o<= =>o =>o o<= o<= =>o

      =>o o<= o<= =>o o<= o<= =>o

      o<= =>o =>o o<= =>o o<= =>o

Then Ogg he roared in high delight, cartwheeling to and fro
(Carts had not been invented, but he did it just to show!),
And Mugg he grinned a shaggy grin and slapped a hairy thigh
And said, “It’s true, as sure as Pterodactyls learned to fly!”

And then they feasted on their wives in unluxorious zest,
Except for one whose skull was rather thicker than the rest,
And she was sent to dig a pit and bury every bone,
While Mugg and Ogg went off to find a flat unsullied stone.

Then Ogg he sharpened up a flint and scratched upon the rock:
First Arithmetic Theorem – by Ogg the son of Mok.
He drew his little diagram, and proved, with QED,
That three times hand-plus-two of x is hand-plus-two times three.

But Mugg the Missing Link grew bored, and left him there alone,
Still scratching with his silly flint upon his silly stone;
And belching, plunged back in the woods on feet toe’s simple fives,
In search of wives and mammoth-meat, particularly wives!

— (c) J. A. Lindon

---

Haikumputation I

The haiku form is simple: a verse of 17 syllables, divided into three lines of five, seven, and five syllables respectively. The Western ear should note that the metrical unit is the syllable (Japanese is a syllabic language) and not, as in Western prosody, the foot composed of one or two syllables. The form of 17 syllables is not chance; it derives from the traditional view of Japanese linguistic philosophy that 17 syllables is the optimum length of human speech to be delivered clearly and coherently in one breathing.

All green in the leaves
I smell dark pools in the trees
Crash the moon has fled

All white in the buds
I flash snow peaks in the spring
Bang the sun has fogged

All starred in the cold
I seize thin trails in the mist
Look the oth has gone

The three examples above were produced by online man-machine interaction at the Cambridge Language Research Unit. The program provides a frame with “slots” in which the operator types words. His choice is constrained by the lists and arrow directions in the thesaurus and diagram (below). These show the semantic center of the poem, with five arrows going to it and one going from it, is situated at slot 5.

Thesaurus

Semantic schema

All [--1--] in the [--2--]
I [--3--] [--4--] [--5--] in the [--6--]
[--7--] the [--8--] has [--9--]

An asterisk * above indicates a double linkage. For the system to be computable, only one arrow may be chosen.

Conclusions

Here are two haiku written by human members of the NPL.

Pattern perception
Is easier to do than
Cerebrate about

Don’t design systems
Of automatic control.
Ride a bicycle.

Adapted from the catalog of the exhibition
Cybernetic Serendipity: the Computers and the Arts,
(c) Studio International, London 1968 p. 53; and NPL News 204, 10, 1967.

---

Haikumputation II

In Japan, they have replaced the impersonal and unhelpful Microsoft error messages with Haiku poetry messages. Haiku poetry has strict construction rules. Each poem has only three lines, 17 syllables: five syllables in the first line, seven in the second, five in the third. Haiku is used to communicate a timeless message often achieving a wistful, yearning and powerful insight through extreme brevity — the essence of Zen:

    A crash reduces
    Your expensive computer
    To a simple stone.

    Chaos reigns within.
    Reflect, repent, and reboot.
    Order shall return.

    First snow, then silence.
    This thousand-dollar screen dies
    So beautifully.

    Having been erased,
    The document you're seeking
    Must now be retyped.

    Out of memory.
    We wish to hold the whole sky,
    But we never will.

    Program aborting:
    Close all that you have worked on.
    You ask far too much.

    Serious error.
    All shortcuts have disappeared.
    Screen. Mind. Both are blank.

    Stay the patient course.
    Of little worth is your ire.
    The network is down.

    The Tao that is seen
    Is not the true Tao -- until
    You bring fresh toner.

    The Website you seek
    Cannot be located, but
    Countless more exist.

    Three things are certain:
    Death, taxes and lost data.
    Guess which has occurred.

    Windows NT crashed.
    I am the Blue Screen of Death.
    No one hears your screams.

    With searching comes loss
    And the presence of absence:
    Your file, not found.

    Yesterday it worked.
    Today it is not working.
    Windows is like that.

    You step in the stream,
    But the water has moved on.
    This page is not here.

    Your file was so big.
    It might be very useful.
    But now it is gone.

---

Hilbert space

We’ve studied oscillators big and small
With pendulum and breathing modes embraced
But there’s another way to grasp it all,
With normal modes as vectors in a space.
This inner product space holds many keys.
Where basis vectors sum to give us clues
‘Bout how to wield our normal modes with ease,
And thus, solutions to our Diff. EQs.
Used often in describing quantum states,
Indeed a daunting subject it subtends.
But with the vector math it correlates,
One sees its usefulness needs no defense.
Without it, where we’d be, I cannot place.
Its name, known far and wide, is Hilbert Space.

— (c) Lee Herron, Haverford College, 2003

---

Hyperbolic orbit

It’s a universal law that bodies are drawn to one another by love (or gravity). But, sometimes, a relationship (or gravitational interaction) just isn’t meant to last, and, after a quick fling, the two involved simply drift apart.

She is just a wanderer with no destination,
And he is just a stop along her way.
Coming near him, she feels a strange attraction,
But it might not be enough to make her stay.

Every part of her is drawn to be near him,
It warps her world, this feeling is so vast!
Perhaps that’s why he thinks the earth revolves around him,
But as for her, she’s moving much too fast…

It’ll never last.

‘Cause she is on a hyperbolic orbit;
No matter how she tries to bend, she knows the dance is going to end.
She is on a hyperbolic orbit,
And she won’t be coming back his way again…

He’s burning up inside, all his friends can see him glowing,
Can his warmth begin to melt her icy heart?
And leave a trail of stars so bright they paint her name across the night?
Or is it written they must always be apart?

Perhaps she wishes that she didn’t have to leave him;
But she knows she’s not the type to stay too long.
There won’t be another chance, but she’s got time for just one dance,
So she’ll whirl around him once, then she’ll move on…

And she’ll be gone.

Cause she is on a hyperbolic orbit;
She has too much energy to be anything but free.
She is on a hyperbolic orbit,
And a wanderer is all she’ll ever be…

— Benjamin Newman

Download this song from Haverford College’s “Physics Songs” page

---

I got physics

(To the tune of “I Got Rhythm”)

In this spinning, complex world, the masses lose their way
But I am never lost, I feel this way because…

I got scalars, I got vectors, I got physics
Who could ask for anything more?
I’ve got good grades, no more bad grades,
I’ve got physics, who could ask for anything more?

Tycho Brahe, I don’t mind him,
You won’t find him ’round my door.
I’ve got wavelengths, I’ve got impulse,
I’ve got physics, who could ask for anything more,
Who could ask for anything more?

I got scalars, I got vectors, I got physics,
Who could ask for anything more?
I’ve got good grades, no more bad grades,
I’ve got physics, who could ask for anything more?

Old man Brahe, I don’t mind him,
You won’t find his gold nose round my door.
Oh, I’ve got Isaac, I’ve got Albert,
I got physics, who could ask for anything more?
Who could ask for anything more?

— William M. Phillian
Montclair State University

---

If

If you can solve a literal equation
And rationalize denominator surds,
Do grouping factors (with a transformation)
And state the factor theorem in words;
If you can plot the graph of any function
And do a long division (with gaps),
Or square binomials without compunction
Or work cube roots with logs without mishaps.
If you possess a sound and clear-cut notion
Of interest sums with P and i unknown;
If you can find the speed of trains in motion,
Given some lengths and “passing-times” alone;
If you can play with R (both big and little)
And feel at home with l (or h) and pi,
And learn by cancellation how to whittle
Your fractions down till they delight the eye.
If you can recognize the segment angles
Both at the center and circumference;
If you can spot equivalent triangles
And friend Pythagoras (his power’s immense);
If you can see that equiangularity
And congruence are two things and not one,
You may pick up a mark or two in charity
And, what is more, you may squeeze through, my son.

— Times Educational Supplement, July 19, (c) 1947.

---

Ions mine

In the dusty lab’ratory,
Mid the coils and wax and twine,
There the atoms in their glory
Ionize and recombine.

Chorus
On my darlings! Oh my darlings!
Oh my darling ions mine!
You are lost and gone forever
When just once you recombine!

In a tube quite electrodeless,
They discharge around a line,
And the glow they leave behind them
Is quite corking for a time.

Chorus

And with quite a small expansion
1 : 8 or 1 : 9,
You can get a cloud delightful,
Which explains both snow and rain.

Chorus

In the weird magnetic circuit
See how lovingly they twine,
As each ion describes a spiral
Round its own magnetic line.

Chorus

Ultraviolet radiation
From the arc or glowing lime,
Soon discharges a conductor
If it’s charged with minus sign.

Chorus

Alpha rays from radium bromide
Cause a zinc-blende screen to shine,
Set it glowing, clearly showing
Scintillations all the time.

Chorus

Radium bromide emanation,
Rutherford did first divine,
Turns to helium, then Sir William
Got the spectrum every line.

On my darlings! Oh my darlings!
Oh my darling ions mine!
You are lost and gone forever
When just once you recombine!
You are lost and gone forever
When just once you recombine!

— J. J. E. Durak

In the heroic days of the Cavandish Laboratory, it was the custom to hold an annual dinner followed by home-made entertainments, usually songs at the piano. These Postprandial Porceedings of the Cavandish Society (Cambridge: Bowes and Bowes 1926) celebrate the discoveries of gas discharge phenomena and the early days of nuclear physics.

Legend has it that this song was penned by J. J. Thompson, the discoverer of the electron.

---

The Kiss Precise

Involves no trigonometry.
‘Tis not so when for circles kiss
Each one the other three.
To bring this off the four must be:
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.

---

Lines inspired by a lecture on extra-terrestrial life

Some time ago my late Papa
Acquired a spiral nebula.
He bought it with a guarantee
Of content and stability.
What was his undisguised chagrin
To find his purchase on the spin,
Receding from his call or beck
At several million miles per sec.,
And not, according to his friends,
A likely source of dividends.
Justly incensed at such a tort
He hauled his vendor into court,
Taking his stand on Section 3
Of Bailey “Sale of Nebulae.”
Contra was cited Volume 4
Of Eggleston’s “Galactic Law”
That most instructive little tome
That lies uncut in every home.
“Cease,” said the sage, “Your quarrel base,
Lift up your eyes to outer space.
See where the nebulae like buns,
Encurranted with infant suns,
Shimmer in incandescent spray
Millions of miles and years away.
Think that, provided you will wait,
Your nebula is Real Estate,
Sure to provide you wealth and bliss
Beyond the dreams of avarice.
Watch as the rolling aeons pass
New worlds emerging from the gas:
Watch as brightness slowly clots
To eligible building lots.
What matters a depleted purse
To owners of a Universe?”
My father lost the case and died:
I watch my nebula with pride
But yearly with decreasing hope
I buy a larger telescope.

— (c) J. D. G. M.
From The Observatory 65, 88 (1943).

---

Lobachevsky

For many years now, Mr. Danny Kaye, who has been my particular idol since childbirth, has been doing a routine about the great Russian director Stanislavsky and the secret of success in the acting profession. And I thought it would be interesting to stea… to adapt this idea to the field of mathematics. I always like to make explicit the fact that before I went off not too long ago to fight in the trenches, I was a mathematician by profession. I don’t like people to get the idea that I have to do this for a living. I mean, it isn’t as though I had to do this, you know, I could be making, oh, $3000 a year just teaching.

Be that as it may, some of you may have had occasion to run into mathematicians and to wonder therefore how they got that way, and here, in partial explanation perhaps, is the story of the great Russian mathematician Nicolai Ivanovich Lobachevsky.

Who made me the genius I am today,
The mathematician that others all quote,
Who’s the professor that made me that way?
The greatest that ever got chalk on his coat.

One man deserves the credit,
One man deserves the blame,
and Nicolai Ivanovich Lobachevsky is his name. Oy!
Nicolai Ivanovich Lobache…

I am never forget the day I first meet the great Lobachevsky. In one word he told me secret of success in mathematics: Plagiarize!

Plagiarize,
Let no one else’s work evade your eyes,
Remember why the good Lord made your eyes,
So don’t shade your eyes,
But plagiarize, plagiarize, plagiarize…
Only be sure always to call it please “research.”

And ever since I meet this man my life is not the same,
And Nicolai Ivanovich Lobachevsky is his name. Oy!
Nicolai Ivanovich Lobache…

I am never forget the day I am given first original paper to write. It was on analytic and algebraic topology of locally Euclidean metrization of infinitely differentiable Riemannian manifold. Bozhe moi! This I know from nothing. But I think of great Lobachevsky and I get idea — ha! ha!

I have a friend in Minsk,
Who has a friend in Pinsk,
Whose friend in Omsk
Has friend in Tomsk
With friend in Akmolinsk.
His friend in Alexandrovsk
Has friend in Petropavlovsk,
Whose friend somehow
Is solving now
The problem in Dnepropetrovsk.

And when his work is done —
Haha! — begins the fun.
From Dnepropetrovsk
To Petropavlovsk,
By way of Iliysk,
And Novorossiysk,
To Alexandrovsk to Akmolinsk
To Tomsk to Omsk
To Pinsk to Minsk
To me the news will run,
Yes, to me the news will run!

And then I write
By morning, night,
And afternoon,
And pretty soon
My name in Dnepropetrovsk is cursed,
When he finds out I published first!

And who made me a big success
And brought me wealth and fame?
Nicolai Ivanovich Lobachevsky is his name. Oy!
Nicolai Ivanovich Lobache…

I am never forget the day my first book is published. Every chapter I stole from somewhere else. Index I copy from old Vladivostok telephone directory. This book, this book was sensational!

Pravda — ah, Pravda — Pravda said…
It stinks.
But Izvestia! Izvestia said…
It stinks.
Metro-Goldwyn-Moskva bought the movie rights for six million rubles,
Changing title to The Eternal Triangle,
With Brigitte Bardot playing part of hypotenuse.

And who deserves the credit?
And who deserves the blame?
Nicolai Ivanovich Lobachevsky is his name.
Oy!

— Tom Lehrer

---

The New Math

(For the stage)

Some of you who have small children may have perhaps been put in the embarrassing position of being unable to do your child’s arithmetic homework because of the current revolution in mathematics teaching known as the New Math. So as a public service here tonight, I thought I would offer a brief lesson in the New Math. Tonight, we’re gonna cover subtraction. This is the first room I’ve worked for a while that didn’t have a blackboard, so we will have to make do with more primitive visual aids, as they say in the ed biz. Consider the following subtraction problem, which I will put up here: 342 – 173. Now, remember how we used to do that: But in the new approach, as you know, the important thing is to understand what you’re doing, rather than to get the right answer. Here’s how they do it now:

You can’t take three from two,
Two is less than three,
So you look at the four in the tens place.
Now that’s really four tens
So you make it three tens,
Regroup, and you change a ten to ten ones,
And you add ’em to the two and get twelve,
And you take away three, that’s nine.
Is that clear?

Now instead of four in the tens place
You’ve got three,
‘Cause you added one,
That is to say, ten, to the two,
But you can’t take seven from three,
So you look in the hundreds place.

From the three you then use one
To make ten ones…
(And you know why four plus minus one
Plus ten is fourteen minus one?
‘Cause addition is commutative, right!)…
And so you’ve got thirteen tens
And you take away seven,
And that leaves five…

Well, six actually…
But the idea is the important thing!

Now go back to the hundreds place,
You’re left with two,
And you take away one from two,
And that leaves…?

Everybody get one?
Not bad for the first day!

Hooray for New Math,
New-hoo-hoo Math,
It won’t do you a bit of good to review math.
It’s so simple,
So very simple,
That only a child can do it!

Now, that actually is not the answer that I had in mind, because the book that I got this problem out of wants you to do it in base eight. But don’t panic! Base eight is just like base ten really — if you’re missing two fingers! Shall we have a go at it? Hang on…

You can’t take three from two,
Two is less than three,
So you look at the four in the eights place.
Now that’s really four eights,
So you make it three eights,
Regroup, and you change an eight to eight ones
And you add ’em to the two,
And you get one-two base eight,
Which is ten base ten,
And you take away three, that’s seven.
Ok?

Now instead of four in the eights place
You’ve got three,
‘Cause you added one,
That is to say, eight, to the two,
But you can’t take seven from three,
So you look at the sixty-fours…

Sixty-four? “How did sixty-four get into it?” I hear you cry! Well, sixty-four is eight squared, don’t you see? (Well, ya ask a silly question, ya get a silly answer!)

From the three, you then use one
To make eight ones,
You add those ones to the three,
And you get one-three base eight,
Or, in other words,
In base ten you have eleven,
And you take away seven,
And seven from eleven is four!
Now go back to the sixty-fours,
You’re left with two,
And you take away one from two,
And that leaves…?

Now, let’s not always see the same hands!
One, that’s right.
Whoever got one can stay after the show and clean the erasers.

Hooray for New Math,
New-hoo-hoo Math!
It won’t do you a bit of good to review math.
It’s so simple,
So very simple,
That only a child can do it!

Come back tomorrow night…we’re gonna do fractions!

Y’know, I’ve often thought I’d like to write a mathematics textbook someday because I have a title that I know will sell a million copies; I’m gonna call it Tropic of Calculus.

— Tom Lehrer

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Ode to Z

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Oh! The places waves go!

— Tom Lehrer

The math version (c) American Mathematical Monthly, 81 (1974), p. 745

The physics version (c) Physics Today, July 2005, p. 58

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The publication system: a jaundiced view

This is the paper that X wrote.

This is the editor, all distraught,
Who tore his hair at the horrible thought
Of printing the paper X wrote.

This is the friend whose help was sought,
By E, the editor, all distraught,
Who tore his hair and groaned at the thought
Of the horrible paper X wrote.

This is the proof, all shiny and new,
Of 2.1 and 2.2
Conceived by F, whose help was sought
By E, the editor, all distraught,
Who tore his hair and groaned at the thought
Of the odious paper X wrote.

This Covering Note pretends to be
Detached about the referee.
(“He doesn’t tell you how to fix
The proof of Theorem 2.6.”)
It quotes the Referee’s Report
Which says Such Things are Better Short
And gives the proof, all shiny and new,
Of 2.1 and 2.2
Proposed by F, whose help was sought
By E, the editor, all distraught,
Who tore his hair and groaned at the thought
Of the pitiful paper X wrote.

“But we are sure it can be mended;
If wholly changed it could be splendid.”
Typing on a new machine,
F answers for his magazine. Signed E₁ = F₂ = X₃.

— Ian Stewart and John Jaworski
From Seven Years of Manifold, 1968-1980
(c) Nantwhich: Shive Publishing, 1981, 90.

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Relativity

(To the tune of “Personality” by James Van Heusen.)

Introductory Dialog, spoken by the English major:
But I think this course underemphasizes the importance of relativity, and I would like you to hear a few words in its behalf.

Relativity

Einstein was the first who stated,
He was the first who dared:
Mass and energy are related,
By E = mc².

When Isaac Newton wrote The Laws that we all quote,
It’s now extremely apparent that he
Neglected to consider: relativity.

What focused our attention on the fourth dimension?
We’d been doing so well with just three:
‘Twas Mr. Einstein’s brainchild: relativity.

Now who would think, and who’d forecast,
That bodies shrink, when they go fast.
It makes old Isaac’s theory
Look weary.

So then if you are near when atom bombs appear,
And you’re reduced to a pile of debris,
You’ll know it’s largely due to… relativity..
Yes, you can place the blame on… relativity.

— Tom Lehrer
(c) American Mathematical Monthly, 81 (1974) 612

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The researcher’s prayer

Grant, O God, Thy benedictions
On my theory’s predictions
Lest the facts, when verified,
Show Thy servant to have lied.

May they make me B.Sc.,
A Ph.D. and then
A D.Sc., and F.R.S.,
A Times Obit. Amen.

O Lord, I pray, forgive me please,
My unsuccessful syntheses,
Thou know’st, of course — in Thy position —
I’m up against such competition.

Let not the hardened Editor,
With referee to quote,
Cut all my explanation out
And print it as a Note.

— Proceedings of the Chemical Society,
(c) January 1963, 8-10

The Proceedings of the Chemical Society record some of the 128 verses submitted in a competition at Christmas 1962 for quatrains in the style of the Fisherman’ Parayer: “God give me strength to catch a fish / So large that even I / When telling of it afterwards / May never need to lie.”

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s is one-half gt-squared

Introductory dialog by the Professor: Now, with your permission, I’ll attempt to summarize the fundamentals that this course has tried to emphasize:

The professor	The student
s is one-half gt-squared PV equals nRT KE is one-half mv-squared and T is 2 pi times the square root of L/g.
	I try my level best, and I cram for every test, And I do all my work without delaying, I’m certainly not lazy but I think I’ll soon go crazy, ‘Cause I don’t understand a word he’s saying. I hope they overlook this course, For why I ever took this course, Will always be a mystery to me. But still I wish I knew, Just what the Dean will do, If I don’t get a C.
The force is mass times acceleration Yes, the force is the mass times the acceleration, or more precisely, the time derivative of the linear momentum.	I think he said the force I know that word, of course It’s all becoming clear But it’s too late, I fear.There he goes again!
s is one-half gt-squared PV equals nRT KE is one-half mv-squared and T is 2 pi times the square root of L/g.	I try my level best, and I cram for every test, And I do all my work without delaying, I’m certainly not lazy but I think I’ll soon go crazy, ‘Cause I don’t understand a word he’s saying. I hope they overlook this course, For why I ever took this course, Will always be a mystery to me. But still I wish I knew, Just what the Dean will do, If I don’t get a C.
By g, by g, Square root of L by g…[ Repeat from second verse]	A C, a C, If I don’t get a C…[ Repeat from second verse]

— Tom Lehrer, 1951

Download this song from Professor Walter Smith’s “Physical Revue” page.

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The slide rule song

Introductory Dialog, spoken by the Professor: Now to get on with the class —
I give you now the Section Man,
This friend of yours and mine’ll
Offer you some very good advice about the final.

Section Man: Be sure to bring your slide rules,
Don’t forget them if you’re sensible.
For if you use them wisely,
You’ll find them indispensable.

The Slide Rule Song

Don’t bring the answers in on bits of paper,
And don’t be crude and write them on your cuff:
The proctors would catch on to such a caper,
And you can bet they’d get you soon enough.

Yes, the proctors would catch on to such a caper
And you can bet they’d get you soon enough.

Don’t write them on your thumbnail — that’s the worst place,
Don’t write them in the lining of your hat:
You really shouldn’t be here in the first place,
If you can’t be more original than that.

No, you really shouldn’t be here in the first place
If you can’t be more original than that.

Against such things they have a justified rule,
They’d expel you without benefit of doubt.
But if you hide the answers in your slide rule,
It’s eight-to-five that no one will find out.

Yes, if you hide the answers in your slide rule,
It’s eight-to-five that no one will find out.

— Tom Lehrer, 1951

Download this song from Professor Walter Smith’s “Physical Revue” page.

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Take away your billion dollars

Upon the lawns of Washington the physicists assemble,
From all the land are men at hand, their wisdom to exchange.
A great man stands to speak, and with applause the rafters tremble.
‘My friends,’ says he, ‘You all can see that physics now must change. Now in my lab we had our plans, but these we’ll now expand,
Research right now is useless, we have come to understand.
We now propose constructing at an ancient Army base,
The best electronuclear machine at any pace. –Oh

‘It will cost a billion dollars, then billion volts ’twill give,
It will take five thousand scholars sever years to make it live.
All the generals approve it, all the money’s now in hand,
And to help advance our program, teaching students now we’ve banned.
We have chartered transportation, we’ll provide a weekly dance,
Our motto’s integration, there is nothing left to chance.
This machine is just a model for a bigger one, of course,
That’s the future road for physics, as I hope you’ll all endorse.’

And as the halls with cheers resound and praises fill the air,
One single man remains aloof and silent in his chair.
And when the room is quiet and the crowd has ceased to cheer,
He rises up and thunders forth an answer loud and clear:
‘It seems that I’m a failure, just a piddling dilettante,
Within six months a mere ten thousand bucks is all I’ve spent,
With love and string and sealing wax was physics kept alive,
Let not the wealth of Midas hide the goal for which we strive. –Oh

‘Take away your billion dollars, take away your tainted gold,
You can keep your damn ten billion volts, my soul will not be sold.
Take away your army generals; their kiss is death, I’m sure.
Everything I build is mine, and every volt I make is pure.
Take away your integration; let us learn and let us teach,
Oh, beware this epidemic Berkeleyitis, I beseech.
Oh, dammit! Engineering isn’t physics, is that plain?
Take, oh take, your billion dollars, let’s be physicists again.’

— Arthur Roberts, 1946

1956: a sequel

Within the halls of NSF the panelists assemble.
From all the land the experts band their wisdom to exchange.
A great man stands to speak and with applause the rafters tremble.
‘My friends,’ says he, ‘we all can see that budgets now must change.
By toil and sweat the Soviets have reached ten billion volts.
Shall we downtrodden physicists submit? No, no — revolt!
It never shall be said that we let others lead the way.
We’ll band together all out finest brains and save the day.

‘Give us back our billion dollars, better add ten billion more.
If your budget looks unbalanced, just remember this is war.
Never mind the Army’s shrieking, never mind the Navy’s pain,
Never mind the Air Force projects disappearing down the drain.
In coordinates barycentric, every BeV means lots of cash,
There will be no cheap solutions, — neither straight nor synoclash.
If we outbuild the Russians, it will be because we spend.
Give, oh give those billion dollars, let them flow without an end.

Written while the Brookhaven National Laboratory was being planned. Folklore records that the brave and solitary scientist who so vigorously defended the purity of science at the original meeting was killed by a beam of hyperons when the Berkeley Bevatron was first switched on.

Download this song from Professor Walter Smith’s “Physics songs” page.

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That’s mathematics!

(To the tune of “That’s Entertainment!”)

Counting sheep
When you’re trying to sleep,
Being fair
When there’s something to share,
Being neat
When you’re folding a sheet,
That’s mathematics!

When a ball
Bounces off of a wall,
When you cook
From a recipe book,
When you know
How much money you owe,
That’s mathematics!

How much gold can you hold in an elephant’s ear?
When it’s noon on the moon, then what time is it here?
If you could count for a year, would you get to infinity,
Or somewhere in that vicinity?

When you choose
How much postage to use,
When you know
What’s the chance it will snow,
When you bet
And you end up in debt,
Oh try as you may,
You just can’t get away
From mathematics!

Andrew Wiles gently smiles,
Does his thing, and voila!
Q.E.D., we agree,
And we all shout hurrah!
As he confirms what Fermat
Jotted down in that margin,
Which could’ve used some enlargin’.

Tap your feet,
Keepin’ time to a beat,
Of a song
While you’re singing along,
Harmonize
With the rest of the guys,
Yes, try as you may,
You just can’t get away
From mathematics!

— Tom Lehrer

This version was written for the July 18, 1993, Fermat Fest, held in San Francisco and presented by the Mathematical Sciences Research Institute to celebrate the fact that Andrew Wiles had proven the famous Fermat’s Last Theorem that had gone unproven for centuries.

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There’s a delta for every epsilon

(Calypso)

There’s a delta for every epsilon,
It’s a fact that you can always count upon.
There’s a delta for every epsilon
And now and again,
There’s also an N.

But one condition I must give:
The epsilon must be positive
A lonely life all the others live,
In no theorem
A delta for them.

How sad, how cruel, how tragic,
How pitiful, and other adjec-
Tives that I might mention.
The matter merits our attention.
If an epsilon is a hero,
Just because it is greater than zero,
It must be mighty discouragin’
To lie to the left of the origin.

This rank discrimination is not for us,
We must fight for an enlightened calculus,
Where epsilons all, both minus and plus,
Have deltas
To call their own.

— Tom Lehrer,
(c) American Mathematical Monthly, 81 (1974) 612.

Download this song from Professor Walter Smith’s “Physical Revue” page.

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The triumph of reason

Behold the mighty dinosaur
Famous in prehistoric lore,
Not only for his weight and length
But for his intellectual strength.
You will observe by these remains
The creature had two sets of brains–
One on his head (the usual place),
The other at his spinal base.
Thus he could reason a priori
As well as a posteriori.
No problem bothered him a bit
He made both head and tail of it.

So wise was he, so wise and solemn,
Each thought filled just a spinal column.
If one brain found the pressure strong
It passed a few ideas along.
If something slipped his forward mind
‘Twas rescued by the one behind.
And if in error he was caught
He had a saving afterthought.
As he thought twice before he spoke
He had no judgment to revoke.
Thus he could think without congestion
Upon both sides of every question.
Oh, gaze upon this model beast
Defunct ten million years at least.

— (c) Bert Liston Taylor
From “A line o’Type or Two,” Chicago Tribune, circa 1920

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The zeros of the Zeta Function

Where are the zeros of zeta of s?
G.F.B. Riemann has made a good guess;
They’re all on the critical line, saith he,
And their density’s one over 2 pi log(t).

This statement of Riemann’s has been like a trigger,
And many good men, with vim and with vigour,
Have attempted to find, with mathematical rigour,
What happens to zeta as mod t gets bigger.

The efforts of Landau and Bohr and Cramer,
Littlewood, Hardy and Titchmarsh are there,
In spite of their effort and skill and finesse,
In locating the zeros there’s been little success.

In 1914 G.H. Hardy did find,
An infinite number do lay on the line,
His theorem, however, won’t rule out the case,
There might be a zero at some other place.

Oh, where are the zeros of zeta of s?
We must know exactly, we cannot just guess.
In order to strengthen the prime number theorem,
The integral’s contour must never go near ’em.

Let P be the function p minus Li,
The order of P is not known for x high,
If square root of x times log x we could show,
Then Riemann’s conjecture would surely be so.

Related to this is another enigma,
Concerning the Lindelöf function mu sigma.
Which measures the growth in the critical strip,
On the number of zeros it gives us a grip.

But nobody knows how this function behaves,
Convexity tells us it can have no waves,
Lindelöf said that the shape of its graph,
Is constant when sigma is more than one-half.

There’s a moral to draw from this sad tale of woe,
which every young genius among you should know:
If you tackle a problem and seem to get stuck,
Use R.M.T., and you’ll have better luck.

— (c) Tom Apostal, with revisions by Chris Hughes

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Xmas Carols

Calculus

(To the melody of Jingle Bells)

Looking for the tangent
It’s really m we seek
With epsilon and delta
Mathematics looks like Greek

Trying to find a limit
Everything gets small
If you can’t determine it
You land in L’Hopital

Calculus, Calculus
Let us celebrate
Riemann Sums, so much fun
We can integrate

Hey!

Calculus, Calculus
Let us celebrate
dx, dy, dz, dt
We love related rates

Calc 2 is coming next
The really fun stuff starts
Integral x sin(x)
Only works in parts

Summing the series harmonic
The terms keep getting small
But isn’t it ironic?
It won’t converge at all

Calculus, Calculus
Put your mind at rest
Most divergent series fail
The Root or Ratio Test

Hey!

Calculus, Calculus
While we sing this song
We can sum (1/2)ⁿ
It won’t take all day long

Now on to Calc 3
Learning spheres and cones
Maximizing functions
With seventeen unknowns

Fancy vector fields
Finding flux of curl
Using Mr. Stokes
Over a circle

Calculus, Calculus
Ain’t it really cool
Gauss and Green are really keen
Don’t take them for fools

Hey!

Calculus, Calculus
Ain’t it really cool
f and g, composing thee
Time for the chain rule

— (c) Deborah Alterman, Martin Mohlenkamp, and Gareth Roberts

Cantor sets

(To the melody of Jingle Bells)

Cantor sets, Cantor sets
Going from naught to one.
Walking on a cantor set
Would be so much fun.

Hey!

Cantor sets, cantor sets
Arc length they have none.
To get across it in no time
I guess you’d have to run.

— Anon

Deck the board with differentials

(To the melody of Deck the Halls)

Fill the boards with differentials,
Fa-la-la-la-la La-la-la-la.
Note that du’s are essential,
Fa-la-la-la-la La-la-la-la.
C’s are constants here before us,
Fa-la-la La-la-la La-la-la.
Integration cannot floor us,
Fa-la-la La-la-la La-la-la.

Quizzes always make us queasy,
Fa-la-la-la-la La-la-la-la.
Max and mins are never easy,
Fa-la-la-la-la La-la-la-la.
Conic volumes we can measure,
Fa-la-la La-la-la La-la-la.
The FTC we’ll always treasure,
Fa-la-la La-la-la La-la-la.

— (c) Dennis Gannon (1940-1991)

O calculus! O calculus!

(To the melody of O Tanunbaum)

Oh calculus; oh calculus!
How tough are both thy branches.
O calculus; O calculus!
To pass, what are my chances?
Derivatives I cannot take,
At integrals my fingers shake.
O calculus; O calculus!
How tough are both thy branches.

O calculus; O calculus!
Thy theorems I can’t master.
O calculus; O calculus!
My proofs are a disaster.
You pull a trick out of the air,
Or find a reason God knows where.
O calculus; O calculus!
Thy theorems I can’t master.

O calculus; O calculus!
Thy problems so distress me.
O calculus; O calculus!
Related rates depress me.
I walk toward lampposts in my sleep,
And running water makes me weep.
O calculus; O calculus!
Thy problems so distress me.

O calculus; O calculus!
My limit I am reaching.
O calculus; O calculus!
For mercy I’m beseeching.
My grades to not approach a B,
They’re just an epsilon from D.
O calculus; O calculus!
My limit I am reaching.

— (c) Dennis Gannon (1940-1991)

Dennis Gannon was the head of the society for Isaac Newton and an inspiring teacher at F. T. Maloney High School in Meriden, CT for 29 years. Each year at the holiday season he bellowed out Calculus Carols that he wrote for his AP students.

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Fission and superstition

This is the Tale of Frederick Wermyss
Whose Parents weren’t on speaking terms.
So when Fred wrote to Santa Claus
It was in duplicate because
One went to Dad and one to Mum —
Both ask for some Plutonium.
See the result: Father and Mother —
Without Consulting one another —
Purchased two Lumps of Largish Size,
Intending them as a Surprise,
Which met in Frederick’s Stocking and
Laid level Ten square Miles of Land.

The moral?

Learn from this Dismal Tale of Fission
Not to mix Science with Superstition.

— H. M. K.
From New Statesman and Nation,
London, January 14 (c) 1950