Let ε < 0.


The kiss precise

Filed under: Harmonic analysis — Travis @

–Frederick Soddy

For pairs of lips to kiss maybe
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.

Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance form the center.
Though their intrigue left Euclid dumb
There’s now no need for rule of thumb.
Since zero’s bend’s a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.

–A final verse by Thorold Gosset

And let us not confine our cares
To simple circles, planes and spheres,
But rise to hyper flats and bends
Where kissing multiple appears.
In n-ic space the kissing pairs
Are hyperspheres, and Truth declares-
As n+2 such osculate
Each with an (n+1)-fold mate.
The square of the sum of all the bends
Is n times the sum of their squares.


If (math version)

Filed under: Harmonic analysis — Travis @

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.

This appeared in the Times Educational Supplement, July 19, 1947.


Haiku generator

Filed under: Harmonic analysis — Travis @

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.


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.


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 teh Arts, Studio International, London 1968 p. 53; and NPL News 204, 10 (1967).


Foundations of arithmetic

Filed under: Harmonic analysis — Travis @

–J. A. Lindon

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 unuxorious zest,
Exept 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!


Hilbert space (a sonnet)

Filed under: Harmonic analysis — Travis @

–Lee Herron, Haverford College, 2003

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.


Euclid drew a circle

Filed under: Harmonic analysis — Travis @

–Vachel Lindsay

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.


Error message haiku

Filed under: CS silliness, Harmonic analysis — Travis @

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.

This was sent to me by Laura “RuneMercury” H.



Filed under: Harmonic analysis, Science humor — Travis @

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

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


The electron in gold

Filed under: Harmonic analysis, Science humor — Travis @

–Arthur H. Snell

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.”

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



Filed under: CS silliness, Harmonic analysis — Travis @

Mathematicians’ cheer

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
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,
e to the x dy dx
Sliderule, slipstick,

Programmers’ cheer

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

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