Posts Tagged math

Forms and Formulae: Proof and Presupposition


A picture of the Sun peeking over the spine of The Princeton Companion to Mathematics as it rests on top of The Princeton Encyclopedia of Poetry & PoeticsThis is the sixth in a series called ‘Forms and Formulae‘ in which I write about articles in the Princeton Companion to Mathematics using poetic forms covered by articles in the Princeton Encyclopedia of Poetry and Poetics. This installment’s mathematics article is entitled ‘Geometry’, and the poetic form is anecdote. This poem tells a true story I was reminded of by the discussion of the many attempts to prove Euclid’s parallel postulate from the other postulates, before people finally considered what would happen if it were false, opening up whole new geometries. This anecdote is not directly analogous, however, since I actually proved a statement to be false rather than proving it to be independent of the other axioms and then investigating what would happen if it were false.

A statement that the learned man had tried for days to prove
was set for students as a test
for four points extra credit,
to boost percentage marks assessed
of anyone to get it.

I mined brain gold with mind-brainpan, but things did not improve.
My efforts could not beat a path
from axiom to conjecture.
I sighed, and then let go of math
and headed to a lecture.

As I was sitting on the can, the shit began to move.
I saw the field with eyes anew
and found a boundary sample
that proved the statement was not true —
an outright counterexample.

To draw for years a foregone plan, for sure does not behoove
explorers hoping quests provide
not just what’s sought, but more.
Perhaps the field was opened wide,
but I scored one-oh-four.

I’ve been sitting on a draft of this one for a while, because, as noted above, disproving something is not the same thing as proving that one axiom can neither be proven nor disproven from the others, and then launching new fields of mathematics in which the axiom is taken to be false. Besides that, it’s a poem mentioning poop (though written before Shit Your Inner Voice Says), and it has a really weird rhyme scheme and awkward rhythm, for no good reason. Then again, I did once credit my short-story-writing success to the mention of toilets.

It is a true story; my abstract algebra professor at university set a couple of problems he hadn’t managed to prove himself for extra credit, and after proving problem number one I happened to think of a counterexample for problem number 2 while doing number 2s, and ended up scoring more than 100% for that class. I felt like I couldn’t make up an entirely fictional anecdote (though that is allowed, according to to the encyclopaedia) and while I’m sure I could write all sorts of other poems about geometry (on top of at least one I already have), I don’t have a lot of anecdotes about it.

Unimpressed as I am by this particular effort, I have to publish this to get onto the next Forms and Formulae, which will be… oh, for the love of Gödel — a national anthem for the development of abstract algebra?! What have I let myself in for?! It will take a while, because I’m heading to a programming conference followed by a translation conference soon, and then I’ll probably have to exercise my fledgling musical skills again.

Meanwhile, you can enjoy the highlights videos from Open Phil, an awesome open mic night in Vienna, where I’ve been practising reciting my poetry for audiences, and other people have been doing amazing musical things and other performances. Also, here‘s a very Vi-Hart-esque video I found while searching to see whether Vi Hart had anything to say on non-Euclidean geometry:

, , , , , , , ,

Leave a comment

Forms and Formulae: Not A Number


A picture of the Sun peeking over the spine of The Princeton Companion to Mathematics as it rests on top of The Princeton Encyclopedia of Poetry & PoeticsThis is the fifth in a series called ‘Forms and Formulae‘ in which I write about articles in the Princeton Companion to Mathematics using poetic forms covered by articles in the Princeton Encyclopedia of Poetry and Poetics. This installment’s mathematics article is entitled ‘From Numbers to Number Systems’ and the poetic form is allegory, making this the third poetic form in a row that isn’t actually a poem.

A long time ago in Greece, there was a community of numbers where everybody lived as one, or two, or three. They were not all equal, because each was unique, but they were all numbers, and that’s what counted. They were the true numbers, and they lived alongside the false, or negative, numbers.

Then One day, which was the day when the number One was celebrated, One Seventh came along. The other numbers looked at it with pity.

“You poor, broken thing,” they said. But the seventh didn’t feel broken.

“I’m not broken. I’m a number, just like you!” said One Seventh.

Seven looked at One Seventh with trepidation. “I don’t think it’s safe to be around a part of seven. What if it wants to take more of my parts?”

Three agreed. “It’s just not wholesome.”

One Seventh pointed to its numerator. “Is this not a one, like the number of the day? How can I not be a number when my very numerator is the purest number of all?”

One was flattered by the description, and in the spirit of the celebration, declared, “One must not only celebrate Oneself, but also display kindness to all those around One. I declare One Seventh to be a number, along with all little Ones like it!” After that, the other numbers were largely kind to the unit fractions, and the fractions always reciprocated.

The next day, Two Fifths came along. Emboldened by the success of One Seventh, Two Fifths said, “I’m a number too! Can I join the celebration?”

Two, whose day it was, said, “But you’re just One Fifth plus One Fifth. It’s just not proper to be going around as if you’re a single number. Split into unit fractions before you scare the little Ones!”

But Two Fifths persisted. “What are you,” it said to Two, “if not One plus One?”

Two did not like the idea two bits, but it could not find a problem with the argument.

Five, who was never any good at acting composed, protested. “This is preposterous! Two, I always knew you weren’t quite as prime as us. Think about it. If we let these two fifths…”

This two fifths,” corrected Two Fifths.

Five shot it an incalculable look. “If we let these two fifths act like a whole number, next we’ll have matrices, or lengths, or linear graphs wanting to be numbers. It’s a steep gradient!”

“That’s not true!” said Two Fifths. “In other cultures I am a perfectly acceptable number. In Mesopotamia, nobody thinks twice about my being a number, but they would never allow One Seventh. It’s all a matter of culture! And graphs are not numbers there either, so you needn’t worry about that.”

Two was divided by Five’s argument. It worried about diluting the number system, of course, but it was aware that even it could have been excluded from the primes using such an argument. Having always felt like an outsider itself, it had pity on Two Fifths, and declared the fraction and others like it to be numbers.

The next day, The Square Root of Two, who could not be expressed as a fraction, decided to join the numbers. Three said, “Don’t be absurd. You’re not really the square root of two; only square numbers have square roots. You’re just a fraction who’s confused. You look like about one and a hundred and sixty nine four hundred and eighths, to me.”

But the square root was resolute. “Look,” it said, holding up a square. “If we say the sides have length one, then the diagonal has length the square root of two. There is no way we can find a unit that can measure both of them as whole numbers. I can prove it to you!” And The Square Root of Two proved it.

“Okay,” said Three. “You’ve shown that the diagonal can’t be measured with the same unit as the sides. But they’re just lengths, not numbers. All you’ve done is show that not all lengths can be measured with numbers. The numbers are not going to be happy about this, you know.”

“But I am a number! I am the number which can measure that diagonal!”

“That’s just irrational. Lengths are not numbers. Either you’re a number, in which case you should show yourself as a fraction instead of wearing that radical outfit, or you’re a length, or a ratio of lengths, and you should go back where you belength. Make up your mind.”

“I told you this would happen!” said Five. “I told you lengths would be next!”

So the Square Root of Two skulked back to geometry, and commiserated, but did not commensurate, with the ratio of a circumference to a diameter.

Meanwhile, Two Fifths told all its new number friends about its adventures in Babylon, and the sexy sexagesimal numbers there. Before long, it became fashionable for numbers to represent themselves using decimal places instead of fractions. Some of them had to use zeros to make sure their digits hung in the right places.

Zero saw its chance, and claimed its right to be considered a number.

“But you’re not a number!” said Four. “You’re just a placeholder that the fractions use when they’re dressing up in their costumes for their unwholesome sexagesimal parties.” Four looked down its slope at a nearby decimal.

“But if I add myself to you, is there not equality? I should be treated the same as you.”

“But,” said One, “numbers have to be able to multiply. If you multiply you only get yourself. Only multiplying with me should do that! I’m the Unit around here, not you.”

“You’re destroying the family Unit!” shouted Five, in defense of its onely other divisor.

“I can’t even tell whether you’re true or false!” cried One Seventh, nonplussed.

So Zero went back to dutifully holding places, quietly adding itself to everyone and everytwo it met, until they were all convinced it held a place in society.

On the Seventh day, which was the day when One Seventh’s acceptance as a number was celebrated, they rested.

On the Tenth day, which was the day when The Tenth was celebrated, The Tenth returned from a vacation in Flanders and declared, “There are no absurd, irrational, irregular, inexplicable, or surd numbers!”

Five and Three cheered, and made obtuse gestures at The Square Root of Two and its friends. “You see? You’re not numbers.”

“All numbers are squares, cubes, fourth powers, and so on. The roots are just numbers. Quantities, magnitudes, ratios… they are all just numbers like us. We can all fit along the same line.”

Five and Three looked at each other in primal disgust. “I’m not a point on a line! I’m a number! A real number!” Five shouted.

“Real numbers,” countered The Tenth, “include everyone, and everyfraction, and everylength in between.”

The Square Root of Two led its friends into their places between the other numbers, and they celebrated with unlimited sines, cosines, and logarithms. Some of the stuffier primes and fractions protested, but they backed down when they realised just how many of these strange new numbers there were.

But even as The Tenth spoke, it knew that not everything it said was true. After all, false numbers were not the square of anything, even though it had seen them act like they were in some delightful formulae.

At Length, which was the day when the acceptance of lengths as numbers was celebrated, somereal wondered what would happen if false numbers were squares of something too. It imagined a new kind of radical, like those the square roots wore, but for false numbers. It imagined a world where every polynomial equation had roots, be they real, false, or imaginary. These were clearly not like all the other numbers The Tenth had listed.

Soon after, the imaginary numbers came out of hiding. “We do exist!” they said. “And we can add and subtract and multiply and divide just like you!”

The other numbers were wary, for they could not work out where the imaginaries fit amongst them. They could not even tell who was bigger. Five was disgusted that such numbers had been secretly adding themselves to real numbers all along.

The real numbers were nonetheless intrigued by and slightly envious of these exotic creatures, and despite having become accustomed to all having equal status as numbers, sought new ways to distinguish themselves from the crowd. The whole numbers had never quite got over the feeling of being generally nicer than the other numbers, so they used the new trend to vaunt their natural wholesomeness. The ratio of a circumference to a diameter, who had taken on the name Pi, discovered that in addition to not being expressible as a fraction, it was so much more interesting than The Square Root of Two that it couldn’t even be expressed in such roots. It called itself ‘transcendental’, and had quite some cachet until most of its admirers realised that they had the same property.

Finally they discovered that instead of trying to organise everynum into a line, they could arrange themselves in two dimensions, with the imaginaries along one axis and the reals along the other, and the vast plane in between filled with complex combinations of both.

Some of the more progressive numbers were so excited by this system that they tried to find new numbers that they could arrange into a three-dimensional volume, but they couldn’t find any. However, during their search they found things called quaternions, which lived in a fourth dimension.

An excited transcendental, whose name is too long to write here, brought a subgroup of quaternions in front of the crowd and announced, “I have travelled to the fourth dimension, and found numbers there just like us. We are not alone!”

Five kept its fury pent up this time, but Four Sevenths called out, “They are not numbers like us. I have seen how they multiply. When two quaternions multiply, they can give different results depending on which comes first!”

The numbers clattered their numerals in shock, and a great amount of whispering about unlikeabel multiplication practices ensued.

A complex transcendental sneered, “And what were you doing watching them multiply, eh?”

“Oh, get real!” retorted Four Sevenths, crudely conveying what the transcendental should do with its complex conjugate.

The pair fought, and disorder spread throughout the dimensions. Some sets of numbers sneaked off into the fields to form their own self-contained communities, sick of the controversy surrounding being or not being numbers. As they did, they found still other communities which functioned much like theirs, and some were communities of functions themselves. Indeed, even matrices and graphs formed structures which the enlightened subgroups found familiar, though rather than trying to be accepted as numbers, these groups took pride in having their own identities. The p-adics were adamant that they were numbers, but did not care to join the rest of the real or complex numbers. The octonions did not associate themselves with such labels, going about their operations however it worked for them, and consenting to be called numbers only when it was useful to act as such.

When peace finally settled, there were more groups of objects than there had been numbers, and still more came about when those groups interacted with each other.  Most no longer cared about being called numbers, and simply communicated which rules they followed before participating in a given system. And if the requisite system turned out not to exist yet, well, it just had to be invented.

Turning this particular article into an allegory did not take much work. It almost seemed like one already, when I read it in that frame of mind. There are a few direct quotes in the story. The Tenth’s proclamations come from The Tenth, in which Simon Stevin introduced decimal notation to Europe. The very last line of the story is paraphrased from the last line of the article. All I really did was rephrase it as a story from the perspective of the numbers, and add in far too many mathematical puns of greatly varying levels of subtlety.

I’m sorry to anyone with ordinal linguistic personification who thinks I’ve given the wrong personalities to the numbers. Also, in case anyone was wondering, the Greek numeral for four does have a slope.

The next Forms and Formulae will be an anecdote about geometry.

, , , , , , , , , ,

1 Comment

Forms and Formulae: Self-Avoiding Walk


A picture of the Sun peeking over the spine of The Princeton Companion to Mathematics as it rests on top of The Princeton Encyclopedia of Poetry & PoeticsThis is the fourth in a series called ‘Forms and Formulae‘ in which I write about articles in the Princeton Companion to Mathematics using poetic forms covered by articles in the Princeton Encyclopedia of Poetry and Poetics. This post’s mathematics article is entitled ‘The General Goals of Mathematical Research‘ and the poetic form is alba, which is a kind of song; I recorded it [direct mp3 link] using my robot choir and some newfound musical knowledge, and there are many notes on that after the lyrics below.

Here are some extracts from the article on the alba, explaining the features that I ended up using:

A dawn song about adulterous love, expressing one or both lovers’ regret over the coming of dawn after a night of love. A third voice, a watchman, may announce the coming of dawn and the need for the lovers to separate. An Occitan alba may contain a dialogue (or serial monologues) between lover and beloved or a lover and the watchman or a combination of monologue with a brief narrative intro.

The alba has no fixed metrical form, but in Occitan each stanza usually ends with a refrain that contains the word alba.

…the arrival of dawn signaled by light and bird’s song…

The watchman plays an important role as mediator between the two symbolic worlds of night (illicit love in an enclosed space) and day (courtly society, lauzengiers or evil gossips or enemies of love)

I based the song on section 8.3 of the article, entitled ‘Illegal Calculations‘. In retrospect, using the word alba in each refrain (are these even refrains?) doesn’t make much sense, since I’m not writing in Occitan, and the casual listener will not know that alba means ‘dawn’ in Occitan. But hey, it kind of rhymes with the start of ‘self-avoiding walk‘. How can I not rhyme an obscure foreign word with an obscure mathematical concept?

Introduction:
Mathematicians struggle even today to learn about the average distance between the endpoints of a self-avoiding walk. French physicist Pierre-Gilles de Gennes found answers by transforming the problem into a question about something called the n-vector model when the n is zero. But since this implies vectors with zero dimensions, mathematicians reject the approach as non-rigorous. Here we find that zero waking up next to its cherished n-vector model after a night of illicit osculation.

Zero:
I am just a zero; I am hardly worth a mention.
I null your vector model figure, discarding your dimension,
and every night I’m here with you I fear the break of day,
when day breaks our veneer of proof, and we must go away.

Here by your side
till alba warns the clock.
Fear’s why I hide
in a self-avoiding walk.

N-vector model:
Let the transformations of De Gennes show your place.
Never let them say we’re a degenerate case.
When I’m plus-two-n there’s just too many ways to move,
But you’re my sweetest nothing and we’ve got nothing to prove.

Here by your side
till alba warms the clock.
Fear can’t divide;
it’s a self-avoiding walk.

Watchman:
The sun has come; your jig is up. It’s time for peer review.
You think your secret union has engendered something new.
You thought you would both find a proof, but is it you’re confusing
The sorta almost kinda-truths the physicists are using?

That’s not rigorous,
says alba’s voice in shock.
All but meaningless
to the self-avoiding walk.

Zero and N-vector model together:
If you say that our results don’t matter,
then go straight to find a better path.
For as long as you insult our data,
Is it wrong to say you’re really math?

Hey there, Rigorous
at alba poised in shock,
you are just like us,
in a self-avoiding walk.

All voices are built-in Mac text-to-speech voices, some singing thanks to my robot choir (a program I wrote to make the Mac sing the tunes and lyrics I enter, which still needs a lot of work to be ready for anyone else to use.) Older voices tend to sound better when singing than the newer ones, and many new voices don’t respond to the singing commands at all, particularly those with non-US accents. So for the introduction I took the opportunity to use a couple of those non-US voices. These are the voices used:

Introduction: Tessa (South African English) and, since I also can’t fine-tune Tessa’s pronunciation of ‘Pierre-Gilles de Genne’, Virginie (French from France)

Zero: Junior

N-vector Model: Kathy

Watchman: Trinoids

Most of the bird noises come from the end of Jonathan Coulton’s ‘Blue Sunny Day‘, and I can use them because they’re either Creative Commons licensed or owned by the birds. The two peacock noises are from a recording by junglebunny. Free Birds!

As I mentioned, I’ve been learning about songwriting from John Anealio, and since the Forms and Formulae project sometimes requires me to write songs, I’m putting the new knowledge into practice sooner than I expected. This song uses several musical things I’ve never tried before, which is quite exciting, but it also means I probably didn’t do them very well, because there’s only so much I can learn in a couple of months of half-hour weekly lessons. I welcome friendly criticism and advice. The new things are: Read the rest of this entry »

, , , , , , , , , , ,

Leave a comment

Forms and Formulae: The Numbers Are Not Enough


A picture of the Sun peeking over the spine of The Princeton Companion to Mathematics as it rests on top of The Princeton Encyclopedia of Poetry & PoeticsThis is the third in a series called ‘Forms and Formulae‘ in which I write about articles in the Princeton Companion to Mathematics using poetic forms covered by articles in the Princeton Encyclopedia of Poetry and Poetics. This post’s mathematics article is entitled ‘Some Fundamental Mathematical Definitions’ and the poetic form is air, which is a kind of song.

This song covers the first few sections of the article, about the development of the various number sets (Natural numbers [which I learnt as not including zero], whole numbers [including zero], integers, rational numbers, real numbers, and complex numbers) and finally a little abstract algebra. I’ve made a recording of it [direct mp3 link] using my robot choir and some instruments in GarageBand. I didn’t follow all the suggestions relating to airs, but one hallmark of an air is ‘illustrative musical devices highlighting specific words’, and I went overboard on that, illustrating each set using the background music. Airs are typically accompanied by a lute or other plucked instrument, but I used a piano instead, to highlight the word ‘Peano‘ in the first line.

[1 2 3]
You can play the Peano axioms.
Your successor will never fail.
But if you ain’t got nothing you ain’t got enough
so you start lower down the scale.

[0 1 2]
Well you’ve now got zero problems.
You can count on every fact.
You can add without an end, but exceed your subtrahend
or you’ll find you can’t subtract.

[-1 0 1]
So you add in the minus integers.
Zero gains another side.
You can add and take away, but not conquer all the way
’cause you can’t always divide.

[⅕,⅓, ¼]
Now your system is highly rational,
no division you can’t deal.
But no matter what you do, you can’t find the root of two
though you know that it must be real.

[ɸ, e, π]
So you fill all the gaps with irrationals.
You have a solid number line.
Solve absurdities at will but you’re out of square roots still
when you start with a minus sign.

[1+⅕i]
So you use your imagination.
You take the square of your mind’s i.
Your calculations never stall, but you wonder if that’s all
that this complex plane can fly.

[triangles, snares, cats]
The operations work on all numbers,
but is that all they can do?
They apply to other things; now you’ve groups and fields and rings
to apply that structure to.

This took longer than my last Forms and Formulae, due to the recording. I made several improvements to my robot choir (an app I wrote one weekend to get my Mac to sing for me) including fixing a silly bug which had thrown the timing of my previous recordings off. I’ve also been taking music lessons over Skype with John Anealio, and I used a few of the things I learnt for this; if you know a bit of music theory you might notice a few music theory puns in there.

It’s not especially funny overall, but I mentioned when I called into Dementia Radio last night that I would submit it to the FuMP Sideshow, so I will. [Edit: and here it is!] Another thing that came up were these Tom Lehrer songs about mathematics, which the host was not aware of. They were some of the first Tom Lehrer songs I heard, and definitely worth a listen if you like Tom Lehrer, maths, or both. I found them in 2005 while looking to replace some pirated Tom Lehrer songs I’d accidentally deleted before listening to them (I did eventually buy all of Tom Lehrer’s albums) and in that same search I came across the MASSIVE database of maths and science songs, which led me to Jonathan Coulton and so many other musicians and friends.

One of those other musicians was Monty Harper, and the first tune I came up with was very similar to the verses of his Silly Song. I changed some parts to make it less similar, but mostly I just made it more repetitive and annoying.  Dammit, Jim, I’m a poet, not a musician.

The article in the Princeton Companion to Mathematics was actually very long, and I haven’t finished reading it yet. Assuming I do get to the next article instead of writing something about the latter parts of this one, the next Forms and Formulae will be an alba (a dawn song about adulterous love!) about the goals of mathematical research. That should be fun. It will probably take a while, since it’s another song. Also, I will be busy next week at the 13th International Conference on the Short Story in English. I will be reading a story on the Thursday afternoon; probably a slightly revised version of Valet de cœur.

, , , , , , , , ,

1 Comment

Forms and Formulae: Linguistics → Mathematics


A picture of the Sun peeking over the spine of The Princeton Companion to Mathematics as it rests on top of The Princeton Encyclopedia of Poetry & PoeticsThis is the second in a series called ‘Forms and Formulae‘ in which I write about articles in the Princeton Companion to Mathematics using poetic forms covered by articles in the Princeton Encyclopedia of Poetry and Poetics. This week’s mathematics article is entitled ‘The Language and Grammar of Mathematics’ and the poetic form is acrostic, which is a superset of last week’s form, the abecedarius.

I’ve already written plenty of apronyms about mathematics that could be considered acrostics, so for this I had to do something else. The following is a double acrostic about the language of mathematics — the first letter of each line spells ‘Linguistics’ and the last letter of each line, read upwards, spells ‘Mathematics’. The line lengths are highly irregular (just as the mapping from linguistics to mathematics can be), which makes that less impressive, but I tried to keep decent enough rhythm and rhyme that it sounds good when read aloud.

Linguistics is mathematics.
Is’ it? Well, that ‘is’ a classic.
Now which ‘is’ is that ‘is’ that you and I
Grammatically understand… wait!
Understand, or understands? It all depends on how that ‘and’ treats data:
I understand ∧ you understand, or you+I is? Are? Am?
Some singular object that understands ambiguous copulae
That may~equivalence relations, ambivalent notations for functions, adjunctions, or ∈ life ∪ death
I ‘am’ and i ‘is’, in a nonempty set?
Cogito, ergo ∀ subjects Ɣ ∈ {sums, numbers, dynamics, …} Ɣ has Grammar s.t. Meaning(s)=Meaning(t)⇔s=t ∀ symbols s,t in Grammar sub gamma.
So, let ‘is’ be a relation where no such equation’s imposed but the intersection of the sets of accepted bijections on the subjects’ grammar sets are nonempty we get (and I don’t have the proof yet to hand, um… It’s trivial, readers with wits understand’em) that linguistics is mathematics, quod erat demonstrandum.

This was a particularly interesting article for me, since I’m very interested in language and grammar in general. It goes into various symbols used in mathematics and talks about which parts of speech they are and how they compare to similar words or parts of speech in English. It turns out mathematics has no adjectives. I had several attempts at different acrostics, and when I figured out the first few lines of this one, I thought I’d move on to explaining a different section of the article every few lines. Then I was inspired to continue it at a time when I didn’t have the book handy, so it ended up focusing on just the first few parts with a nod to something mentioned in a later section. One nice thing I found in the article was:

  1. Nothing is better than lifelong happiness.
  2. But a cheese sandwich is better than nothing.
  3. Therefore, a cheese sandwich is better than lifelong happiness.

Soon after, we get the haiku I found earlier:

For every person
P there exists a drink D
such that P likes D.

It’s really a fun book to read. Next week’s Forms and Formulae will be an air on some fundamental mathematical definitions, which should be interesting because I’m not certain I fully understand the requirements for an air. I may have to dust off the robot choir.

In other news, I got some copies of the They might not be giants poster printed locally, and they look great, even when accidentally printed at twice the intended size. The English pronoun poster is quite readable at about 42x42cm, which is a little less than the size it’s on Zazzle at.

, , , , , , , ,

Leave a comment

Forms and Formulae: Y Lines About X Letters of the Alphabets (an Abecedarius of Math(s))


A picture of the Sun peeking over the spine of The Princeton Companion to Mathematics as it rests on top of The Princeton Encyclopedia of Poetry & PoeticsThis is the first in a series called ‘Forms and Formulae‘ in which I write about articles in the Princeton Companion to Mathematics using poetic forms covered by articles in the Princeton Encyclopedia of Poetry and Poetics, even though the Companion already contains plenty of poems. The first entry in the former is entitled ‘What is Mathematics About?’ and the first entry in the latter is abecedarius.

The following is an abecedarius of what mathematics is about — an ABC of mathematics, if you like. You can also try reading it along to  ’88 Lines About 44 Women’ (which you might be familiar with from The Brunching Shuttlecocks’ ‘88 Lines About 42 Presidents‘ or the great Luke Ski’s ‘88 Lines About 44 Simpsons‘) though the rhyme scheme is different. It only coincidentally has a similar meter, but once I saw it I decided to go along with it.

Axioms are how you ask ‘what if’; just pick some — you decide.
Break it down and every branch of math(s) depends on these.
Calculus will help you count the branches that you can’t divide,
Differentiating the conditions at the boundaries.

Elements of Euclid was a textbook for millennia.
Functions follow formulae to map domain to range.
Gödel showed some true things can’t be proven, but still many are,
Held without theology as truths that never change.

Inconsistent axioms will prove all and its opposite,
Jeopardising hopes the formal system will be sending forward
Knowledge for deriving knowledge-prime or knowledge-composite.
Logic’s only limits are the ones that something’s tending toward.

Manifold(s) are ways to bring such limits to geometry.
Numerous are non-numeric methods that we use.
Often are two manifolds the same, up to isometry,
Proving that(,) there’s gobs of generality to lose.

Quod Erat Demonstrandum quoth inerrant understander,
Rigorously rational and rooted in the real,
Symbol-shuffling spanning such solution sets with candor,
Theorem after theorem or conjecture from ideal.

Universal sets have mathematicians quite inside themselves;
Vector spaces set a basis they can build upon.
Wolfram’s Weisstein’s MathWorld’s website rivals books on many shelves.
X rules the domain that functions are dependent on.

Y‘s home on the range is the solution set that many seek.
Zeno cuts each line in half so drawing it is undefined.
Alphabet is insufficient;
Beta hurry onto Greek.
Gamma raises complex powers.
Delta changes Zeno’s mind.

Epsilon‘s so small that
Zeta covers the prime landscape sole.
Eta‘s very many things;
Theta‘s varied just by one
Iota in the calculus where
Kappa played a founding role.
Lambda has a calculus.
Mu (micron)’s small, but not-none.

Nu math(s) is Tom Lehrer’s nightmare.
Xi‘s that universal set.
Omicron‘s a small big-O.
Pi squares circles’ radii.
Rho‘s a row (zeros-out) rank.
Sigma sum is all you get.
Tau is sometimes phi, 2pi.
Upsilon, we wonder, ‘Y?’

Phi‘s the golden ratio.
Chi-squared ballpark’s on the ball.
Psi‘s a polygammous one.
Omegahd, there is no end;
Aleph-null can yet extend;
Aleph one is still too small;
Beth one, too, still isn’t all;
Beth-two, one can yet transcend.

Gimel still can bring you some,
Daleth beats continuum.

Now you know your ABC(-Omega-Aleph-NOP)
Out you go to maybe see (oh, mathematicality!)
That math(s) is an infinity (for all things there exists a key!)
And cast it as a trinity (a singular plurality!)

When I decided to do this, I don’t think I realised how many Greek letters there were. In the time it would have taken to finish a normal abecedarius, I was only halfway there, and further motion seemed impossible. Luckily, Zeno was there to sympathise. I also didn’t realise any Hebrew letters after bet were used in mathematics. Apparently Cantor used gimel and daleth for yet bigger infinities. I hope to write a new Forms and Formulae each week, so the later forms had better not be this long. I didn’t always stick to things from the ‘What is Mathematics About’ article, or even that subject. However, I think I conformed to the abecedarius form fairly well; the abecedarius is often used for religious purposes, and I was able to work in that mathematics requires no faith (‘held without theology’) and extends beyond alpha and omega, and also that the differing ways of abbreviating the word in different countries (with or without ‘s’) makes it similar to the three-in-one Christian trinity.

Read the rest of this entry »

, , , , , , , , , ,

1 Comment

Drabble: I sure appreciate the way you’re working with me.


“I… I th… thought you’d left,” I stammered.

“I came back,” he replied nonchalantly. “It’s not as if I died.” He looked at me accusingly.

“Well, I…”

Such lively eyes staring at me from a deathly face were unnerving. I gave in, and went to get some textbooks.

“Let’s work on something together,” he suggested. “My brain is open.” Indeed it was, but I tried not to look.

Uncertain though I was about the feasibility of living and undead working together, I could not refuse his offer of collaboration. And that’s how I got a late Erdős number of one.

Read the rest of this entry »

, , , , , , , , ,

Leave a comment

Unintentional Haiku in the Princeton Companion to Mathematics


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
automorphism

We 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 infinite

It 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

, , , , , , , ,

3 Comments

VI of Hearts: The Cantor Ternary Set Cantor Ternary Set


It’s self-referential! It’s self-similar! It would give Jonathan Coulton nightmares! It’s the Cantor Ternary Set Cantor Ternary Set: a representation of six steps in the construction of the Cantor ternary set using sped-up and slowed-down samples of Jonathan Coulton singing ‘Cantor ternary set’ in his song Mandelbrot Set, in which he professes to fear said set. I suppose you could say Jonathan Coulton is the cantor, but would it make him turn a reset?

Six of Hearts on a Jonathan Coulton Mandelbrot Floyd shirtI added a Wilhelm scream to the end, because that seemed appropriate. Here’s the audio-only version.

The Cantor ternary set is what you get if you take a line (technically a line segment, but we’ll call it a line), cut out the middle third, then cut out the middle third of the lines that remain, then cut out the middle thirds of those, and so on. If you continue doing this forever, you end up with just as many points as you started with (isn’t infinity grand?) but they’re nowhere near each other. I made the ‘lines’ at each stage out of clips of Jonathan singing ‘Cantor ternary set’ at different speeds; first at one 27th normal speed, then at one ninth, then one third, then normal speed, then three, nine, and 27 times normal speed. Then I put all the lines (i.e. audio clips) from the different steps on top of each other, positioned according to where each line came from in the original line, to make the full canticle (cantorcle?) You can see how it works in the video. To make it easier to differentiate the different layers, I put the second and fifth layers (counting from the slowest one at the bottom) toward the left ear and the third and sixth toward the right, leaving the other three (1/27-speed, original-speed, and 27-times speed) in the centre.

This didn’t take very long to make, in the end, but there were a lot of false starts. A long time ago I decided to make some kind of song about mathemusician Vi Hart using snippets of the various source tracks I have of Jonathan Coulton songs — a Hart-shaped box, on the table, and far too late you see the one inside the box is Vi Hart, who’s not a real heart but is a real bad-ass mathematician… that kind of thing. I realised some time ago that it would have to be the six of hearts, because in Roman numerals that’s the vi of Hearts. But that didn’t stop me from putting off starting it till about a week after the last minute. It’s a good thing I set my own deadlines.

A couple of days ago I finally started to actually work on this. I cut some sounds together (‘my heart’ and a ‘v’ sound from When You Go) to make Jonathan sing Vi Hart’s name, and collected relevant phrases from other songs. But I needed some kind of musical background track to tie it all together (like the Mr. Fancy Pants choir I used in my ‘Code Monkey Like…’ thingy.) I had considered using Vi’s piano music that she played on JoCo Cruise Crazy 2, but in the moment I didn’t feel like looking for it, and also didn’t feel like I could do it justice; I’ve just recently started listening to a basic and hilariously over-dramatic audio course on music theory, but most of what little I know about music, I learnt from Douglas HofstadterLeonhard Euler, Leon Harkleroad, and Vi Hart herself. While I’m okay with the mathematical side of things, I don’t think I remember enough to make a fitting musical tribute. So I asked myself, as I often do, what would Vi Hart do? Probably something symmetrical, mathematical, brilliant. So I hit on the idea of making a Cantor ternary set of Jonathan Coulton singing Vi Hart’s name, and using that as a backing track for the song.

Well, that was interesting, but it sounded terrible. The gap in the middle (the middle middle, not all the other gaps in middles which make the Cantor set what it is) sounded like a lawnmower, most of the rest sounded like a bad choir being massacred by a possessed lawnmower, and the 3x-speed ‘Vi Harts’ were more prominent and understandable than the ones at the original speed. I fiddled with levels for a while, and tried to make the lawnmower sound better by adding more words from other songs, but no dice. The fact was, using a Cantor ternary set of Vi Hart’s name (sung in that particular way) as a backing track was a terrible idea. And now that I think of it, I seem to recall that Hofstadter mentioned experimenting with fractal music and finding it didn’t work very well. That’s fine, though; I’m no musician, so I figured I could make it work to my low standards eventually. But just to take a break, on a whim I decided to try making a Cantor ternary set out of Jonathan singing ‘Cantor ternary set’.

Five minutes later, I discovered that the greater variety in syllables and pitches makes this sound quite interesting even without added lyrics, and you can fairly easily hear the words at several different speeds, so you can tell what’s happening well enough for it to be a demonstration of the Cantor ternary set in itself rather than just a backing track. Plus it’s a Cantor ternary set made up of the words ‘Cantor ternary set’. Why on Earth did I not think of that in the first place? Sorry Vi Hart; you’ll get your tribute song some day, and hopefully from someone better at music than I am.

On the subject of Vi Hart, last weekend I was at my physicist friend Aidan‘s place and noticed he had made some pretty neat things with Geomag, so I asked him to explain it all on video. He did mention Vi at one point. Here’s the video, in which we talk about RF cavities, conservation of angular momentum, triangles, and various kinds of pole, among other things:

Aidan also makes a lot of videos explaining particle physics; you should check them out.

Also on the subject of physics, and cool people I’ve met on JoCo Cruise Crazy (Vi Hart being one of them) here’s an LHC-related comic that Randall Munroe from xkcd drew for me on JoCo Cruise Crazy 3.

, , , , , , , ,

1 Comment

Six of Clubs: Birthday Monduckenen-duckenen


First, check out Vi Hart‘s video about the Thanksgiving turduckenen-duckenen:

Now have a look at Mike Phirman‘s song, Chicken Monkey Duck:

Okay, there are monkeys instead of turkeys, and the mathematics isn’t quite as explicit, but it’s pretty similar, don’t you think? Now, let’s imagine that Mike Phirman is actually singing the recipe for a fractal turducken, or rather, monducken. You can imagine all the monkeys are turkeys if you’d rather eat the result than present it to some pretty thing to please them. (Note: Please do not kill any actual monkeys.) Monkeys, like birds, belong in trees, so I wrote an AppleScript to draw binary trees in OmniGraffle based on the text of the song. You can try it for yourself if you like; all you need is a Mac, OmniGraffle, and a text file containing some words. See the bottom of this post for links and instructions.

If Mike’s reading the binary tree recipe layer by layer, like the first example in Vi’s video, one possible tree for the first stanza of Chicken Monkey Duck looks like this, where the orange ovals are monkeys, blue hexagons are chickens and green clouds are ducks. You can click it (or any other diagram in this post) for a scalable pdf version where you can read the words:

First stanza breadth-first tree

I added numbers so you can easily tell the chickens, monkeys and ducks apart and see which way to read the tree. It’s simple enough now, but the numbers will be useful for reading later trees which are not in such a natural reading order. This is called a breadth-first traversal of the tree, in case you’re interested. Now, what do birds and monkeys do in trees? They nest! So I wrote another script that will take any tree-like diagram in OmniGraffle and draw what it would look like if the birds, monkeys, or whatever objects they happen to be (the drawing is pretty abstract) were nested inside each other, just like the quails inside the chickens inside the ducks inside the turkey. This is what the monducken described by the first stanza of Chicken Monkey Duck, in the tree structure shown above, would look like:

First stanza breadth-first tree nested

The Monducken script allows using a different shape for each animal as redundant coding for colourblind people, even though it already chooses colours which most colourblind people should be able to distinguish. But that makes the nested version look a little messy, so here’s the above diagram using only ovals:

First stanza breadth-first tree nested ovals only

If you named this particular recipe in the other way, going down the left side of the tree and then reading each branch in turn in what is known as a pre-order traversal, it would be called a Monenmonenduckduckmon-monmonducken-enenmonduckmon-enmonduck-enduckmonducken-enmonen-duckenenmon-monenmon. It doesn’t sound nearly as nice as Turduckenailailenailail-duckenailailenailail because Mike Phirman didn’t take care to always put smaller animals inside large ones. I’m not holding that against him, because he didn’t realise he was writing a recipe, and besides, it’s his birthday. For reasons I’m not sure I can adequately explain, it’s always his birthday.

But what if I completely misunderstood the song, and his recipe is already describing the fractal monducken as a pre-order traversal, always singing a bird or monkey immediately before the birds and monkeys inside it? Well, don’t worry, I added a ‘pre-order’ option to the script, so you can see what that would look like. Here’s the tree:


First stanza preorder tree

and here’s how the actual birds/monkeys would look if you cut them in some way that showed all the animals, dyed them the correct colours, and looked through something blurry (here’s the version with different shapes):

First stanza preorder tree nested ovals only

Okay, but that’s only the first stanza. What if we use the whole song? If we pretend the recipe is breadth-first, this just means all the extra monkeys and birds will be at the bottom levels of the tree, so the outer few layers of our monducken will be the same, but they’ll have a whole lot of other things inside them:

Entire song breadth-first

Entire song breadth-first nested

Here’s a close-up. Isn’t it beautiful?

breadth-first close-up

If the entire song were treated as a pre-order monducken recipe, we’d still have the same monkey on the outside, but the rest would be quite different:

Entire song preorder

Entire song preorder nested

We could also read the birds and monkeys from left to right, as Vi did in her video. That’s what’s called an in-order tree traversal. But as delicious as they are mathematically, none of these orderings make much sense from a culinary perspective. Even if the monkeys were turkeys, it’s obvious that a nice big goose should be the outer bird. Vi suggested that herself. Of course, we could put the goose on the outside simply by reversing the song so it started with goose. But it would be much more fun and practical to pretend that Mike is naming the two inner birds before the one that contains them. This is called a post-order traversal, because you name the containing bird after the two birds or monkeys it will contain. It makes sense for a recipe. First you prepare a monkey (or turkey) and a chicken, then you immediately prepare a chicken and put them into it. You don’t have your workspace taken up with a whole lot of deboned birds you’re not ready to put anything into yet. Here’s one way the recipe could be done:

entire song postorder

Note that no matter what kind of traversal we use, there are actually several ways the recipe could be interpreted. If Mike says ‘monkey chicken chicken’ you know you should take a monkey and a chicken and put them in a chicken. But if the next words are ‘monkey chicken’, do you take that stuffed chicken and a monkey and put them inside a chicken? Do you debone the monkey and the chicken and wait for the next bird to find out what to put them into? What if there’s no next bird? What if there’s only one more bird (let’s say a duck) and you end up with a stuffed chicken, a stuffed duck, and nothing to stuff them into? You’d have to throw one of them out, because obviously your oven only has room for one monducken. Assuming you want two things in each thing, and you don’t know how long the song’s going to be, the best way to minimise this kind of problem is to always take your latest stuffed thing and the next, unstuffed thing, and put them inside the thing after that. The worst that’ll happen is you’ll have to throw out one unstuffed bird or monkey. But then you end up with a really unbalanced monducken, with a whole lot of layers in one part and lonely debonely birdies floating around in the rest.

It helps to have a robot chef on hand to figure out how many full layers of monducken you can make without it being too asymmetric. Mine makes the trees completely balanced as deeply as possible, and then does whatever was easiest to program with the remaining birds and monkeys. In this case it was easiest for my program to stuff a whole lot of extra animals into that one monkey on the left. This is what it looks like, with the varied shapes this time. Luckily, geese are rectangular, so they fill your oven quite efficiently:

entire song postorder nested

I like how you can see the explosion of duck radiating out from the inner left, engulfing all the other birds and monkeys before itself being swallowed by a goose. Such is life.

IFIf you would like to make diagrams like this yourself, there are two AppleScripts you can use. Both of them require OmniGraffle 5 for Mac, and if you want to make trees with more than 20 nodes you’ll probably need to register OmniGraffle.

The first is Monducken diagrammer, which you can download either as a standalone application (best if you don’t know what AppleScript is) or source code (if you want to tweak and critique my algorithms, or change it to use OmniGraffle Professional 5 instead of OmniGraffle 5.) Because it’s AppleScript, it works by telling other applications what to do, rather than doing things itself. So when you run it, TextEdit will ask you to open the text file you want to turn into a tree. Once you’ve opened one, OmniGraffle will start up (you may need to create a new document if it’s just started up) and ask you two things. First it will ask what kind of tree traversal the text file represents. Then it will ask you what kinds of shapes you want to use in your tree. You can select several shapes using the shift and command keys, just as you would for selecting multiple of just about anything on your Mac. Then you can sit back and watch as it creates some shapes and turns them into a tree.

The other one is Tree nester (standalone application/source code) You should have an OmniGraffle document open with a tree-like diagram in it (I suggest a tree generated using Monducken diagrammer; it has not been tested on anything else, and will probably just duplicate most of the shapes that aren’t trees or end up in an infinite loop if there’s a loopy tree) before you run this. It won’t ask any questions; it’ll just create a new layer in the front OmniGraffle document and draw nested versions of any trees into that layer.

If you’re looking at the source code, please bear in mind that I wrote most of this while on a train to Cologne last weekend, based on some code I wrote a while ago to draw other silly diagrams, and I really only dabble in AppleScript, and I forgot about the ‘outgoing lines’ and ‘incoming lines’ properties until I’d almost finished, so it probably isn’t the best quality AppleScript code. Not the worst either though. I welcome any tips.

, , , , , , , , , , , , , , ,

1 Comment

%d bloggers like this: