Posts Tagged math

NanoRhymo


November was National Novel Writing Month, which is a global event (‘National’ in this context means ‘More Official Than Other Made-Up Events, I Swear! Also, I Forgot There’s More Than One Nation’) in which people attempt to write 50000 words in a month. This is often abbreviated NaNoWriMo. I decided to go for an easier homophone, NanoRhymo.

Every day, I tweeted a short poem inspired by a rhyme I’d found using the ‘Random‘ button on the rhyming dictionary I made, rhyme.science. I’ve been tweeting interesting rhymes from this for a while, so some days I generated a new random rhyme, other days I looked through the @RhymeScience Twitter feed to find one that inspired me. Try it yourself! You might need to click a few times to get an accurate, interesting rhyme; I’m working on a new version that has fewer incorrect words or pronunciations in it, but don’t have enough spare time right now to get that up and running.

Some of them were fun little ditties, others were more inspired by current events or politics than my writing usually is. I’m not informed enough on most such issues to write on them, but that’s where the rhymes led me, and I have just enough of an informed opinion for a rhyming tweet. There are a few references to sexual assault and one to the Holocaust, so proceed with caution if reading such things is likely to be traumatic for you.

You can see all the NanoRhymos by clicking on the tweet above and reading the whole thread. But for those who prefer reading a blog, here they are. I’m considering continuing to write a NanoRhymo every day until I get bored of it, or until I have enough to publish a poem-a-day calendar. What do you think?

Day 1, inspired by the rhyme cloned and unowned:

It didn’t seem so wrong —
you wanted to prolong
the perfection of their hearts.
A collection of spare parts
was made when they were cloned.
They’re fêted and unowned,
but I’m a pair, impaired them-prime;
what’s mine is theirs, and theirs to mine.

Day 2, inspired by the rhyme no sin and close in:

A Jewish saviour way back
was questioned about payback,
and bade that those with no sin
grab stones and gather close in
to pelt a sinful other,
but then, the saviour’s mother
(whose immaculate conception
had made her an exception)
came forward — oh, Lord!

I would have ended that with ‘came forward, and Lord guffawed’, but I wanted to keep it in one tweet. The short version is probably better.

Day 3, inspired by the rhyme disguise and FBI’s:

This guy’s disguise will fool your eyes, but not the FBI’s.
They prize the wise who recognise through lies, despise the spies.

Day 4, inspired by the rhyme DVD on and neon:

Noise and darkness, stink and heat
Senses strained to find a beat
Shout to strangers lit by neon

Go home, put a DVD on.

Day 5, inspired by the rhyme young can and drunken:

Drunken groping long ago
Young can learn it’s wrong, and grow
Admit mistakes and make amends
Ensure the waking nightmare ends

Day 6, inspired by the rhyme enlighten’d and bite and, the election in the USA happening that day, and apparently also dogs:

Frantic voters running frightened,
Heavy hearts, but heads enlightened,
Two years on, they’ve seen his bite and
barking mad demeanour heightened.
Doggedly, they’ll put things right and
chase a future slowly brightened.

Day 7 (actually posted on November 10, because I missed a few days and made up for them by posting a few rhymes in quick succession) inspired by the rhyme promise’s and Thomas’s:

Don’t believe every Tom, Dick, and Harry;
Thomas’s promises won’t hold true;
Richard’s switch’d leave you hanging;
Trust Henry? Then regret that too.

Day 8, inspired by the rhymes rushes, crushes, and (in non-rhotic accents) ushers:

The wise one rushes,
Telling all their crushes,
Finding out the answers: yes or no?
No, they’ll only blush, or
Yes, follow the usher,
Make some more advances in the back row.

Day 9, inspired by the rhyme cynically and clinically:

After loving declarations,
you should not yet have relations,
but evaluate the information cynically.
Ask for terms and motivations,
and when provided with citations,
then concur, and place your arms around them clinically.

Day 10, inspired by the rhyme flattery and battery:

Some fake care to prime their victim:
soften and enlarge with flattery,
and while most juries don’t convict ‘em,
often they are charged with battery.

Day 11, inspired by the rhyme aghast and gassed:

See the harrowed and harrassed
Seething horrors of the past
Browse and mull, and be aghast
Thousands, millions, people gassed

This one has a different rhyme scheme in different accents: ABBA with the trap-bath split or AAAA without. It’s probably about the worst way I could have demonstrated the trap-bath split.

Day 12, inspired by the rhyme borders and marauder’s:

Halt, ye marauders,
attacking our borders!
Just following orders —
you can’t come in here!
If we don’t mistreat you,
we’re worried that we, too,
will soon have to flee to
another frontier.
(But they’d give us shelter;
they’d surely do well to!
We’re good folk, just dealt a
harsh blow; we’re sincere!)

Day 13, inspired by the rhymes (in non-rhotic accents) built a, kilter, and still to:

We’re all a bit off-kilter now;
we’ve lost a then and built a now,
but later’s what we’ve still to know,
and do we have the will to? No.

Day 14, inspired by the rhyme mathematician and proposition, and also Rudyard Kipling’s poem If—:

If⟹

If you see a proposition,
apply to it your intuition,
at length, exerting full cognition,
come to trivial fruition,
and restart with a new suspicion,
then you’ll be a mathematician.

Day 15 (actually posted on November 17, because I missed a few days), inspired by the rhyme compiling and unsmiling:

Robotic faces, cold, unsmiling,
hypnotic glaze of code compiling,
illuminating status lights
as humans wait for access rights.
Once it’s done, they wake anew —
the robots and their coders too.

Day 16, inspired by the rhymes amnesia and anesthesia, and whiskey and frisky:

Dull the pain with anaesthesia,
hoping for a brief amnesia
of harm sustained while he was frisky
(lapses blamed on too much whiskey)
Clear dark thoughts to fit some leisure:
dull clear thought with too much whiskey.

Day 17 (actually posted on November 18, because when catching up the missed days on November 17, I missed a day, so everything after this is a day late), inspired by the rhyme predestined and unstressed and:

If your plans have not progressed, and
stasis leaves you quite depressed, ind-
eed, you must stick out your chest, and
pray that you’ll be always blessed, and
loaf around, remain unstressed and
wait for that which comes predestined.

Day 18, inspired by the rhyme prick’d and afflict:

If Santa wanted Christmas cheer
to fill the Northern Hemisphere,
he’d send the elves to spread the word
to save yourselves and save the herd:
Inoculate yourselves; get prick’d!
Be those the flu will not afflict!

Day 19, inspired by the rhyme trustee can and deacon, although I ended up using 15 other rhymes for deacon and not trustee can:

Sea can weaken a Puerto Rican deacon.
We can be concerned, and we can sneak unspeakin’
or be concertedly connected shriekin’
so s/he can see kin be a freakin’ beacon.

Day 20, inspired by the rhyme Wozniak and Bosniak:

Once upon a time I met Steve Wozniak,
Who bought me a replacement when I lost me Mac
My life got weirder still; I saw a tokamak 🤷🏻‍♀️
And now I code for iOS to clothe me back
While Android code is handled by a Bosniak

It’s a pretty silly poem, but it is also 100% true (you can read the Wozniak story in even cheesier rhymes, if you like), and how could I ignore it when my random rhyme generator comes up with ‘Wozniak’?

Day 21, inspired by the rhyme xylem’s and asylums:

I’m committed to squeezing out a daily poem —
soon committed to a poetry asylum?
But pull enough good water up a xylem
and some day something sweet comes down the phloem.

Day 22, inspired by the rhyme Dulles’s and portcullises, and that time when I had about three hours to change planes in Washington Dulles airport but still only just made it, with an airport staff member running while pushing me in a wheelchair:

A change of planes is always an adventure
and no ride is as perilous as Dulles’s —
with far-off gates and queues there to prevent’ya
you’re sliding under plummeting portcullises.

This one is best read with low rhoticity and yod coalescence, to make adventure rhyme with prevent’ya.

Day 23, inspired by the rhymes routeddisputed, and (in non-rhotic accents) untutored:

In the untutored,
the pronunciation of ‘routed’
can be disputed,
so if you doubted
the pronunciation of ‘routed’
you’ve been outed.
(It depends on where you’re rooted,
so feud no more about it.)

Day 24, inspired by the rhyme fantoccini and Jeannie:

Jeannie, Jeannie, fantoccini
pulled the strings of her bikini,
made a pervy man look up it,
then she moved him like a puppet.

Day 25, inspired by the rhyme unstudied and ruddied, and the first image sent back from Mars by NASA’s InSight lander:

Freckled surface, still and ruddied,
sweeping spaces still unstudied,
far in space is InSight near;
insight can begin right here.

The freckles were actually dust on the lens, though.

Day 26, inspired by the rhyme Bernard would and hardwood:

A tree would never leave you;
it’s your steadfast, loyal bud.
A tree would never leave you;
it will root for you, come hail or flood.
A tree would never leave you;
it’s as solid as hardwood.
A tree would never leave you,
but Bernard would.

Day 27, inspired by the rhymes ultimata, weight a, and (in non-rhotic accents) eliminator:

The real eliminator
is bearing all the weight a
gazillion ultimata
can have on the enforcer.
Can they just divorce a
person they adore so
much because they made a
gaffe they said would force it?

Day 28, inspired by the rhyme conceal’d and kneeled:

They saw the violence, and they kneeled,
did not kowtow, and did not yield.
A shout of silence that conceal’d
the loudest power they could wield.

Day 29, inspired by the rhymes miss a, Alissa, and (in non-rhotic accents) kisser:

🙂: “Why would you miss a
girl named Alissa?”
🙃: “She’s a good kisser;
don’t you dismiss her!”
🙂: “But so is Melissa,
also Clarissa,
even Idrissa,
why just one Miss, huh?”
🙃: “Not gonna diss ya,
just not down with this, yeah.
I’ll follow my bliss, you
kiss them and I’ll miss ‘lissa.”

Day 30, inspired by the rhymes becharmed, unharmed, in non-rhotic accents calm’d, and in non-rhotic accents with the father-bother merger, glommed:

Onto magic guild they glommed,
vibes they quivered, nerves they calm’d.
Vicariously thus becharmed,
they came through escapades unharmed.

NaNoWriMo is over, but tune in on Twitter to see if NanoRhymo continues!

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In which I appear content with content in which I appear


I’ve been having a pretty relaxed month, but my life is ridiculous, therefore so far in September I have appeared in a music video, a radio broadcast, and a podcast.

The music video is Molly Lewis’s ‘Pantsuit Sasquatch‘, for which I recorded my feet walking up to a tortoise sculpture on a playground:

This joins the six other official music videos I have contributed to, and five unofficial music videos I’ve made. I guess I just like being in music videos.

The radio broadcast (which you can also listen to online) was episode #9 of the Open Phil Broadcast on Radio Orange. The broadcast mostly features regulars at the Open Phil open mic in Vienna. Each episode features an interview with and performance by two acts; I shared this one with Adrian Lüssing, also known as The Cliff.

It was an honour to be invited to participate in the broadcast, and it was made extra awesome by the fact that it happened while Joey Marianer, who has been setting a lot of my poetry to music, was visiting Vienna, so he participated too. I recited They Might Not Be Giants, then he sang his version of it, then we sang I Love Your Body, with Joey singing the first part and me singing the second part. Yes, me singing. This is about the first time I’ve sung for an audience, and the third time Joey and I had sung that song together, and it went on the radio. I think it went pretty well, though! We performed it again a few days later on the Open Phil stage, and I’ll post video of that once I’ve uploaded it.

The podcast was episode #60 of Wrong, but Useful, a recreational mathematics podcast by @icecolbeveridge (Colin in real life) and @reflectivemaths (Dave in real life). I was invited to be a special guest cohost. I’m not sure I contributed very much, but I once again recited They Might Not Be Giants, because the hosts had heard me perform that at the MathsJam Annual Gathering last year. I have to admit, I had not actually listened to the podcast until I was invited to be on it — podcast listening is something I usually do while commuting, and lately I’ve been noncommutative. However, before episode #60 was recorded, Joey and I listened to episode #59 together, and I’m happy to report that the answer we came up with for the coin-flipping puzzle was correct.

In hindsight, I wish I’d mentioned my linguistics degree while we were chatting about English and poetry and such. I also wish I’d said something about the fact that nobody on episode #59 noticed that the diameter of the Fields medal in millimetres happened to round up to the number of the podcast (that is, 64, not 59. You don’t expect mathematicians to give each podcast episode only a single number do you?)

This reminds me, I need to register for the MathsJam Annual Gathering soon. You should too, if you can get to it. It’s a lot of fun! And who knows? Maybe if you go, you’ll end up co-hosting a podcast.

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≥3 (a poem and song)


A while ago I wrote a poem about love, and how much more complicated it is than mathematics, and how the <​​3 heart symbol is a little oversimplified, or at least misleading to any mathematicians such as myself who come to believe that love is a strict inequality. I didn’t publish it here but I did perform it at my show in Café Concerto, while Johanna Van Tan improvised backing music:

I also performed it at A Bunch of Monkeys Read Some Stuff on JoCo Cruise 2017.

This is one of those poems that was always secretly a song in my head, so while we were on a train to Minneapolis I told Joey how the tune went, and when he was back in stationary accommodation he sang it to a slightly better tune:

So in a sense that’s two (which is less than three) musical versions of it! I can barely come up with anything coherent to say about this. ❤️

Here are the words:

Read the rest of this entry »

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They Might Not Be Giants: now a song!


The other day I discovered that the ukuletrically charged Joey Marianer has once again set something I wrote to music! Truly, a Joey is an exciting kind of friend to have. (No, not a joey. Not everybody‘s got a baby kangaroo.) This time it’s They Might Not Be Giants.

On the subject of people who could conceivably be called Joey, and who make music, my friend Joseph will be singing a parody of a song I wrote on his patreon some time soon. I’m looking forward to it! If you support him on patreon you’ll see it as soon as it comes out — check out some of his recently-unlocked older posts to get an idea of what you’re in for. The patreon is his only source of income at the moment, so your contribution would mean a lot to him, as well as being good value for you.

On the subject of They Might Not Be Giants, I recited it at the MathsJam Annual Gathering last weekend. It was my first time at a MathsJam and it was great fun. At MathsJam, anyone can give a five-minute talk about anything mathematical, and newcomers were especially encouraged to, so I decided to present The Duel, a more mathematical poem than I would usually do at open mics. I even made some slides depicting what was going on. Eventually, though, I started to think The Duel wasn’t very good and I should do They Might Not Be Giants instead. After reciting both to a focus group of order two a few hours before my talk, I made the switch. With my remaining talk time, I showed some of the haiku I found in the Princeton Companion to Mathematics. It seemed to go down well. I had brought along a few of my posters in case people would be interested in them, and came back with none.

The rest of MathsJam was amazing, and I’m sure I’ll be back. There were all sorts of talks, including another mathematical poet, as well as magic, coin-floating, robotic cube-solving, juggling, puzzles, balloon animals, fancy yarn spinning, mathematical song parodies (I also sang Tom Lehrer’s Derivative Song for the people at the MathsJam Jam, since they hadn’t heard it), mathematical cakes, and a competition competition!

I won an origami double-stellated tetrahedron in a competition competition competition. It might not technically be a double-stellated tetrahedron, but the competition was to name it, and, inspired directly by the talk by the shape’s creator (Kathryn Taylor), that’s what I named it.

I was a bit worried that it was going to be a pain to get that home without damaging or losing it, since it would get crushed in my bag and I’m not used to carrying something in my hands constantly. At first it had a string or rubber band around it which had been used to tie it to the competition box, so I tied it to a belt hook. At some point it fell off and partly came apart, but I was having dinner with other MathsJam attendees at the time, and one of them knew enough modular origami to fix it (Kathryn had run a table devoted to modular origami on the Saturday night.) After that I held it by hand, until I realised that it could be suspended quite securely in the Acme Möbius scarf I was wearing.

I heard, repeatedly, that there’s a magazine called chalkdust which I should really be submitting some of my mathematical writing to, so I’ll do that. First, though, I will read the copies I picked up at MathsJam.

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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:

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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.

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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 »

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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.

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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.

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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.

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