The Lumber Room

"Consign them to dust and damp by way of preserving them"

Colliding balls approximate pi

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Found via G+, a new physical experiment that approximates \pi, like Buffon’s needle problem: The Pi Machine.

Roughly, the amazing discovery of Gregory Galperin is this: When a ball of mass M collides with one of ball m, propelling it towards a wall, the number of collisions (assuming standard physics idealisms) is \pi \lfloor\sqrt{M/m}\rfloor, so by taking M/m = 10^{2n}, we can get the first n+1 digits of \pi. Note that this number of collisions is an entirely determinstic quantity; there’s no probability is involved!

Here’s a video demonstrating the fact for M/m = 100 (the blue ball is the heavier one):

The NYT post says how this discovery came about:

Dr. Galperin’s approach was also geometric but very different (using an unfolding geodesic), building on prior related insights. Dr. Galperin, who studied under well-known Russian mathematician Andrei Kolmogorov, had recently written (with Yakov Sinai) extensively on ball collisions, realized just before a talk in 1995 that a plot of the ball positions of a pair of colliding balls could be used to determine pi. (When he mentioned this insight in the talk, no one in the audience believed him.) This finding was ultimately published as “Playing Pool With Pi” in a 2003 issue of Regular and Chaotic Dynamics.

The paper, Playing Pool With π (The number π from a billiard point of view) is very readable. The post has, despite a “solution” section, essentially no explanation, but the two comments by Dave in the comments section explain it clearly. And a reader sent in a cleaned-up version of that too: here, by Benjamin Wearn who teaches physics at Fieldston School.

Now someone needs to make a simulation / animation graphing the two balls in phase space of momentum. :-)

I’d done something a while ago, to illustrate The Orbit of the Moon around the Sun is Convex!, here. Probably need to re-learn all that JavaScript stuff, to make one for this. Leaving this post here as a placeholder.

Or maybe someone has done it already?

Written by S

Mon, 2014-06-23 at 23:03:18

Posted in mathematics, unfinished

The Indian theory of aesthetic appreciation (rasa)

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Here’s a great, simple write-up aimed at a Western audience, from the Clay Sanskrit Library edition of Kālidāsa’s The Recognition of Shakuntala, written by Somadeva Vasudeva:

Imagine that you find yourself going to see a performance of “Romeo and Juliet.” You are in the right mood for the play, no mundane worries preoccupy your mind, you have agreeable company, and the theatre, the stage, the director and the actors are all excellent—capable of doing justice to a great play. Your seat in the theatre is comfortable and gives an unobstructed view.

The play begins and you find yourself drawn into the world Shakespeare is sketching. The involvement deepens to an immersion where the ordinary, everyday world dims and fades from the center of attention, you begin to understand and even share the feelings of the characters on stage—under ideal conditions you might reach a stage where you
begin to participate in some strange way in the love being evoked.

Now, if at that moment you were to ask yourself: “Whose love is this?” a paradox arises.

It cannot be Romeo’s love for Juliet, nor Juliet’s love for Romeo, for they are fictional characters. It cannot be the actors’, for in reality they may despise one another. It cannot be your own love, for you cannot love a fictional character and know nothing about the actors’ real personalities (they are veiled by the role they assume), and, for the same reasons, it cannot be the actors’ love for either you or the fictional characters. So it is a peculiar, almost abstract love without immediate referent or context.

A Sanskrit aesthete would explain to you that you are at that moment “relishing” (āsvādana) your own “fundamental emotional state” (sthāyi-bhāva) called “passion” (rati) which has been “decontextualised” (sādhāraṇīkṛta) by the operation of “sympathetic resonance” (hṛdayasaṃvāda) and heightened to become transformed into an “aesthetic sentiment” (rasa) called the “erotic sentiment” (śṛṅgāra).

This “aesthetic sentiment” is a paradoxical and ephemeral thing that can be evoked by the play but is not exactly caused by it, for many spectators may have felt nothing at all during the same performance. You yourself, seeing it again next week, under the same circumstances, might experience nothing. It is, moreover, something that cannot be adequately explained through analytic terms, the only proof for its existence is its direct, personal experience.

[…]

It is, moreover, a blissful experience. The fact that sensitive readers often weep while reading poetry does not mean that they are suffering, rather the tenderness of the work has succeeded in melting the contraction of their minds or hearts.

The non-ordinary nature of such aesthetic sentiments makes it possible for the spectator or reader to derive a pleasurable experience even from what in ordinary life would be causes of grief.

The Indian scholarly tradition has a lot more, including some very thoughtful deliberation and perceptive observation, but it seems good to start a discussion of rasa with an example like this, than to start with the technical details.

[Another good start may be via film. See for instance:
How to Watch a Hindi Film: The Example of Kuch Kuch Hota Hai by Sam Joshi, published in Education About Asia, Volume 9, Number 1 (Spring 2004).
and perhaps (and if you have a lot of time):
Is There an Indian Way of Filmmaking? by Philip Lutgendorf, published in International Journal of Hindu Studies, Vol. 10, No. 3 (Dec., 2006), pp. 227-256.
Previously on this blog: On songs in Bollywood]

Written by S

Fri, 2014-06-06 at 23:46:41

The rest is commentary

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Famous verses appear in many variants. Thanks to Google, it is easy to find many of them. For “paropakāraḥ puṇyāya, pāpāya parapīḍanam”, Google throws up a lot of variants for the first half.

The Vikramacarita has:

śrūyatāṃ dharmasarvasvaṃ, yad uktaṃ śāstrakoṭibhiḥ /
paropakāraḥ puṇyāya, pāpāya parapīḍanam

Other variants are:

saṅkṣepāt kathyate dharmo janāḥ kiṃ vistareṇa vaḥ |
paropakāraḥ puṇyāya pāpāya para-pīḍanam ||Panc_3.103||

or

aṣṭādaśapurāṇeṣu vyāsasya vacanadvayam /
paropakāraḥ puṇyāya pāpāya parapīḍanam //

or

ślokārdhena pravakṣyāmi yaduktaṃ grantha-koṭibhiḥ
paropakāraḥ puṇyāya pāpāya parapīḍanam

Going by the first line gives other verses:

śrūyatāṃ dharmasarvasvaṃ śrutvā caivāvadhāryatām | (or caiva vicāryatām ।)
ātmanaḥ pratikūlāni pareṣāṃ na samācaret ||

[Cāṇakya-nīti, Pañcatantra, Subhāṣitāvalī etc.]

See also:

https://en.wikipedia.org/wiki/Golden_Rule

prāṇā yathātmano ‘bhīṣṭā bhūtānām api te tathā |
ātmaupamyena gantavyaṃ buddhimadbhir mahātmabhiḥ

तस्माद्धर्मप्रधानेन भवितव्यं यतात्मना ।
तथा च सर्वभूतेषु वर्तितव्यं यथात्मनि ॥ Mahābhārata Shānti-Parva 167:9
(http://blog.practicalsanskrit.com/2013/05/do-unto-others-golden-rule-of-humanity.html)

05,039.057a na tatparasya saṃdadhyāt pratikūlaṃ yadātmanaḥ
05,039.057b*0238_01 ātmanaḥ pratikūlāni vijānan na samācaret
05,039.057c saṃgraheṇaiṣa dharmaḥ syāt kāmād anyaḥ pravartate

As Hillel says, the rest is commentary.

For (some) commentary, go here.

Written by S

Fri, 2014-06-06 at 23:38:30

Posted in quotes, sanskrit, unfinished

Prefatory apprehension

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Robert Recorde’s 1557 book is noted for being the first to introduce the equals sign =, and is titled:

The Whetstone of Witte: whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers.

Its title page (see http://www.maa.org/publications/periodicals/convergence/mathematical-treasures-robert-recordes-whetstone-of-witte, see also the full book at https://archive.org/stream/TheWhetstoneOfWitte#page/n0/mode/2up) contains this verse:

Original spelling

Though many ſtones doe beare greate price,
The
whetſtone is for exerſice
As neadefull, and in woorke as ſtraunge:
Dulle thinges and harde it will ſo chaunge,
And make them ſharpe, to right good vſe:
All arteſmen knowe, thei can not chuſe,
But uſe his helpe: yet as men ſee,
Noe ſharpeneſſe ſemeth in it to bee.

The grounde of artes did brede this ſtone:
His vſe is greate, and moare then one.
Here if you lift your wittes to whette,
Moche ſharpeneſſe thereby ſhall you gette.
Dulle wittes hereby doe greately mende,
Sharpe wittes are fined to their fulle ende.
Now proue, and praiſe, as you doe finde,
And to your ſelf be not vnkinde.

   
Modern spelling

Though many stones do bear great price,
The whetstone is for exercise
As needful, and in work as strange:
Dull things and hard it will so change
And make them sharp, to right good use:
All artsmen know they cannot choose
But use his help; yet as men see,
No sharpness seemeth in it to be.

The ground of arts did breed this stone;
His use is great, and more than one.
Here if you lift your wits to whet,
Much sharpness thereby shall you get.
Dull wits hereby do greatly mend,
Sharp wits are fined to their full end.
Now prove and praise as you do find,
And to yourself be not unkind.

Apparently the full title contains a pun (see http://www.pballew.net/arithm17.html): “the cossike practise” in the title refers to algebra, as the Latin cosa apparently meaning “a thing” was used to stand for an unknown, abbreviated to cos — but the Latin word cos itself means a grindstone.

The author again reminds readers not to blame his book, at the end of his preface:

To the curiouſe ſcanner.

If you ought finde, as ſome men maie,
That you can mende, I ſhall you praie,
To take ſome paine ſo grace maie ſende,
This worke to growe to perfecte ende.

But if you mende not that you blame,
I winne the praiſe, and you the ſhame.
Therfore be wiſe, and learne before,
Sith ſlaunder hurtes it ſelf moſte ſore.

Authors are either anxious about how their book is received, or make sure to be pointedly uncaring.

Sir Arthur Conan Doyle, in a mostly forgettable volume of poetry (Songs of the Road, 1911), begins:

If it were not for the hillocks
   You’d think little of the hills;
The rivers would seem tiny
   If it were not for the rills.
If you never saw the brushwood
   You would under-rate the trees;
And so you see the purpose
   Of such little rhymes as these.

Kālidāsa of course begins his Raghuvaṃśa with a grand disclaimer:

kva sūryaprabhavo vaṃśaḥ kva cālpaviṣayā matiḥ /
titīrṣur dustaram mohād uḍupenāsmi sāgaram // Ragh_1.2 //

mandaḥ kaviyaśaḥ prārthī gamiṣyāmy upahāsyatām /
prāṃśulabhye phale lobhād udbāhur iva vāmanaḥ // Ragh_1.3 //

atha vā kṛtavāgdvāre vaṃśe ‘smin pūrvasūribhiḥ /
maṇau vajrasamutkīrṇe sūtrasyevāsti me gatiḥ // Ragh_1.4 //

But the most nonchalant I’ve seen, thanks to Dr. Ganesh, is this gīti by Śrīkṛṣṇa Brahmatantra Yatīndra of the Parakāla Maṭha, Mysore:

nindatu vā nandatu vā
mandamanīṣā niśamya kṛtim etām
harṣaṃ vā marṣaṃ vā
sarṣapamātram api naiva vindema

Screw you guys. :-)

Written by S

Wed, 2014-05-28 at 23:56:11

Posted in history, mathematics

Big O() notation: a couple of sources

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This post contains, just for future reference, a couple of primary sources relevant to the O (“Big O”) notation:

  1. Some introductory words from Asymptotic Methods in Analysis by de Bruijn
  2. An letter from Donald Knuth on an approach to teaching calculus using this notation.

Read the rest of this entry »

Written by S

Thu, 2014-03-13 at 16:33:20

Visualizing product of permutations

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A simple pedagogical trick that may come in handy: represent a permutation \sigma using arrows (curved lines) from k to \sigma(k) for each k. Then, the product of two permutations can be represented by just putting the two corresponding figures (sets of arrows) one below the other, and following the arrows.

Representing permutations and products of permutations.

Representing permutations and products of permutations.

The figure is from an article called Symmetries by Alain Connes, found via the Wikipedia article on Morley’s trisector theorem (something entirely unrelated to permutations, but the article covers both of them and more).

I’m thinking how one might write a program to actually draw these: if we decide that the “height” of the figure is some h, then each arrow needs to go from some (k, 0) to (\sigma(k), h) (using here the usual screen convention of x coordinate increasing from left to right, and y coordinate increasing from top to bottom). Further, each curve needs to have vertical slope at its two endpoints, so that successive curves can line up smoothly. The constraint on starting point, ending point, and directions at the endpoints defines almost a quadratic Bezier curve, except that here the two directions are parallel. So it’s somewhere between a quadratic and the (usual) cubic Bezier curve, which is given by the start point, end point, and derivatives at the start and end point. (Here we only care about the direction of the derivative; we can pick some arbitrary magnitude to fix the curve: the larger we pick, the more smooth it will look at the ends, at the cost of smoothness in the interior.)

Even knowing the curve, how do we generate an image?

Written by S

Thu, 2014-03-06 at 23:15:44

The letter in Roister Doister

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An early English example of resegmentation for change of meaning: Merygreeke’s letter to the Dame Custance, from Ralph Roister Doister (c. 1553), the first comedy to be written in the English language. [Aside: Shakespeare born in 1564 started had written all his plays before 1613, only 60 years from then.]

The example also shows that enjambment (no pause at end of line) did indeed exist in early English as well.

Scene 4

Sweete mistresse where as I loue you nothing at all,
Regarding your substance and richesse chiefe of all,
For your personage, beautie, demeanour and wit,
I commende me vnto you neuer a whit.
Sorie to heare report of your good welfare.
For (as I heare say) suche your conditions are,
That ye be worthie fauour of no liuing man,
To be abhorred of euery honest man.
To be taken for a woman enclined to vice.
Nothing at all to Vertue gyuing hir due price.
Whersore concerning mariage, ye are thought
Suche a fine Paragon, as nere honest man bought.
And nowe by these presentes I do you aduertise
That I am minded to marrie you in no wise.
For your goodes and substance, I coulde bee content
To take you as ye are. If ye mynde to bee my wyfe,
Ye shall be assured for the tyme of my lyfe,
I will keepe ye ryght well, from good rayment and fare,
Ye shall not be kepte but in sorowe and care.
Ye shall in no wyse lyue at your owne libertie,
Doe and say what ye lust, ye shall neuer please me,
But when ye are mery, I will be all sadde,
When ye are sory, I will be very gladde.
When ye seeke your heartes ease, I will be vnkinde,
At no tyme, in me shall ye muche gentlenesse finde.
But all things contrary to your will and minde,
Shall be done: otherwise I wyll not be behinde
To speake. And as for all them that woulde do you wrong
I will so helpe and mainteyne, ye shall not lyue long.
Nor any foolishe dolte, shall cumbre you but I.
Thus good mistresse Custance, the lorde you saue and kepe,
From me Roister Doister, whether I wake or slepe.
Who fauoureth you no lesse, (ye may be bolde)
Than this letter purporteth, which ye haue vnfolde.

Scene 5

Sweete mistresse, where as I loue you, nothing at all
Regarding your richesse and substance: chiefe of all
For your personage, beautie, demeanour and witte
I commende me vnto you: Neuer a whitte
Sory to heare reporte of your good welfare.
For (as I heare say) suche your conditions are,
That ye be worthie fauour: Of no liuing man
To be abhorred: of euery honest man
To be taken for a woman enclined to vice
Nothing at all: to vertue giuing hir due price.
Wherfore concerning mariage, ye are thought
Suche a fine Paragon, as nere honest man bought.
And nowe by these presents I doe you aduertise,
That I am minded to marrie you: In no wyse
For your goodes and substance: I can be content
To take you as you are: yf ye will be my wife,
Ye shall be assured for the time of my life,
I wyll keepe you right well: from good raiment and fare,
Ye shall not be kept: but in sorowe and care
Ye shall in no wyse lyue: at your owne libertie,
Doe and say what ye lust: ye shall neuer please me
But when ye are merrie: I will bee all sadde
When ye are sorie: I wyll be very gladde
When ye seeke your heartes ease: I will be vnkinde
At no time: in me shall ye muche gentlenesse finde.
But all things contrary to your will and minde
Shall be done otherwise: I wyll not be behynde
To speake: And as for all they that woulde do you wrong,
(I wyll so helpe and maintayne ye) shall not lyue long.
Nor any foolishe dolte shall cumber you, but I,
I, who ere say nay, wyll sticke by you tyll I die.
Thus good mistresse Custance, the lorde you saue and kepe.
From me Roister Doister, whether I wake or slepe,
Who fauoureth you no lesse, (ye may be bolde)
Than this letter purporteth, which ye haue vnfolde.

http://www.gutenberg.org/files/21350/21350-h/21350-h.htm

See also https://plus.google.com/+MohanKV/posts/MXGPBQfAsMz

Written by S

Tue, 2014-02-25 at 07:30:28

Posted in language, literature

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