Colliding balls approximate pi

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

Roughly, the amazing discovery of Gregory Galperin is this: When a ball of mass collides with one of ball , propelling it towards a wall, the number of collisions (assuming standard physics idealisms) is , so by taking , we can get the first digits of . 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 (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?

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The Indian theory of aesthetic appreciation (rasa)

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]

The rest is commentary

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.