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*From*: John Denker <jsd@av8n.com>*Date*: Fri, 06 Jun 2014 13:25:45 -0700

Scaling laws have been central to physics since Day One (1638)

and remain tremendously important. They tend to get little

coverage in typical courses nowadays, which seems crazy, since

they are easier to learn, easier to teach, more powerful, and

more age-appropriate than a lot of the stuff that is covered.

It pays to always be on the lookout for scaling relationships.

http://www.av8n.com/physics/scaling.htm

I mention this because on 06/03/2014 06:17 PM, I wrote:

You can plot the probability of having at least one

collision as a function of N (the number of pigeons) and

H (the number of holes). It goes from p=0 at N=1 up to

p=1 at N=1+H. The distribution is Boolean at the ends of

the range, but probabilistic in the interior.

I should have pointed out that there is a nice scaling law

here. For any given probability of collision, the number of

items you can have (N) scales like the square root of the

number of slots (H).

In more detail, to a good approximation:

P_collision(N,H) ≈ N(N−1)/2H

≈ N^2/2H

That is to say, you can plot the whole probability-of-collision

curve, for (almost) all N and H, as a function of the single

variable N(N−1)/2H. This is an example of universality.

This is somewhat more esoteric than the simple scaling laws we

use to introduce the subject, such as surface-to-volume ratios,

but it has plenty of real-world applications, including computer

science (e.g. compiler design) and cryptography. The idea that

the number of items scales like the square root of the number

of slots is definitely a nontrivial result. Non-experts tend to

guess that N should scale in proportion to H. That's what simple

notions of dimensional analysis would suggest, but it's nowhere

near correct.

For details, see

http://www.av8n.com/physics/scaling.htm#main-birthday

**References**:**Re: [Phys-L] Faraday paradox +- pedagogy +- critical thinking***From:*Bernard Cleyet <bernard@cleyet.org>

**Re: [Phys-L] Faraday paradox +- pedagogy +- critical thinking***From:*Philip Keller <pkeller@holmdelschools.org>

**[Phys-L] probability Was: Re: Faraday paradox +- pedagogy +- critical thinking***From:*Bernard Cleyet <bernard@cleyet.org>

**Re: [Phys-L] pigeon holes***From:*John Denker <jsd@av8n.com>

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