Scaling: Why Giants Don’t Exist

Scaling: Why Giants Don’t Exist

Michael Fowler, UVa  10/12/06

 

Galileo begins Two New Sciences with the striking observation that if two ships, one large and one small, have identical proportions and are constructed of the same materials, so that one is purely a scaled up version of the other in every respect, nevertheless the larger one will require proportionately more scaffolding and support on launching to prevent its breaking apart under its own weight.  He goes on to point out that similar considerations apply to animals, the larger ones being more vulnerable to stress from their own weight (page 4):

Who does not know that a horse falling from a height of three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  … and just as smaller animals are proportionately stronger and more robust than the larger, so also smaller plants are able to stand up better than the larger.  I am certain you both know that an oak two hundred cubits high would not be able to sustain its own branches if they were distributed as in a tree of ordinary size; and that nature cannot produce a horse as large as twenty ordinary horses or a giant ten times taller than an ordinary man unless by miracle or by greatly altering the proportions of his limbs and especially his bones, which would have to be considerably enlarged over the ordinary.

 

For more of the text, click here.

 

To see what Galileo is driving at here, consider a chandelier lighting fixture, with bulbs and shades on a wooden frame suspended from the middle of the ceiling by a thin rope, just sufficient to take its weight (taking the electrical supply wires to have negligible strength for this purpose).  Suppose you like the design of this particular fixture, and would like to make an exactly similar one for a room twice as large in every dimension.  The obvious approach is simply to double the dimensions of all components.  Assuming essentially all the weight is in the wooden frame, its height, length and breadth will all be doubled, so its volume—and hence its weight—will increase eightfold.   Now think about the rope between the chandelier and the ceiling.  The new rope will be eight times bigger than the old rope just as the wooden frame was.  But the weight-bearing capacity of a uniform rope does not depend on its length (unless it is so long that its own weight becomes important, which we take not to be the case here).  How much weight a rope of given material will bear depends on the cross-sectional area of the rope, which is just a count of the number of rope fibers available to carry the weight.  The crucial point is that if the rope has all its dimensions doubled, this cross-sectional area, and hence its weight-carrying capacity, is only increased fourfold.  Therefore, the doubled rope will not be able to hold up the doubled chandelier, the weight of which increased eightfold.  For the chandelier to stay up, it will be necessary to use a new rope which is considerably fatter than that given by just doubling the dimensions of the original rope.

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Why do we believe in electrons, but not in fairies?

Why do we believe in electrons, but not in fairies?

by Benjamin Kuipers

No one has directly observed either electrons or fairies. Both of them are theoretical constructs, useful to explain observations that might be difficult to explain otherwise. The “theory of fairies” can actually explain more things than the “theory of electrons”. So why do we believe in electrons, but not in fairies?

Is the issue a political one, where the “electron” fans got the upper hand in the nineteenth century, so by the twentieth century the “fairy” fans were a scorned and persecuted minority? Or, have we proved for sure that fairies don’t exist?

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The 500 mile email

I read the following story some years ago but lost track of the original, it just resurfaced in my inbox. I think this is one of the best tourbleshooting stories around.

The imposssible problem

Here’s a problem that *sounded* impossible… I almost regret posting
the story to a wide audience, because it makes a great tale over
drinks at a conference. :-) The story is slightly altered in order to
protect the guilty, elide over irrelevant and boring details, and
generally make the whole thing more entertaining.

I was working in a job running the campus email system some years ago
when I got a call from the chairman of the statistics department.

“We’re having a problem sending email out of the department.”

“What’s the problem?” I asked.

“We can’t send mail more than 500 miles,” the chairman explained.

I choked on my latte. “Come again?”

“We can’t send mail farther than 500 miles from here,” he repeated.
“A little bit more, actually. Call it 520 miles. But no farther.”

“Um… Email really doesn’t work that way, generally,” I said, trying
to keep panic out of my voice. One doesn’t display panic when
speaking to a department chairman, even of a relatively impoverished
department like statistics. “What makes you think you can’t send mail
more than 500 miles?”

“It’s not what I *think*,” the chairman replied testily. “You see,
when we first noticed this happening, a few days ago–”

“You waited a few DAYS?” I interrupted, a tremor tinging my voice.
“And you couldn’t send email this whole time?”

“We could send email. Just not more than–”

“–500 miles, yes,” I finished for him, “I got that. But why didn’t
you call earlier?”

“Well, we hadn’t collected enough data to be sure of what was going on
until just now.” Right. This is the chairman of
*statistics*. “Anyway, I asked one of the geostatisticians to look
into it–”

“Geostatisticians…”

“–yes, and she’s produced a map showing the radius within which we
can send email to be slightly more than 500 miles. There are a number
of destinations within that radius that we can’t reach, either, or
reach sporadically, but we can never email farther than this radius.”

“I see,” I said, and put my head in my hands. “When did this start?
A few days ago, you said, but did anything change in your systems at
that time?”

“Well, the consultant came in and patched our server and rebooted it.
But I called him, and he said he didn’t touch the mail system.”

“Okay, let me take a look, and I’ll call you back,” I said, scarcely
believing that I was playing along. It wasn’t April Fool’s Day. I
tried to remember if someone owed me a practical joke.

I logged into their department’s server, and sent a few test mails.
This was in the Research Triangle of North Carolina, and a test mail
to my own account was delivered without a hitch. Ditto for one sent
to Richmond, and Atlanta, and Washington. Another to Princeton (400
miles) worked.

But then I tried to send an email to Memphis (600 miles). It failed.
Boston, failed. Detroit, failed. I got out my address book and
started trying to narrow this down. New York (420 miles) worked, but
Providence (580 miles) failed.

I was beginning to wonder if I had lost my sanity. I tried emailing a
friend who lived in North Carolina, but whose ISP was in Seattle.
Thankfully, it failed. If the problem had had to do with the
geography of the human recipient and not his mail server, I think I
would have broken down in tears.

Having established that — unbelievably — the problem as reported was
true, and repeatable, I took a look at the sendmail.cf file. It
looked fairly normal. In fact, it looked familiar.

I diffed it against the sendmail.cf in my home directory. It hadn’t
been altered — it was a sendmail.cf I had written. And I was fairly
certain I hadn’t enabled the “FAIL_MAIL_OVER_500_MILES” option. At a
loss, I telnetted into the SMTP port. The server happily responded
with a SunOS sendmail banner.

Wait a minute… a SunOS sendmail banner? At the time, Sun was still
shipping Sendmail 5 with its operating system, even though Sendmail 8
was fairly mature. Being a good system administrator, I had
standardized on Sendmail 8. And also being a good system
administrator, I had written a sendmail.cf that used the nice long
self-documenting option and variable names available in Sendmail 8
rather than the cryptic punctuation-mark codes that had been used in
Sendmail 5.

The pieces fell into place, all at once, and I again choked on the
dregs of my now-cold latte. When the consultant had “patched the
server,” he had apparently upgraded the version of SunOS, and in so
doing *downgraded* Sendmail. The upgrade helpfully left the
sendmail.cf alone, even though it was now the wrong version.

It so happens that Sendmail 5 — at least, the version that Sun
shipped, which had some tweaks — could deal with the Sendmail 8
sendmail.cf, as most of the rules had at that point remained
unaltered. But the new long configuration options — those it saw as
junk, and skipped. And the sendmail binary had no defaults compiled
in for most of these, so, finding no suitable settings in the
sendmail.cf file, they were set to zero.

One of the settings that was set to zero was the timeout to connect to
the remote SMTP server. Some experimentation established that on this
particular machine with its typical load, a zero timeout would abort a
connect call in slightly over three milliseconds.

An odd feature of our campus network at the time was that it was 100%
switched. An outgoing packet wouldn’t incur a router delay until
hitting the POP and reaching a router on the far side. So time to
connect to a lightly-loaded remote host on a nearby network would
actually largely be governed by the speed of light distance to the
destination rather than by incidental router delays.

Feeling slightly giddy, I typed into my shell:

$ units
1311 units, 63 prefixes

You have: 3 millilightseconds
You want: miles
* 558.84719
/ 0.0017893979

“500 miles, or a little bit more.”

Original author unknown