h a l f b a k e r yGuitar Hero: 4'33"
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Does anyone these days use slide projectors? Or, better, overhead
projectors (the old ones
that used acetate films).
Anyway.
This is a pointless idea, but illustrates a well-known fact about the
prime numbers which I,
nevertheless, always find amazing.
Here we go.
Prime numbers can
be generated by Eratosthene's sieve. In its
simplest (and least-efficient)
form, you can imagine all the numbers (2 to infinity) written out
in a long line, evenly spaced.
First you take a lot of black markers, and tie them to a stick so
that they are two spaces
apart, like a comb; you use this to cross out every second number
(starting with 4). Then you
adjust the pens so they're three spaces apart, and use this to cross
out every third number
(starting with 6, even though it's already been crossed out once).
Then adjust again, cross out every fourth
number (starting with 8; this is
redundant, of course, but hey), and so on.
Yes yes, you all know this. And you also all know that the pattern
of primes (the numbers left
un-crossed out) is effectively irregular.
But it ALWAYS amazes me that a series of regularly-spaced
crossings out can leave behind an
irregular pattern. It's a kinda magic.
TO THE IDEA! I hear you shout.
OK. The idea is this. Print a series of transparencies, each with
evenly-spaced dark lines. On
the first transparency, the lines are spaced at 0.2mm apart; on the
next transparency, 0.3mm
apart, etc. All the transparencies have holes down one edge to
hold them in to a binder.
Just lay the first transparency on the overhead projector, then
overlay it with the second, the
third, and so on, until only the primes (up to, say 2000) are left as
lines of light.
There is no point to this. Yet I would love to see the irregular and
unpredictable pattern of
primes being formed from a series of utterly regular, tangible
patterns.
[link]
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// then overlay it with the second, the third, and so on, until only the primes (up to, say 2000) are left as lines of light.//
And by this time, how much light is getting through your ND filters? |
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And if your transparency is 20cm wide, then for 2000 numbers the width of most of the light stripes will be 0.1mm - what kind of binding holds the transparencies in horizontal alignment so accurately? |
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//An ingenious one.// [+] |
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Re the first link - yes, but that's not what I have in mind at
all. It shows the end result (the primes) arranged in 2D,
and fails to convey the thing that intrigues me: that an
irregular pattern arises from the regular superposition of
regular patterns. (In contrast, the 2D representation of
primes emphasizes what little degree of regularity there is
in the broadly irregular sequence. The interesting thing is
that it also works for many different widths of pattern.
But that's another matter entogether.) |
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Re the second link - as far as I can tell, it's a picture of a
driverless musical bulldozer in front of what seems to be a
very inaccurate yet recognisable sketch of the old Norman
church in Yattsby-cum-Clitteris. |
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(By a quite extra-ordinary coincidence - or perhaps one
should say 'correlation' - I spent most of this weekend on a
smaller version of the same vehicle, re-scaping my land. I
now own the second-highest peak in East Anglia, as well as
Cambridgeshire's only fjord.) |
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Oh no, not again. [MB], the Great Auk is EXTINCT. That means
they are all, without exception, dead. |
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You are not going to be able to enliven this Autumn's shooting
parties on your estate by enticing Great Auks into your artificial
fijord. No more Auks, all gone, versteh? Your great-uncle Jasper
may have been a philanthropist, but that doesn't relieve him of
the guilt of having rendered an entire species extinct. |
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What is it with the Buchanan family and Auks, anyway? It's just
a sick obsession. |
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And NO MORE GENE SPLICING. "Jurassic Park" was a movie, OK?
Remember what happened Bob Peck when the velociraptors got
out? Yes; and how long did it take you to hunt down that last
clutch of Pterodactyls? "They'll come when I call …" - yes, well
that was nearly Famous Last Words, wasn't it? |
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There are not going to be any more Auks for your family and
friends to slaughter, and you are NOT to try and make any ever
again. You promised, remember? You seemed genuinely sorry
about your cousin Eustacius- what is it, seven years now and he,s
still hiding under that bed ? What sort of a life is that? |
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Look, it's nothing personal. Yes, everyone knows you can re-
create extinct species, but it's not big, and it's not clever. Just
because you can do something doesn't mean you should. Like the
slug farm; "The french will never notice, we can add plastic
shells". Well, they did notice. |
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Great Auks extinct?? Have they tried looking on one of my
Hebrides? Anyway, they taste like chewy fish. "Great" my
arse. |
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They've looked everywhere. And that business you tried with the
painted puffins didn't fool anyone except Bill Oddie. |
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And if they taste vile, why do you hunt them so relentlessly ?
Pandas, yes, panda with cashew nuts, sweet and sour panda, ery
tasty. Nothing wrong with Chinese food. What have Auks ever
done to you ? |
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The guards would have said if they'd been. |
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And we don't hunt them relentlessly. We just try to keep
them under control in the areas where the passenger pigeons
roost. |
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[MB], please sit down and prepare yourself for some bad news … |
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I know this isn't what you mean and i very much admire the relatively low-tech approach, but it occurs to me that a PostScript program could manage this quite admirably on a transparency. |
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//But, but, thats exactly what it does show. Especially for
your primes example.// |
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But, but no. At least not if I'm looking at the right link the
right way. To take your two-dimensional example, if I
understand your point, it puts the integers into a 2D grid
(like
consecutive lines of numbers, bookwise), and the prime
numbers tend to lie on diagonal lines. In other words, it
takes the apparently chaotic (yes, OK, 8th) pattern of the
primes, and brings out some level of order (the neat
diagonal stripes). |
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It's pretty, but it's the opposite of what I want: from order
(a series of striped patterns, each very regular, each one a
little more widely spaced than the last in a predictable
way), we generate chaos (the unpredictable pattern of the
primes). |
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The same applies to your spiral link. |
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Unless I've missed something. |
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[[8th], I always take my bad news standing up. It's difficult
reloading in a seated position.] |
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[[Nineteenthly] yes, graphical software could do it. But
somehow I like the tangibility of OHPs; the fact that each
one is regular, yet together they give an irregular pattern,
and there's no suspicion of any electronic trickery.] |
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I misread the idea text as implying that 2000 acetates would be piled atop one another as part of this performance and began to wonder where you might find a drummer with sufficient endurance to perform the necessary constant drum roll. |
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No, not 2000. To reveal the primes up to 2000, you'd need
about sqrt-2000 (something like 45) acetates. |
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I must be missing something:,after 2, wouldn't you need all the odd filters up to just under a half of 1000?
Edit: half of 2000 - slip of the thumb |
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No, because if N is composite, then at least one of its prime factors is less than sqrt(N). So if you've established that N has no divisors less than sqrt(N), it must be prime. |
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Ah, there you are. I was right. I was missing something... |
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If you wanted to be efficient, you could miss out all the
composite Ns. However, I want to include them (so, lines
spaced at 2, 3, 4, 5, 6...even though 4 and 6 are redundant).
Otherwise it looks like some sort of jiggery-pokery is going
on. |
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