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Prime slides

Or Eratosthenes' OHPs
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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.

MaxwellBuchanan, May 07 2011

[link]






       // 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?
AbsintheWithoutLeave, May 07 2011
  

       Sufficient.
MaxwellBuchanan, May 07 2011
  

       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?
pocmloc, May 07 2011
  

       An ingenious one.
MaxwellBuchanan, May 07 2011
  

       //An ingenious one.// [+]
Klaatu, May 07 2011
  

       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.)   

       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.   

       (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.)
MaxwellBuchanan, May 08 2011
  

       Oh no, not again. [MB], the Great Auk is EXTINCT. That means they are all, without exception, dead.   

       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.   

       What is it with the Buchanan family and Auks, anyway? It's just a sick obsession.   

       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?   

       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?   

       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.
8th of 7, May 08 2011
  

       Great Auks extinct?? Have they tried looking on one of my Hebrides? Anyway, they taste like chewy fish. "Great" my arse.
MaxwellBuchanan, May 08 2011
  

       They've looked everywhere. And that business you tried with the painted puffins didn't fool anyone except Bill Oddie.   

       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 ?
8th of 7, May 08 2011
  

       The guards would have said if they'd been.   

       And we don't hunt them relentlessly. We just try to keep them under control in the areas where the passenger pigeons roost.
MaxwellBuchanan, May 08 2011
  

       [MB], please sit down and prepare yourself for some bad news …
8th of 7, May 09 2011
  

       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.
nineteenthly, May 09 2011
  

       // irregular //   

       <pedantry>   

       Sp. 'chaotic'   

       </pedantry>
8th of 7, May 09 2011
  

       //But, but, thats exactly what it does show. Especially for your primes example.//   

       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).   

       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).   

       The same applies to your spiral link.   

       Unless I've missed something.   

       [[8th], I always take my bad news standing up. It's difficult reloading in a seated position.]   

       [[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.]
MaxwellBuchanan, May 09 2011
  

       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.
calum, May 11 2011
  

       No, not 2000. To reveal the primes up to 2000, you'd need about sqrt-2000 (something like 45) acetates.
MaxwellBuchanan, May 11 2011
  

       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
Ling, May 11 2011
  

       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.
Wrongfellow, May 11 2011
  

       Ah, there you are. I was right. I was missing something...
Ling, May 11 2011
  

       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.
MaxwellBuchanan, May 11 2011
  
      
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