A Brain That Works ?2% of the Time

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I’m a bit cha­grinned to admit that hav­ing been mired in details I failed to see the obvious.

In stan­dard paper sizes (ISO 216) and pen-point sizes (ISO 128 &c), the square root of two is used as the inter­val between sizes, and between the short and long sides in the case of paper. This makes good sense for the pur­pos­es of mechan­i­cal enlarge­ment or reduc­tion. If you draw or write some­thing with a #2 pen (0.70mm) on A4 paper (210×297mm) which is then reduced to A5 paper (148×210mm) you may write on the reduc­tion with a pen (0.50mm) while retain­ing con­sis­ten­cy with the marks put down ear­li­er. All these neat pro­por­tions, very nice.

It did­n’t occur to me until sev­er­al min­utes ago that using a 1:?2 pro­por­tion as the inter­val is just a fan­cy way of say­ing even steps between sizes where two steps up means twice the orig­i­nal. The sin­gle incre­ments then are «halfway» to dou­ble (though of course not lit­er­al­ly half-way by lin­ear measure).

It’s an obvi­ous fact that has been star­ing me in the face for the bet­ter part of two decades. The more com­pli­cat­ed prod­ucts of that obvi­ous fact have been evi­dent to me all along. It feels a bit like real­iz­ing that words are made up of letters.

Published by splicer

3 Replies to “A Brain That Works ?2% of the Time”

  1. Paper sizes actu­al­ly dou­ble
    Paper sizes actu­al­ly dou­ble the area for each step, not two steps. (Same with cam­era f/stops, which are the same ratio.) Did you know that A0 paper is one square meter?

    Dad

    1. I did know that A0 is a

      I did know that A0 is a square meter, yes. 

      I’m going to have to think about the cam­era f/stops, and try to remem­ber the stuff I was taught back in the Ice Age col­lege. My hazy mem­o­ry was that each inte­ger incre­ment dou­bled or halved the aper­a­ture, not every oth­er inte­ger. It’s been so long since I’ve worked with an inci­den­tal light meter that I haven’t real­ly giv­en it much thought lately. 

      And yes, paper area dou­bles for each step but the length of a giv­en side dou­bles in two steps. A4 is 297×210mm (1÷(24) m²) and A6 148×105mm (1÷(26) m²)

      1. f/stops are based on the
        f/stops are based on the diam­e­ter of the aper­ture divid­ed by the focal length (which is con­stant,) Since the area of the aper­ture is ?r² (Pi x r^2 if you don’t do Uni­code,) dou­bling the aper­ture radius quadru­ples the area. So going from f/1.4 to f/2 halves the amount of light, f/2 to f/2.8 halves it again. f/4, f/5.6, f/8, f/11, f/16, f/22, f/32, etc. con­tin­ue the halv­ing There are oth­er f/stops used, par­tic­u­lar­ly with dig­i­tal cam­eras, but these are the doublers.

        You can ver­i­fy this quick­ly with­out doing any math if you just set your cam­era on aper­ture pri­or­i­ty and run through the choic­es, check­ing what it selects for the match­ing shut­ter speed.

        It is good we can BOTH be right about the paper ratio!

        Dad

        Dad

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