Perhaps a rough look-up table for (say) each 10 degrees of azimuth around the observing point that gives the altitude to solve for? Finally a couple of iterations to find what azimuth the Sun will be nearer the actual setting time, perhaps taking the 'flat horizon' setting time as a starting value?
I am very frustrated that I keep spinning and getting countries that are "a harder starting point" or "a much harder starting point" compared to the USA. Which I suppose is the point. (And it's more biased towards the worse-off countries than I expected, because I'm thinking in terms of population and births are biased that way more than population is.)
Maybe you could start off with a long string and then mark off its midpoint, the midpoints of the halves, and so on. Then you wouldn't have to cut a new string every time you want to measure something.
When working on a project where you need a bunch of things to be the same, you take a stick and mark on it at various points the dimensions you're using -- when working on a house, it might be things like the heights of outlet boxes and switches, the width and height of rough opening for doors, the height of window sills, etc etc.
Then, you just use the stick as the reference, using the marking for outlets to position all of the outlets instead of measuring the height of the floor in inches or millimeters or cubits or whatever each time. It's kind of like a measuring jig.
("Measure once, cut twice" is a superior methodology which has been unfairly maligned for generations.)
This works fantastic for building furniture as well, where the absolute dimension doesn’t matter as much as all of the pieces having matching dimensions. A cabinet with drawers, for example. The story stick captures the spacing between the drawers, the width of the drawer, the slightly smaller height of the drawer face, etc.
It feels really imprecise the first time you set the fence on a table saw based on a marking on a stick instead of a precise specific value but the results are hard to argue with.
With carpentry in particular, it is extremely powerful to make multiple cuts at the same time -- set a fence once and then cut everything that needs to match at the same time, or stack multiple pieces together, or cut a board to length before ripping it into several pieces that need identical lengths.
Sure, check your measurements to be sure they're correct, but the more times you can cut based on the same measurement, the less measurement error can creep in.
Was going to mention that too. 100% agree. If I mess up and end up needing to make matching cuts later on, I'll often set the fence using one of the existing pieces too instead of trying to re-measure. The story stick works great but lining up the teeth on the blade with the cut edge of an existing piece works fabulously well.
A similar strategy I've used when I've known that there was going to be cuts that I couldn't sequence like that is to cut "as built" story sticks with scrap dimensional lumber and write what they are right on the board.
What, and use the same thing to measure stuff anytime you measure something? Like that's ever gonna catch on! Next you're gonna tell me to use my lower arm as a measuring stick!
As an American I have done this with 8.5 x 11 "letter" paper. I wonder if there's some way one can take advantage of the special properties of A[n] paper.
1000% yes! An 8.5x11" paper is effectively a 12" ruler accurate to 2 decimal places.
Fold an 8.5" into a square (right triangle) and the long edge is exactly 12.02"
Fold that in half and you can measure 6.01", and 3.005" (exactly). You get 1.5" for free, and can fairly accurately get exactly 1" by rolling the other 3" side into thirds.
If you want to get an exact 1", you can technically get there via 11"-8.5"-1.5", and that gives you the full imperial (fractional) measurement basis, all from folding a (presumably accurate) 8.5x11" piece of paper.
As a long time European I never thought I’d come to see the sense of American ways, but having lived here now for a couple of years, it actually is easier for it to just be straight up 8.5 x 11 rather that a ratio that includes a square root.
Everyone makes paper the same hypothetical way, by producing large sheets and cutting them in half, and ANSI E (34"x44" or 864mm x 1118mm) isn't that different than A0 (841mm x 1189mm), but the slight starting difference means that there are two aspect ratios for ANSI (17/22 and 11/17). On the one hand, they're simple fractions and not irrational numbers; on the other, they're different, so you can't just double the size of something printed on ANSI A/letter sized to fill ANSI B/tabloid size, the way you can go from A4 to A3.
Only a small subset of users will ever want to do that (since if you're printing text you probably need to re-typeset it to keep the type a good size for reading), but only a small subset of users actually care about the aspect ratio or exact dimensions of their paper at all, so whether it is 8.5 or 8.11 or 8.314159... inches doesn't really matter.
Many, many people want to double or halve documents.
Teachers at school would print (or photocopy) A4 in half to save paper, or doubled for the blind girl in my class.
I'd do it myself at university to save paper (money).
I don't print much nowadays, but I use this feature occasionally to print something as a booklet. I printed some lost board game rules on A3, since it was an A4 PDF.
Sorry, I should have specified "and have it look perfect".
People do that all the time with US letter paper, print two to a sheet, you just end up with slightly wonky margins and usually everything being more like 40-45% the size it would be doubling up A4 paper. That use case isn't really hindered.
That's not a difference between ANSI and ISO paper sizes.
ANSI A (US letter) is a half sheet of ANSI B (ledger/tabloid) is a half sheet of C is a half sheet of D is a half sheet of E. When producing the paper, there is no waste of material or time, its the same process just starting with a slightly differently sized starting sheet (hypothetically; I am assuming that paper production has advanced beyond shaking screens of the largest handleable size by hand).
The difference is that ANSI A, C and E have aspect ratios of 17/22 (0.77) and ANSI B & D have aspect ratios of 11/17 (0.65), while all ISO sizes have aspect ratios of 1/sqrt(2) (0.71).
The waste comes in when scaling between adjacent sizes.
Going from A4 to A3, you can enlarge a document to 141% of the original size, and the margins will match.
Going from US letter to tabloid (ANSI A to B), the width of the paper is 129% larger and the height is 154% larger, so you can only enlarge your document to 129% the original size, and you have larger vertical margins, which is waste.
(But if you double it, from A to C, the problem goes away, because the aspect ratio is the same; so you can produce posters of multiple sizes, just not on every ANSI paper size at once.)
So, regarding books, why do you think the methods of printing books varies based on the size of the printing sheets?
Regardless of the size of your printing sheet, you choose a page size that's based on dividing your printing sheet in half N number of times, typeset your document to that page size (which you can't even skip for ISO paper sizes, because you pick your font size independent of the paper sizes), print 2^N pages per printing sheet in a particular pattern, fold and/or cut the sheet up, and bind.
There's no difference in waste or time regardless of your paper size choices, unless you do something silly or artistic, like choosing to print a square book or some shape not derived from halving your paper size.
I've been working with paper sizes a lot for the last year, and I've rarely thought about the square root of two ratio and when I've, it has been just to amuse myself. However, knowing that to get an A5 piece of paper I just need to cut/fold in half an A4, and that I can get to A3 and A2 by adding A4s, has been really useful. If I were in USA, didn't have that, and instead would have to install yet another new size system in my head[^1], I would despair.
What bothers me mostly about American papersizes (I’m also a European immigrant) is that the ratio is not consistent between sizes. So if I design a poster, but want a couple of letter sized printouts for some reason, I have to create a whole new design, rather then just shrink everything down. Otherwise the margins get all wonky.
One nice thing with Letter size is you can fit 80 columns of 10 dpi text with standard LaserJet margins. With A4 you have to squeeze the characters together slightly to make that fit.
A[n] sizes are useful when enlarging or shrinking documents. Enlarge or shrink by muliples of sqrt(2) and there's always a fitting paper size available. Or you can put two A5s together on an A4, or two A4s on an A3.
> I wonder if there's some way one can take advantage of the special properties of A[n] paper.
Not as a consumer. As a paper producer, you take advantage of it by cutting large sheets of paper in half to produce smaller sheets. Since you never sell any sizes that aren't clean multiples of each other like this, you've minimized the amount of paper you waste. That's the "advantage".
This was once the standard way of making paper; I don't know if it still is.
As a consumer I used to use it all the time, though it matters a lot less these days. Two A4 pages at 50% zoom (A5) fit on one A4. You could cut your printing cost for drafts in half by doing that, back when we had to actually print to check the layout. Same went for posters etc, and since the aspect ratio was preserved it was really handy to preview at home on A4 sheet before taking it to the print shop.
I’m sure you can do that on other size systems, but ISO paper sizing gives you accurate scaling.
Same goes for photocopies, photocopiers can scale copies so two A4 sheets copy to one, if you don’t need the same size.
This assumes there are no errors anywhere in the sizes or alignments of the A4 base page or either A5. Otherwise, you'll have an A5 running over an edge of the A4 or both A5s overlapping in the middle.
If your pages are designed with margins on the assumption that errors in the paper are common, this issue disappears because the margins cover for it. But still, if I wanted to do a display of two 8.5" x 11" sheets of paper, I'd want a board that was bigger than 17" x 11".
Sizing errors are essentially unheard of, and I've never seen anyone having any trouble with joining or folding ISO paper to go one size up/down. It's a completely normal operation, which people working in printing and publishing will routinely do without a second thought.
For commercial printing, there's the SRA paper series (Supplementary Raw) which is designed to accommodate bleed and alignment bars. An A4 glossy magazine, for example, might be printed on SRA3 and will be trimmed, folded, and stapled automatically at the end of the printing process. But that's a technical detail for the printer to care about - the publisher or designer might specify "folded A3 with bleeds", and the printer will choose the correct raw format to provide that within their printing system.
As the other commenter said, alignment issues have never been a problem.
If you're manually aligning the sheets on the photocopier bed maybe, but the edges are set up for that so it's never been an issue for me. However every photocopier I've used that was made since the late 90s lets you do the sheets individually so you can use the copier bed to align each one.
Because the ability to scale like this is so ubiquitous we're just all used to doing it.
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