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What you see above is a tool to reorder the pages of a PDF file, into the order as described by Brent Yorgey here (called “Möbius double-sided printing” here). Basically, it reorders the n pages [1, 2, …, n] of a PDF file into the order [1, m+1, 2, m+2, 3, m+3, … m, 2m], where m = ⌈n/2⌉, and the final page 2m is omitted if it doesn't exist (i.e., if n is odd).
To use this tool, just upload a PDF file above, then click on the link to download a new PDF file with reordered pages.
(In short, so that it requires less mental effort to turn pages, and so that it's easier to look at the next/previous page(s) simultaneously. It's explained by Brent Yorgey on the two pages mentioned above; reproducing part of the explanation here along with his illustrations.)
Consider the standard/default method of two-sided printing: for a document with 8 pages, the result looks like:
That is, the first sheet of paper has pages 1 and 2 on its front and back, the second sheet has the next two pages, and so on.
If you instead print the re-ordered PDF (as generated by this tool above), the resulting sheets of paper will look like this:
When printed out as a bunch of loose sheets, this order has two (very minor) advantages over the standard order:
When you're done reading a certain page, the way to proceed to the next page is the same regardless of whether you're on an odd-numbered page or an even-numbered page: you don't have to remember, and so at page boundaries you don't have to shift your attention from the contents of what you're reading to figuring out page mechanics. What you do is always the same, namely flip the current sheet of paper and move it to the bottom of the stack:
The code rearranges pages using pdf-lib (GitHub) (v1.3.0) by Andrew Dillon. I tried to use Mozilla's PDF.js, but its documentation on these “core” matters (as opposed to just rendering a PDF in a canvas or whatever) seems poor/nonexistent compared to pdf-lib, which is excellent.
The idea of printing documents in this way, and the analysis and pictures above, are due to Brent Yorgey (here and here, as mentioned).