LEdoian's Blog

Print your stuff on Möbius bands!

I found a fun and useful way of printing stuff to ~~both~~all sides of a paper. I just need to find the right printer!

Quick recap: how to conventionally print stuff two-sided

A typical way is just sending the page to get printed two-sided (with setting the correct way of flipping pages). That is, on the other side of page 1 is page 2, next sheet contains pages 3,4, then 5 & 6, …

This is usually trivial to print on duplex printers, a bit hard to simulate on one-sided printers (but some drivers can do that) and has drawbacks when you need to look at stuff on other pages at the same time – you need to flip the sheet often, as you only can put half of the pairs of pages next to each other (even one and the following odd one).

Ordinary two-sided printing. Red arrows show sheet flips between consecutive pages.

A slight improvement hack is putting two pages on the same side of the paper (works well with A-series of papers, I don't know for Letters &co.) – you can put up to four pages of the original document next to each other, if they are the right ones, but there are still pairs of pages that need turning sheets. Also only works if the original pages do not have too tiny features on them. [1]

Booklets are fun and approachable, but still suffer from the same issues as the conventional duplex print. They might be a bit hard to print, but programs like pdfbook or paperjam make it easy to prepare for the classic duplex printing. Also, it is maybe hard to tell which page ends up where, as the order is: last+first, second+penultimate, third-from-end+third, … until the pages meet in the middle.

The most common booklet order with two pages per side for landscape orientation. (Note that we show more pages, and thus more sheet-flips; the number of sheet-flips is in fact the same as for two-sided printing.)

The improvement for seeing multiple consecutive pages

In order to be able to look simultaneously at many consecutive pages of the original, I think the order of first+first-past-half, second+second-past-half, … middle+last is much better (or maybe even the best). Since consecutive pages end up on different sheets (whenever there are at least three pages), if the original has e.g. figures on different page or long code listing, you can see it all!

The "Möbius order" of pages.

And this is really easy to use: You read a page and when you don't need it anymore, you flip it and put to the end of the page stack [2]. If you need to look at several pages, just rotate them in the same order as they go the first time. [3]

Need to print this? For one-sided printers this is rather easy, too: just print the first half (the bigger one) on the sheets, then put them back into the tray and print the rest on them. You might need to experiment which side the sheets should be put in and whether you need to print the rest in reverse order, but that is it.

Got the pages shuffled? Sort them by the first half, as if the print was one-sided.

The only annoying thing for me is that there is not much software that could reorder the pages for two-sided printing, so that you don't need to re-insert the sheets back in the tray. So I patched paperjam to enable this. [4]

And the best part? If you would try to glue consecutive pages side-to-side, you'd end up with a Möbius band! So if you get a Möbius paper, you can just print this one-sided (duh :-D)

Honorable mention: leporello

Printing leporellos (aka concertina folded) also has many of the same benefits, since there is only one pair of consecutive pages that need a page flip. The order is first+last, second+penultimate, … and the original pages can be shuffled this way with paperjam or simply using the other order for the second side printing, than for the Möbius band. But there is a bit of fun topology missing here :-)

A leporello order is also quite good, with only one sheet-flip in the entire document.

Is this the best order?

Yes, if "best" means "the minimum difference of numbers of pages that get put on the same sheet is as big as possible". The proof is left as an exercise for the reader.

Of course, this holds for a set of pages with no additional assumptions. In ordinary print, having a sheet-turn between chapters is fine and under similar guarantees other approaches may yield better results.

Cheat sheet: paperjam commands

Various commands for ordering pages for duplex printing with paperjam.
Order Command
Classic two-sided null
Two pages per side nup(2)
Booklet book (follow with nup(2) for actual booklet print)
Leporello modulo(2) {1 2} modulo(1,half) {1 -1} (The first modulo just adds blank pages to the end.)
Möbius (with patch) interleave(2)
Möbius (known page count) select {1..5 10..6} modulo(1,half) {1 -1}
Multiple Möbius bands, odd-even modulo(4) {1 3 2 4}
Multiple bands, "modulo 3" modulo(6) {1 4 2 5 3 6}
Second half (smaller) of pages in reverse order modulo(1,half) {-1}
Second half (smaller) of pages in normal order modulo(1,half) {-1} modulo(1) {-1}
First half (bigger) of pages modulo(2) {1 2} modulo(1,half) {1}

I might create more patches for avoiding the weird modulo commands…

[1]Most of the figures in this article are drawn with a single page per a side of a sheet. I consider putting more pages on a single side of paper to be an implementation detail, because it is not always possible (e.g. with too small font) and sometimes you could put more than two pages on a single side of paper, which leads to the fact that if you put everything on one side of the paper, you can see everything at once and save the other side. Not very useful though… My only exception is the booklet printing below, because that one seems to be rather common.
[2]See how this neatly puts the first-past-half page right after the half of the stack? Awesome!
[3]Also, if you flip the page around the wrong edge, you can just rotate the rest of the stack and end up with the correct orientation.
[4]A slight variation for which I can generate the order with upstream paperjam is using this order on small subsets of pages. For example, if you only want to be able to see any two consecutive pages, you can do this for just four pages – the order is then 1+3, 2+4, 5+7, 6+8,… Since each sheet either contains two odd or two even pages, the following page is on different sheet than the previous one. And you can do this "modulo 3" to see three pages: 1+4, 2+5, 3+6, 7+10, … This "simulates" multiple smaller Möbius bands, but will be probably harder to use.