blog/content/mobius-print.rst

Print your stuff on Möbius bands!
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:slug: mobius-print
:date: 2024-03-02 18:07
:modified: 2024-03-03 14:59
:tags: lifehack, print
:category: technology
:keywords: Möbius band, two-sided printing,
:lang: en
:translation: false
:status: published

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).

.. figure:: {static}/images/mobius-print/twoside.svg
   :width: 66%
   
   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.
[#illustrations]_

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.

.. figure:: {static}/images/mobius-print/booklet.svg
   :width: 66%
   
   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!

.. figure:: {static}/images/mobius-print/mobius.svg
   :width: 66%
   
   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 [#ordering]_. If you
need to look at several pages, just rotate them in the same order as they go the
first time. [#mistake]_

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
<{static}/images/mobius-print/interleave.patch>`__ `paperjam
<https://mj.ucw.cz/sw/paperjam/>`__ to enable this. [#multi-mobius]_

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 :-)

.. figure:: {static}/images/mobius-print/leporello.svg
   :width: 66%
   
   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.

.. Hint: you cannot pair the middle page to anything else to get a better
   result.

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
==============================

.. list-table:: Various commands for ordering pages for duplex printing with paperjam.
   :header-rows: 1
   :widths: auto

   * - 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…

----

.. [#illustrations] 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.

.. [#ordering] See how this neatly puts the first-past-half page right after
   the half of the stack? Awesome!

.. [#mistake] 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.

.. [#multi-mobius] 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.