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If I understand correctly, the main text box on a page is a special box, since in addition to the normal paragraph typesetting, text can also overflow into the next page. Is it possible to, say, have two minipages (or any sort of boxes that I can position arbitrarily in my document) so that the text in the first box that overflows gets put into the second box?

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You could play around with multicol. Columns aren't too arbitrarily positioned though. –  doncherry Sep 5 '11 at 0:27
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Another package to look at is mdframed. –  Alan Munn Sep 5 '11 at 0:58
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Also see flowfram with documentation in ffuserguide.pdf. –  Will Robertson Sep 5 '11 at 15:57
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up vote 2 down vote accepted

With a lot of manual intervention it's possible, but not with the standard minipage environment.

A "reduced version" of TeX's page builder is available to split vertical boxes:

\newbox\mybigbox \newbox\mysplitbox
\setbox\mybigbox=\vbox{<material>}
\setbox\mysplitbox=\vsplit\mybigbox to 10cm

Now the box register \mysplitbox will contain 10cm high text from \mybigbox and \mybigbox will contain the rest.

Careful reading of the TeXbook or TeX by topic is necessary, as this low level mechanism has never been canned into high level macros.

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