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I wasn't able to find an answer to this question myself - are there any built-in page designs (styles) in memoir class? If so, what are they and how do I set them in the document to see how each one of them looks?

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See §6.5 of the memoir manual, which discusses chapter styles, and has examples. –  Alan Munn May 31 '11 at 22:49
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up vote 10 down vote accepted

Section 7.2 PAGE STYLES of the memoir manual describes the pre-defined page styles:

The page styles provided by the class are:

empty The headers and footers are empty.

plain The header is empty and the folio (page number) is centered at the bottom of the page.

headings The footer is empty. The header contains the folio at the outer side of the page; on verso pages the chapter name, number and title, in slanted uppercase is set at the spine margin and on recto pages the section number and uppercase title is set by the spine margin.

myheadings Like the headings style the footer is empty. You have to specify what is to go in the headers.

simple The footer is empty and the header contains the folio (page number) at the outer side of the page. It is like the headings style but without any title texts.

ruled The footer contains the folio at the outside. The header on verso pages contains the chapter number and title in small caps at the outside; on recto pges the section title is typeset at the outside using the normal font. A line is drawn underneath the header.

Ruled This is like the ruled style except that the headers and footers extend into the fore- edge margin.

companion This is a copy of the pagestyle in the Companion series (e.g., see [MG+ 04]). It is similar to the Ruled style in that the header has a rule which extends to the outer edge of the marginal notes. The folios are set in bold at the outer ends of the header. The chapter title is set in a bold font flushright in the verso headers, and the section number and title, again in bold, flushleft in the recto headers. There are no footers.

book This is the same as the plain pagestyle.

chapter This is the same as the plain pagestyle.

cleared This is the same as the empty pagestyle.

part This is the same as the plain pagestyle.

title This is the same as the plain pagestyle.

titlingpage This is the same as the empty pagestyle.

EDIT: You can see them in action using this simple document

\documentclass{memoir}
\usepackage[english]{babel}
\usepackage{blindtext}
\usepackage{background}

\SetBgContents{empty}
\begin{document}
\pagestyle{empty}
\blinddocument
\clearpage
\SetBgContents{plain}
\pagestyle{plain}
\blinddocument
\clearpage
\SetBgContents{headings}
\pagestyle{headings}
\blinddocument
\clearpage
\SetBgContents{simple}
\pagestyle{simple}
\blinddocument
\clearpage
\SetBgContents{ruled}
\pagestyle{ruled}
\blinddocument
\clearpage
\SetBgContents{Ruled}
\pagestyle{Ruled}
\blinddocument
\clearpage
\SetBgContents{Companion}
\pagestyle{companion}
\blinddocument
\clearpage
\end{document}

Each group of four pages illustrates one style (I didn't include the myheadings style); the background package was used to inform the corresponding style that is being used.

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Thanks, this was extremely useful!! –  Peteris Krumins Jun 1 '11 at 16:32
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