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I am not sure whether linguistics board is the right place to ask this question, but since I couldn't find any better place here is the question:

Most (all?) of the writing systems are using the vector-like/linear alignment of text. It may be left to right, right to left, top to bottom, etc. But are there any writing systems that use multi-dimensional e.g. matrix-like alignment? Were there any real historical examples or how would, in theory, such a system look like?

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    I'm not sure what a matrix-like arrangement would look like; a page of text could be already seen as a matrix, with the letters filling columns and the lines being the rows. (A curious variant were boustrophedon systems, in which the direction of filling changed for each row.) – melissa_boiko Jul 21 '19 at 14:24
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    Cross-writing was used to save paper and postage costs en.wikipedia.org/wiki/Crossed_letter – Owain Jul 21 '19 at 15:37
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    To the extent that writing systems represent language, they have to be mostly one-dimensional because language is time-dependent -- speech events happen with timing implicit. There are parasitic dimensions, like intonation and stress, correlated gestures, facial expressions, and eye movements, which could be represented -- and occasionally are, in special contexts like ballet transcriptions -- but aren't normal parts of orthography, which usually restricts itself to phonemic segments in order. – jlawler Jul 21 '19 at 22:16
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    Islamic calligraphy is an interesting edge case – Luke Sawczak Jul 22 '19 at 2:50
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    Not a real-life example, but probably an interesting note if you aren't aware of it: a multi-dimensional writing system is an important plot point in the science fiction novella Story of Your Life, and in the blockbuster movie Arrival which is based on it. – Federico Poloni Jul 29 '19 at 20:02
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It is a bit of an edge case, and much depends on what you mean by "dimension", but the "hidden meaning" usage of furigana above kanji in Japanese may qualify as one extra dimension. The use of the extra dimension varies. It is certainly not part of the usual Japanese writing system, but rather an extension of it into a separate dimension, and it is only really used for its entertainment value.

SignWriting would certainly count, when/if it becomes more widely adopted. With the nature of most signemes requiring at least 2D space as well as time (and I'm sure many sign language users employ 3D space too), the notation tries to encompass all of it on a 2D page. Unicode has encoded some of its characters, but being able to render it correctly is another matter entirely.

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    In Mandarin, there's a similar use of pinyin above hanzi. My impression is it's influenced by the Japanese practice, though I'm not 100% sure. – WavesWashSands Jul 23 '19 at 0:12
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The Dongba script of Naxi comes to mind:

Example of the script

As you may have guessed from the picture, the Dongba script is generally left-to-right. But you can also have glyphs stacked together, and this gives it an extra vertical dimension. In addition, it's not always read in the same order. The ensuing discussion is from Fu (1981/2012).

Exhibit A:

Two vertical sequences of characters

The two phrases above are read as 3su 2xe and 3su 'khu respectively. The 3su glyph appears on top in the first phrase, but on the bottom in the second; in other words, the phrase is read from top to bottom in the first case and bottom to top in the second.

Exhibit B:

enter image description here

The three characters, from top to bottom, are 3khu, 2mɯ, 2thu. Together they are read as 23khu 2thu. So in this case you start from the middle character.

Exhibit C:

enter image description here

The house is a female buddha's temple. (The context is that the protagonist of the story, a white bat, is looking for the buddha.) There are three characters inside the house, which when put together mean 'unclean'. The bat was saying that the road leading to the front door of the temple was unclean, and refused to enter from that path.

Exhibit D:

enter image description here

The character for 'spear' is on the top left. The character for village is broken up by the spear. The whole character is pronounced 3phu and means to break into the village with force.

From these examples I think the script is pretty unambiguously multidimensional.

Fu, M. (1981/2012). Naxi Zu Tuhua Wenzi 'Bai Bianfu Qujing Ji' Yanjiu [A study of the Naxi pictogram text 'The Tale of the White Bat Obtaining the Scripture', 纳西族图画文字《 白蝙蝠取经记》 研究]. Beijing: The Commerical Press.

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    Similar "creative sign arrangement" can also be found in Egyptian hieroglyphs and in early forms of Sumerian cuneiform (in some respects at least up to the Ur III period, although later cuneiform tends to be highly linear). And the Mayans, of course, used a distinctive zig-zag writing order (with an even more complex and variable ordering of component signs within glyphs), although it's arguably still just a peculiar way of arranging one-dimensional text on a two-dimensional surface. – Ilmari Karonen Jul 31 '19 at 21:56
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Inspired by another answer, you could argue that comic books are an example of multidimensional writing, because:

  1. The reading direction goes left, right, up and down:

    enter image description here

  2. When there is a clear reading direction, words may still be intended to be read simultaneously, so to speak. In this example, all text except the top left box should be considered simultaneous:

    enter image description here

Examples were taken from Walsh, 2012. 'Comic Book Markup Language: An Introduction and Rationale', Digital Humanities Quarterly 6 (1).

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In Korean hangul, each character represents a syllable, which is made of several elements arranged vertically or horizontally. A bit like writing successive syllables from left to right but with each syllable written (in part) vertically.

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A possibly case of 2-D writing is Amerindian (Mexican) codices. They look like elaborate drawings but they actually also encode language.

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Hypertext

(uncountable) Digital text in which the reader may navigate related information through embedded hyperlinks.

Layout

A structured arrangement of items within certain limits.

In principle, language is hyper. There is not one intended order of reading in a shelf of books. Discurse is not hierarchically structured, except when it is. Take three books of one author next to each other, and finding the relations between them will require someone to write a fourth book, repeat ad infinitum.

Caligraphy is a good example when it comes to the spatial dimensions, but painting in general counts as speech in some sense, too. For example, the German term Schrift (cp. "script") covers film et cetera, in legalese.

Writing

(uncountable) Graphism of symbols such as letters that express some meaning

Even the choice of a font gives a typeset document certain notes of accentuation. Not to mention the depth of expressivity that a good hand script has. Effectively, it depends entirely on what you mean by dimension. Surely a letter is a grapheme. Only morse code is somewhat one-dimensional, and not even really, because the meaning of the symbols has to be confered through a side-channel.

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