2.1 Text Direction

While European languages are written from left to right (LTR), Arabic is written from right to left (RTL). Unicode supports these scripts by not only defining codepoints for the individual characters of these languages, but by recording the directionality of each character.

When a mixture of LTR and RTL characters appear in text (ie. bidirectional or BiDi text, such as an English text that includes Arabic words), Unicode's bidirectional algorithm[UnicodeBiDi] describes the order in which the characters will be displayed. All adjacent strongly-typed RTL characters (such as a in a single Arabic word) will be presented in right-to-left order, and vice versa for strongly-typed LTR characters. A cluster of characters with the same directionality is called adirectional run.

Within any given "paragraph", directional runs are then ordered according to the overalldirectional context. The bidirectional algorithm allows for higher-level protocols to determine whichsegments of a structured text constitute "paragraphs" in this sense. For example, in HTML block-level elements are taken as the paragraph segments. The top-levelhtml tag determines the directional context which can be changed on lower-level elements using thedir attribute.

For a gentle introduction to bidirectional text, see[UnicodeBiDiIntro].

2.2 Glyph Shaping

As Arabic is a calligraphic script, letters within words are typically joined together. When text in such calligraphic scripts is specified by character sequences, a process calledshaping is used to blend, or connect the character glyphs. In Arabic words consisting of a single character, that character is drawn in the "isolated" style. In multi-character words, alternative shapes are generally used depending on position: the first (rightmost) character is drawn in its "initial" shape, the last (leftmost) character gets its "final" shape, and any characters in the middle are of the "medial" shape.

Compare the isolated characters غ ي ر to the result of glyph shaping غير.

2.3 Mirroring

Some characters, viewed abstractly, have the same meaning in many languages, but the form used in RTL languages are the roughly the mirror image of the form used in LTR languages. Parentheses and quotation marks are such characters. Unicode deals with these cases by marking some codepoints as mirrored, meaning that an alternate glyph will be used for the character if it appears in a RTL context.

Note that mirrored symbols are not required by Unicode (SeeMirroring in[UnicodeBiDi], section 6) to be literally the exact mirror image. Indeed, it is considered an important point of Arabic calligraphy that they are not: the feather's head (kalam) is a flat rectangle. The writer holds the pen so that the largest side makes an angle of approximately 70° with the baseline. This orientation is kept throughout the process of drawing the character. Furthermore, as Arabic writing goes from right to left, some boldness is produced around segments running from top left toward the bottom right and conversely, segments from top right to the bottom left will rather be slim. Thus, the Arabic sum symbol, for example, is not simply the mirror image of sigma.

2.4 Number Systems

There are several decimal numeral systems in use in Arabic:

System	Unicode	Digits										Image	Regions
European	U0030-U0039	0	1	2	3	4	5	6	7	8	9		Maghreb Arab (eg. Morocco), as well as European
Arabic-Indic	U0660-U0669	٠	١	٢	٣	٤	٥	٦	٧	٨	٩		Machrek Arab (eg. Egypt)
Eastern Arabic-Indic	U06F0-U06F9	۰	۱	۲	۳	۴	۵	۶	۷	۸	۹		Iran

3.1 Arabic Notation; Moroccan Style

The current way of writing mathematical expressions in Morocco, is closely related to the French style:

Style	Image	MathML
Moroccan		<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mrow> <mi>f</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mtable> <mtr> <mtd> <mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>s</mi> </munderover> <mo>⁡</mo> <msup> <mi>x</mi> <mi>i</mi> </msup> </mrow> </mtd> <mtd> <mrow> <mtext>إذاكان </mtext> <mi>x</mi> <mo><</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mo>∫</mo> <mn>1</mn> <mi>s</mi> </msubsup> <mo>⁡</mo> <mrow> <msup> <mi>x</mi> <mi>i</mi> </msup> <mo>⁢</mo> <mi>d</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mrow> </mtd> <mtd> <mrow> <mtext>إذاكان </mtext> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">E</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>tg</mi> <mo>⁡</mo> <mi>π</mi> </mrow> </mtd> <mtd> <mrow> <mtext>غيرذلك </mtext> <mrow> <mo>(</mo> <mi>π</mi> <mo>≃</mo> <mn>3,141</mn> <mtext>مع</mtext> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mrow></math>

Although the mathematics would be embedded within a RTL language (Arabic), its directionality is still LTR. The connecting words and phrases within the math, however, are RTL Arabic, andshould be subject toglyph shaping(although some current MathML renderers are not doing this).Thus these phrases should appear as "إذاكان" (for "if"), "غيرذلك" (for "otherwise") and "مع" (for "with").

Also, the indication is that the bidirectional algorithm[UnicodeBiDi] should beapplied to individual text and token elements, rather than at a higher level as in HTML;that is, the token elements act as paragraph segments.Even with these considerations, the ordering of phrases within the last clause(for "otherwise (with pi=3.141)") is problematic. The obvious markup sandwichinganmrow for "pi=3.141" between twomtext's for "otherwise (with" and ")", respectively,would yield an incorrect ordering. A correct rendering seems to require the possibilityof embeddingmath withinmtext, which is not possible in MathML 2.0.But even then, the desired ordering would need to be marked up as two separatemtext elements:one for "otherwise", and one for "(with pi=3.141)". The Math Interest Group is currentlyconsidering the possibilities of such embedding. The example above was marked up byartificially placing the Arabic word for "with"after the "pi=3.141".

Given such issues, it is sometimes advantageous to minimize the use ofconnecting phrases, with preference to simple punctuation, such as:

Style	Image	MathML
Moroccan		<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mrow> <mi>f</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mtable> <mtr> <mtd> <mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>s</mi> </munderover> <mo>⁡</mo> <msup> <mi>x</mi> <mi>i</mi> </msup> </mrow> </mtd> <mtd> <mrow> <mtext>; </mtext> <mi>x</mi> <mo><</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mo>∫</mo> <mn>1</mn> <mi>s</mi> </msubsup> <mo>⁡</mo> <mrow> <msup> <mi>x</mi> <mi>i</mi> </msup> <mo>⁢</mo> <mi>d</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mrow> </mtd> <mtd> <mrow> <mtext>; </mtext> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">E</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>tg</mi> <mo>⁡</mo> <mi>π</mi> </mrow> </mtd> <mtd> <mrow> <mtext>; </mtext> <mrow> <mo>(</mo> <mi>π</mi> <mo>≃</mo> <mn>3,141</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mrow></math>

3.2 Arabic Notation; Maghreb Style

The Maghreb style of notation is widely used in North Africa:

Style	Image	MathML
Maghreb		Not yet attempted

Here, the most striking difference is that the overall mathematical layout is the mirror image of the preceding examples, that is,the mathematical directionality is RTL. Further, some symbols(eg ∑, <, ∈) are mirrored as well.Thus, we need a means of specifying the mathematical directionality, and assuring that the appropriate symbols are available in Unicode and are marked as mirrored.

The remaining differences are due to a more pronounced use of Arabic symbols:DAL (as the initial of = "function" in Arabic);the Arabic letter BEH,and the letters of the function name abbreviationfor tangent (without dots). Again, these differences fall into the category of localization,but reinforce the idea that the Unicode bidirectional algorithm, along with glyph shaping, should apply individuallyto token elements.

3.3 Arabic Notation; Machrek Style

As the final Arabic example, we consider the Machrek style generally used in the Middle East.

Style	Image	MathML
Machrek		Not yet attempted

Most differences between the Machrek and Maghreb styles are essentially due to localization:a specifically Arabic symbol is used for the summation(initial of = "sum" in Arabic);a different letter is used for the function(initial of, also "function" in Arabic);the letters of the elementary function name abbreviation are with dots;and a number format using Arabic-Indic digits and a comma for the decimal separator (but notthe same as the Arabic comma used in text).

Note that the symbol used for summation should probably be a mathematical symbolwith a codepoint distinct from the Arabic letter, as the European summation symbol isdistinct from the Greek Sigma. This point also applies to the Arabic product.

3.4 Additional Arabic Notations

Two additional unique notations involve combinatorics, namely the factorial andbinomial coefficients:

Style	Image	MathML
English		<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow><mn>12</mn><mo>!</mo></mrow></math>
Arabic		<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" dir="rtl"> <menclose notation="madruwb"> 12 </menclose></math>

The argument to the factorial must be wrapped in a form similar to the character LAM (ل), which must be stretched in both directions to accommodate. A newmenclose notation,madruwb is proposed for this case.

Style	Image	MathML
English		<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mo>(</mo><mtable><mtr><mtd>5</mtd></mtr><mtr><mtd>12</mtd></mtr></mtable><mo>)</mo> </mrow></math>
Arabic		<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" dir="rtl"> <mmultiscripts><mo>ل</mo> <mn>12</mn><none/> <mprescripts/> <none/><mn>5</mn> </mmultiscripts></math>

Finally, although stacked fractions are rendered the same way in both European and Arabic,bevelled fractions in RTL Arabic will appear, as one would expect, with the terms in RTL order,i.e. A divided by B would appear as "B/A".In some locales, the preference is for the slash to also be mirrored, as "B\A". For these cases,we suggest that authors employ explicit markup using the REVERSE SOLIDUS \, such as<mrow><mi>A</mi><mo>\</mo><mi>B</mi></mrow>.

3.5 Persian

Persian languages generally use the Arabic script (written RTL), but withthe mathematical directionality LTR, similar to the Moroccan style.We are aware of only one mathematical notation unique to Persian writing, the notation usedfor limits:

Style	Image	MathML
English		<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mrow> <munder> <mo movablelimits="false">lim</mo> <mrow> <mi>x</mi> <mo>→</mo> <mfrac bevelled="true"> <mi>π</mi> <mn>10</mn> </mfrac> </mrow> </munder> <mo>⁡</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mrow> <mo>=</mo> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>⁢</mo> <mrow> <mo>(</mo> <msqrt> <mn>5</mn> </msqrt> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mrow></math>
Persian		<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mrow> <munder> <mo movablelimits="false">حد</mo> <mrow> <mi>x</mi> <mo>→</mo> <mfrac bevelled="true"> <mi>π</mi> <mn>۱۰</mn> </mfrac> </mrow> </munder> <mo>⁡</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mrow> <mo>=</mo> <mrow> <mfrac> <mn>۱</mn> <mn>۴</mn> </mfrac> <mo>⁢</mo> <mrow> <mo>(</mo> <msqrt> <mn>۵</mn> </msqrt> <mo>-</mo> <mn>۱</mn> <mo>)</mo> </mrow> </mrow> </mrow></math>

While the overall notation is similar to the Moroccan model (LTR), it uses theEastern Arabic-Indic digits. The word "حد" (for "limit"), is used; this word should not only be affected byglyph shaping,but should be stretched horizontally to match the length of the underscript.

4.1 Clarification of bidirectional Algorithm for MathML

The following summarizes how directionality should be applied to MathML and, in particular, describes how the bidirectional algorithm should be applied (it falls into class HL4; SeeHigher Level Protocols: HL4 in[UnicodeBiDi], section 4.3).

• The overallmathematical directionality should be determined by a (new)dir attribute on the outermostmath element which takes one of the valuesltr orrtl; the default isltr. If this attribute isrtl the layout of all Layout, Script, Limit, Table and Matrix schemata should proceed from right to left. This includes such effects as the surd of anmroot starting from the right. When the mathematical directionality isltr, the layout should conform to the current MathML specification.

• The text content of each Token element should be treated as a separate directional segment and the bidirectional algorithm should be applied to each independently. The initial directional context for each Token element is determined by the mathematical directionality. This latter property should assure that individual mirrored symbols are treated correctly.

As an example, consider the MathML fragment:

<mn>1</mn><mo>+</mo><mi></mi><mo>-</mo><mn>2</mn>

Some browsers mis-apply the bidirectional algorithm to the expression as a whole, as in HTML.Applying the HTML algorithm would set the first two items LTR, but then switch directions uponencountering the letter; thus the last three items are reversed.

Style	Image	MathML
Right		<math xmlns="http://www.w3.org/1998/Math/MathML" display="display"> <mn>1</mn><mo>+</mo><mi>ب</mi><mo>-</mo><mn>2</mn></math>
Wrong

4.2 Glyph Shaping

Glyph shaping rules apply not only to the textual content of anmtext,but also to Arabic character sequences used as mathematical symbols (particularly inmi andmo). This shaping is the visual cue thatdistinguishes a single symbol from a sequence of symbols, perhaps representing a product.This is analogous to the use of roman font in European mathematics, to distinguish for example

<math xmlns="http://www.w3.org/1998/Math/MathML" display="display"><mi>sin</mi></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="display"><mi>s</mi><mi>i</mi><mi>n</mi></math>

Thus, implementors should apply shaping to each character sequence within the text content ofany token elements.

Certain Arabic characters (ا د ذ ر ز و)have no unique initial or medial shapes. Their use in the middle of a mathematical symbolwould tend to make the symbol look like the product of two shorter symbols.Thus, to avoid confusion, authors should avoid using these charactersin the middle of mathematical symbols.

4.3 Additional Mathvariants

For single character tokens, additional styles, besides isolated, are usedto enlarge the set of available distinct symbols, just as the bold and Fraktur styles areused in European mathematics. The styles used in Arabic mathematicsare "tailed", "looped" and "stretched", in addition to the "initial" style applied tothe individual character. Furthermore, the "double-struck" style is commonly used.The following table shows the character JEEM in the various styles, in bothdotted and undotted forms (see below):

	isolated	initial	tailed	looped	stretched	double-struck
dotted
undotted

It is proposed to consider themathvariant "normal", when applied to Arabic, to mean the result of glyph shaping, and in particular, the "isolated" style for single character tokens. It is also proposed to add the following values allowed formathvariant: "initial", "tailed", "looped" and "stretched".

It is not expected to be meaningful to apply the "bold", "italic", "fraktur", "script", "sans-serif" or "monospace" mathvariants (or combinations) to Arabic (although there is some sentiment for allowing "bold" and "italic"). Nor is it meaningful to apply any mathvariant other than "normal" to multicharacter tokens, which should have glyph shaping applied. The current MathML specification points out that the only combinations of characters and mathvariant that have an unambiguous interpretation are those that correspond to the SMP Math Alphanumeric Symbols. An analogous argument is to be made for Arabic and the proposed Arabic Math Alphabetic Symbols[UnicodeProposition] (not yet part of Unicode).

Both dotted and undotted alphabetic symbols are encountered in this Note. The choice of which type to use is up to local preferences, however; documents use either dotted or undotted symbols, but not a mixture, and in particular, the dots are not used to indicate semantic distinctions. Thus, it is not felt that dotting is a good candidate for a mathvariant value, but rather should be accommodated by the choice of symbol fonts available to user's browser, or possibly through CSS.

4.4 Mirroring

The MathML attributeslspace,rspace,lquote andrquote should be interpreted as opening and closing, rather than strictly left and right. This historical anomaly is analogous to the standard Unicode names for the parentheses: TheLEFT PARENTHESIS andRIGHT PARENTHESIS are marked asmirrored and are taken to representOPENING PARENTHESIS andCLOSING PARENTHESIS, respectively.

The Math Working Group, and other interested parties, should work to assure that the necessary codepoints for Arabic mathematics are not only available, but appropriately marked for mirroring. It is also to be hoped that available fonts will be available, and will respect the calligraphic qualities regarding mirroring.

4.5 Horizontal Stretchiness

In Arabic mathematics, the sum, product and limit are commonly stretched horizontally to the same width as the limits (over or under) that apply to them. Such stretching does occasionally appear, but is rare, in European mathematics. InHorizontal Stretching Rules of MathML ([MathML22e] section 3.2.5.8.3), standard allows for such horizontal stretching of some symbols at the discretion of the rendering agent. In this Note, we simply encourage developers to implement this feature for the appropriate Arabic symbols.

4.6 Additional Constructs

The Arabic notation for factorial is a sort of enclosure.We propose to add an additional allowed valuemadruwb (transliterationof the Arabic مضروب for factorial) forthenotation attribute ofmenclose.

A.1 Number Systems

Assuming that the text content ofcn elements can be unambiguouslyinterpreted as a number, the locale selection must be able to choose not only the set ofdigits to use, but what set of decimal and thousands separators.Generally, the comma is used as a decimal separator with both the European and Arabic-Indic digits,but note that such a comma is distinct from theArabic comma "،"used to separate items in a list.

A.2 Symbols Choice

There are two kinds of symbols: literal and mirrored symbols used according to the local area:

• the sum operator is presented in the two ways: and;

• the product operator is presented in the two ways: and;

• the limit operator is presented in the two ways: and. This last notation is used in Persian.

• the factorial operator is presented in the two ways: and !12.

These stretched operators can be compared to the mathematical stretchy accents, only the roles are reversed. We can also think of something similarto the square root construction.

B.1 Character Encoding

Even though some local symbols, used in mathematics written in an Arabic notation, can be obtained via mirroring of already existing symbols, there are many symbols found in Arabic mathematical handbooks that are not yet part of the Unicode Standard and cannot be obtained through a simple mirroring[ArabicMathUnicode]. Some of such special characters are submitted for inclusion into the Unicode Standard[UnicodeProposition].

B.2 Mathematical Fonts

Some font families are designed to meet with the requirements of typesetting mathematical documents in an Arabic notation.The RamzArab Arabic mathematical font[RamzArab] aims to provide a complete and homogeneous Arabic font family, in the OpenType format, respecting Arabic calligraphy rules.

Although letters in "tailed" and "stretched" forms are semantically distinctfrom the "initial" forms, they can be simulated by connecting with a particular finalform of HEH and the final form of ALEF, respectively, and applying glyph shaping. This techniquemay be useful when an insufficient variety of fonts is available.

Implementors are encouraged to make it feasible for users tochoose dotted or undotted mathematical symbol fonts easily in accord with local tastes.

B.3 Symbol Stretching

In the cases where operators need to be stretched to matchthe width of sub- or superscripts, the lengthening should be done usingcurves rather than straight lines.This curve lengthening is called curvedkashida. It is one of the most important aspects of the Arabic calligraphy.

Good	Bad

These curvilinear extensible symbols were generated by the CurExt application for the system T_EX with a PostScript font generator[RamzArab].

Although horizontal stretching of sum and product operators is rare in European mathematics: and,this stretching is more common, and more desired, in Arabic mathematics: and.

[Note: the broken corner in these symbolsis a known flaw to be repaired in a future version of RyDArab[RyDArab]].

B.4 Software Tools

The Dadzilla system, an adapted version of Mozilla, allows using MathML for Arabic mathematical notation[Dadzilla].