Notation and symbols have proved very important for mathematics. Mathematics has grown in part because its notation continually changes toward being succinct and suggestive. There have been many new signs developed for use in mathematical notation, and mathematicians have not held back from making use of many symbols originally introduced elsewhere. The result is that mathematics makes use of a very large collection of symbols. It is difficult to write mathematics fluently if these characters are not available for use. It is difficult to read mathematics if corresponding glyphs are not available for presentation on specific display devices.
The W3C Math Working Group therefore took on directly the task of specifying part of the full mechanism needed to proceed from notation to final presentation, and started collaboration with organizations undertaking specification of the rest.
This chapter of the MathML specification contains a listing of character names for use with MathML, recommendations for their use, and warnings to pay attention to the correct form of the corresponding code points given in the UCS (Universal Character Set) as codified in Unicode and ISO 10646 [see [Unicode] and the Unicode Web site]. For simplicity we shall refer to this character set by the short name Unicode. Though Unicode changes from time to time so that it is specified exactly by using version numbers, unless this brings clarity on some point we shall not use them. The specification of MathML 2.0 [MathML2] used to make use of some characters that were not part of Unicode 3.0 but which had been proposed to the Unicode Technical Committee (UTC), and thus for inclusion in ISO 10646. They have been included in the revisions Unicode 3.1 and 3.2. As of the publication of the MathML 2.0 (2nd Edition) the current version is Unicode 4.0. (For more detail about this see Section 6.4.4 Status of Character Encodings.)
While a long process of review and adoption by UTC and ISO/IEC of the characters of special interest to mathematics and MathML is now complete there remains the possibility of some further modification of the lists of characters accepted. To make sure any possible corrections to relevant standards are taken into account, and for the latest character tables and font information, see the W3C Math Working Group home page and the Unicode site (see, for instance, Unicode Work in Progress).
A MathML token element Section 3.2 Token Elements, and Section 4.4.1 Token Elements takes as content a sequence of MathML
Characters. MathML Characters are defined to be either
Unicode characters legal in XML documents or mglyph
elements.
The latter are used to represent
characters that do not have a Unicode encoding, as described in
Section 3.2.9 Accessing glyphs for
characters from MathML
(mglyph).
Because the Unicode UCS provided
approximately one thousand special alphabetic characters for the use
of mathematics with Unicode 3.1, and over 900 further
special symbols in Unicode 3.2, the need for
mglyph
should be rare.
As always in XML, any character allowed by XML may be used in MathML in an XML document. The legal characters have the hexadecimal code numbers 09 (tab = U+0009), 0A (line feed = U+000A), 0D (carriage return = U+000D), 20-D7FF (U+0020..U+D7FF), E000-FFFD (U+E000..U+FFFD), and 10000-10FFFF (U+010000..U+10FFFF). The notation, just introduced in parentheses, beginning with U+ is that recommended by Unicode for referring to Unicode characters [see [Unicode], page xxviii]. The exclusions above code number D7FF are of the blocks used in surrogate pairs, and the two characters guaranteed not to be Unicode characters at all. U+FFFE is excluded to allow determination of byte order in certain encodings.
There are essentially three different ways of encoding character data.
Using characters directly: For example, an A may be entered as 'A' from a keyboard (character U+0041). This option is only available if the character encoding specified for the XML document includes the character. Most commonly used encodings will have 'A' in the ASCII position. In many encodings, characters may need more than one byte. Note that if the document is, for example, encoded in Latin-1 (ISO-8859-1) then only the characters in that encoding are available directly. Using UTF-8 or UTF-16, the only two encodings that all XML processors are required to accept, mathematical symbols can be encoded as character data.
Using numeric XML character references: Using this notation, 'A' may be represented as A (decimal) or A (hex). Note that the numbers always refer to the Unicode encoding (and not to the character encoding used in the XML file). By using character references it is always possible to access the entire Unicode range. For a general XML vocabulary, there is a disadvantage to this approach: character references may not be used in XML element or attribute names. However, this is not an issue for MathML, as all element names in MathML are restricted to ASCII characters.
Using entity references: The MathML DTD defines internal entities that expand to character data. Thus for example the entity reference é may be used rather than the character reference "é or, if, for example, the document is encoded in ISO-8859-1, the character é. An XML fragment that uses an entity reference which is not defined in a DTD is not well-formed; therefore it will be rejected by an XML parser. For this reason every fragment using entity references must use a DOCTYPE declaration which specifies the MathML DTD, or a DTD that at least declares any entity reference used in the MathML instance. The need to use a DOCTYPE complicates inclusion of MathML in some documents. However, entity references are very useful for small illustrative examples, and are used in most examples in this document.
For special purposes, one may need to use a character which is not in
Unicode.
In these cases
one may use the mglyph
element for direct access to a glyph from some font and creation of
a MathML substitute for the corresponding character.
All MathML token elements that accept character data also accept an
mglyph
in their content.
Beware, however, that the font chosen may not be available to all MathML processors.
A noticeable feature of mathematical and scientific writing is the use of single letters to denote variables and constants in a given context. The increasing complexity of science has led to the use of certain common alphabet and font variations to provide enough special symbols of this letter-like type. These denotations are in fact not letters that may be used to make up words with recognized meanings, but individual carriers of semantics themselves. Writing a string of such symbols is usually interpreted in terms of some composition law, for instance, multiplication. Many letter-like symbols may be quickly interpreted by specialists in a given area as of a certain mathematical type: for instance, bold symbols, whether based on Latin or Greek letters, as vectors in physics or engineering, or fraktur symbols as Lie algebras in part of pure mathematics. Again, in given areas of science, some constants are recognizable letter forms. When you look carefully at the range of letter-like mathematical symbols in common use today, as the STIX project supported by major scientific and technical publishers did, you come up with perhaps surprisingly many. A proposal to facilitate mathematical publishing by inclusion of mathematical alphabetic symbols in the UCS was made, and has been favorably handled.
The additional Mathematical Alphanumeric Symbols provided in Unicode 3.1 have code points in Plane 1, that is, in the first plane with Unicode values higher than 2^{16}. This plane of characters is also known as the Secondary Multilingual Plane (SMP), in contrast to the Basic Multilingual Plane (BMP) which was originally the entire extent of Unicode. Support for Plane 1 characters in currently deployed software is not always reliable, and in particular support for these Mathematical Alphanumeric Symbol characters is not likely to be widespread until after public fonts covering the characters adopted for mathematics are available.
As discussed in Section 3.2.2 Mathematics style attributes common to token
elements, MathML offers an
alternative mechanism to specify mathematical alphabetic
characters. This alternative spans the gap between the
specification of Unicode 3.1 and its associated deployment in software and
fonts.
Namely, one uses the mathvariant
attribute on the surrounding token element, which will most commonly
be mi
. In this section we detail the
correspondence that a MathML processor should apply between certain
characters in Plane 0 (BMP) of Unicode, modified by the
mathvariant
attribute, and the Plane 1
Mathematical Alphanumeric Symbol characters.
The basic idea of the correspondence is fairly simple. For example, a Mathematical Fraktur alphabet is in Plane 1, and the code point for Mathematical Fraktur A is U+1D504. Thus using these characters, a typical example might be
<mi>𝔄</mi>
However, an alternative, equivalent markup would be to use
the standard A and modify the identifier using the
mathvariant
attribute, as follows:
<mi mathvariant="fraktur">A</mi>
The exact correspondence between a mathematical alphabetic character and an unstyled character is complicated by the fact that certain characters that were already present in Unicode are not in the 'expected' sequence.
The detailed correspondence is shown in the tables given in Section 6.3.6 Mathematical Alphanumeric Symbols.
Mathematical Alphanumeric Symbol characters should not be used for styled text. For example, Mathematical Fraktur A must not be used to just select a blackletter font for an uppercase A. Doing this sort of thing would create problems for searching, restyling (e.g. for accessibility), and many other kinds of processing.
Some characters, although important for the quality of print or alternative rendering, do not have glyph marks that correspond directly to them. They are called here non-marking characters. Their roles are discussed in Chapter 3 Presentation Markup and Chapter 4 Content Markup.
In MathML 2 control of page composition, such as line-breaking, is
effected by the use of the proper attributes on the mspace
element.
The characters below are not simple spacers. They are especially important new additions to the UCS because they provide textual clues which can increase the quality of print rendering, permit correct audio rendering, and allow the unique recovery of mathematical semantics from text which is visually ambiguous.
Character name | Unicode | Description |
---|---|---|
⁢ |
02062 | marks multiplication when it is understood without a mark (Section 3.2.5 Operator, Fence, Separator or Accent (mo) |
⁣ |
02063 | used as a separator, e.g., in indices (Section 3.2.5 Operator, Fence, Separator or Accent (mo) |
⁡ |
02061 | character showing function application in presentation tagging (Section 3.2.5 Operator, Fence, Separator or Accent (mo) |
The Universal Character Set (UCS) of Unicode and ISO 10646 continues to evolve, see Section 6.4.4 Status of Character Encodings. At the time of writing the standard is Unicode 4.0. As before, we can only reiterate that for latest developments on details of character standards as far as they influence mathematical formalism the home page of the W3C Math Activity should be consulted.
The characters are given with entity names as well as Unicode numbers. To facilitate comprehension of a fairly large list of names, which totals over 2000 in this case, we offer more than one way to find to a given character. A corresponding full set of entity declarations is in the DTD in Appendix A Parsing MathML. For discussion of entity declarations see that appendix.
The characters are listed by name, and sample glyphs provided for all of them. Each character name is accompanied by a code for a character grouping chosen from a list given below, a short verbal description, and a Unicode hex code drawn from ISO 10646.
The character listings by alphabetical and Unicode order in Section 6.3.7 MathML Character Names are in harmony with the ISO character sets given, in that if some part of a set is included then the entire set is included.
To begin we list separately a few of the special characters which MathML has introduced. These now have Unicode values. Rather like the non-marking characters above, they provide very useful capabilities in the context of machinable mathematics.
Entity name | Unicode | Description |
---|---|---|
ⅅ |
02145 | D for use in differentials, e.g. within integrals |
ⅆ |
02146 | d for use in differentials, e.g. within integrals |
ⅇ |
02147 | e for use for the exponential base of the natural logarithms |
ⅈ |
02148 | i for use as a square root of -1 |
The first table offered is a very large ASCII listing of characters considered particularly relevant to mathematics. This is given in Unicode order. Most, but not all, of these characters have MathML names defined via entity declarations in the DTD. Those that do not are usually symbols which seem mathematically peripheral, such as dingbats, machine graphics or technical symbols.
A second table lists those characters that do have MathML entity names, ordered alphabetically, with a lower-case letter preceding its upper-case counterpart.
The tables in this section detail Unicode code points (displayed with 256 code points per table) that have mathematically significant characters. The sample glyph images link to the table of characters ordered by Unicode given in the previous section. The names of the blocks are those of the Unicode blocks included in the numerical range given; bracketing indicates glyphs for characters of that type are not shown in these tables.
Block Range | Description |
---|---|
00000 - 000FF | Controls and Basic Latin, and Latin-1 Supplement |
00100 - 001FF | Latin Extended-A, Latin Extended-B |
00200 - 002FF | IPA Extensions, Spacing Modifier Letters |
00300 - 003FF | Combining Diacritical Marks, Greek [and Coptic] |
00400 - 004FF | Cyrillic |
02000 - 020FF | General Punctuation, Superscripts and Subscripts, Currency Symbols, Combining Diacritical Marks for Symbols |
02100 - 021FF | Letter-like Symbols, Number Forms, Arrows |
02200 - 022FF | Mathematical Operators |
02300 - 023FF | Miscellaneous Technical |
02400 - 024FF | Control Pictures, Optical Character Recognition, Enclosed Alphanumerics |
02500 - 025FF | Box Drawing, Block Elements, Geometric Shapes |
02600 - 026FF | Miscellaneous Symbols |
02700 - 027FF | Dingbats |
02900 - 029FF | Supplemental Arrows, Miscellaneous Mathematical Symbols |
02A00 - 02AFF | Supplemental Mathematical Operators |
03000 - 030FF | CJK Symbols and Punctuation, [Hiragana, Katakana] |
0FB00 - 0FBFF | Alphabetic Presentation Forms |
0FE00 - 0FEFF | [Combining Half Marks, CJK Compatibility Forms, Small Form Variants, Arabic Presentation Forms-B] |
1D400 - 1D4FF | Mathematical Styled Latin (Bold, Italic, Bold Italic, Script, Bold Script begins) |
1D500 - 1D5FF | Mathematical Styled Latin (Bold Script ends, Fraktur, Double-struck, Bold Fraktur, Sans-serif, Sans-serif Bold begins) |
1D600 - 1D6FF | Mathematical Styled Latin (Sans-serif Bold ends, Sans-serif Italic, Sans-serif Bold Italic, Monospace, Bold), Mathematical Styled Greek (Bold, Italic begins) |
1D700 - 1D7FF | Mathematical Styled Greek (Italic continued, Bold Italic, Sans-serif Bold), Mathematical Styled Digits |
In addition to the Unicode Characters so far listed, one may use the combining characters U+0338 (/), U+20D2 (|) and U+20E5 (\) to produce negated or canceled forms of characters. A combining character should be placed immediately after its 'base' character, with no intervening markup or space, just as is the case for combining accents.
In principle, the negation characters may be applied to any Unicode character, although fonts designed for mathematics typically have some negated glyphs ready composed. A MathML renderer should be able to use these pre-composed glyphs in these cases. A compound character code either represents a UCS character that is already available, as in the case of U+003D+0038 which amounts to U+2260, or it does not as is the case for U+2202+0338. The common cases of negations, of the latter type, that have been identified are listed in the table
Note that it is the policy of the W3C and of Unicode that if a single character is already defined for what can be achieved with a combining character, that character must be used instead of the decomposed form. It is also intended that no new single characters representing what can be done by with existing compositions will be introduced. For further information on these matters see the Unicode Standard Annex 15, Unicode Normalization Forms [UAX15], especially the discussion of Normalization Form C.
Unicode attempts to avoid having several character codes for simple font variants. For a code point to be assigned there should be more than a nuance in glyphs to be recorded. To record variants worth noting there is a special character in Unicode 3.2, U+FE00 (VARIATION SELECTOR-1), which acts as a postfix modifier. However the legally allowed combinations with this variation selector are restricted to a list recorded as part of Unicode. The VARIATION SELECTOR-1 character may only be applied to the characters listed here. The resulting combination is not regarded by Unicode as a separate character, but a variation on the base character. Unicode aware systems may render the combination as the base if the available fonts do not support the variant glyph shape.
Here we list the special mathematical alphabets. Note that the names for these alphabetic runs should be regarded as conventions resulting from recent tradition in the typesetting of mathematical formulas, rather than as fixing exactly and forever the styles which are to be used. Of course, they do correspond to the styles presently most common. But, for instance, there may be font variations in the glyphs from double-struck, open-face or blackboard bold fonts, all of which would naturally be used for the characters in the range here labelled Double-struck. Similar considerations would apply to appellations such as fraktur and gothic, or script and calligraphic.
As discussed above, the use of these characters is formally equivalent
to the use of characters in Plane 0, together with a suitable value
for the mathvariant
attribute. The
correspondence is given in the character tables. Most of these
characters come from the additions to Plane 1, however a few
characters (such as the double-struck letters N, P, Z, Q, R, C, H
representing common number sets) were already present in Unicode 3.0
and retain their original positions. These characters are highlighted
in the tables.
This section corresponds closely with the entity definitions in the DTD described in Appendix A Parsing MathML. All of the entity sets except the last correspond to entity sets defined by ISO 8879 or ISO 9573-13.
ISO Handle | Description |
---|---|
ISOAMSA | Added Mathematical Symbols: Arrows |
ISOAMSB | Added Mathematical Symbols: Binary Operators |
ISOAMSC | Added Mathematical Symbols: Delimiters |
ISOAMSN | Added Mathematical Symbols: Negated Relations |
ISOAMSO | Added Mathematical Symbols: Ordinary |
ISOAMSR | Added Mathematical Symbols: Relations |
ISOBOX | Box and Line Drawing |
ISOCYR1 | Cyrillic-1 |
ISOCYR2 | Cyrillic-2 |
ISODIA | Diacritical Marks |
ISOGRK3 | Greek-3 |
ISOLAT1 | Latin-1 |
ISOLAT2 | Latin-2 |
ISOMFRK | Mathematical Fraktur |
ISOMOPF | Mathematical Openface (Double-struck) |
ISOMSCR | Mathematical Script |
ISONUM | Numeric and Special Graphic |
ISOPUB | Publishing |
ISOTECH | General Technical |
MMLEXTRA | Extra Names added by MathML |
We have excluded a very few other characters that may have appeared in
the corresponding lists in MathML 1. Those characters thus
lost will be found to be used very infrequently in the
experience of mathematical publishers, or simply to be completely
unacceptable for inclusion in Unicode. However MathML 2 does provide
the mglyph
element to accommodate new
characters that authors may wish to introduce.
It used to be in MathML 1.0 that there were a number more
non-marking character entities listed. These were concerned with
composition control, such as line-breaking. In MathML 2 such control
is effected by the use of the proper attributes on the mspace
element.
The character listings by alphabetical and Unicode order in Section 6.3.7 MathML Character Names have now been brought more into line with the corresponding ISO character sets than was the case in MathML 1.0, in that if some part of a set is included then the entire set is included. In addition, the group ISOCHEM has been dropped as more properly the concern of chemists. All the ISO mathematical alphabets are listed, since there are now Unicode characters to point to, in particular the bold Greek of ISOGRK3. These changes have also been reflected in the entity declarations in the DTD in Appendix A Parsing MathML.
A significant change after MathML 1.0 occurred in the movement toward adoption of more characters for mathematics in the UCS and availability of public fonts for mathematics. The encoding of characters in the UCS is done jointly by the Unicode Technical Committee and by ISO/IEC JTC1/SC2/WG2. The process of encoding takes quite some time from the deliberation of first proposals to the final approval. The characters mentioned in this chapter and listed in the associated tables have been though the various stages of this approval process.
At the time of the preparation of the MathML 2.0 Specification [MathML2] the characters relevant to mathematics fell into three categories: fully accepted characters, characters in final (JTC1) ISO/IEC ballot, and characters before the final ISO/IEC ballot.
Fully accepted characters included a large number of Latin, Greek, and Cyrillic letters, a large number of Mathematical Operators and symbols, including arrows, and so on. Fully accepted characters were exactly those that are in both Unicode 3.0 [Unicode] and ISO/IEC 10646-1:2000 [ISOIEC10646-1], which are identical code point by code point. Those of obvious special interest to mathematics numbered over 1,500, depending on how you count.
In April 2001, the Mathematical Alphanumeric Symbols came up for a final ballot together with a large number of ideographs and other characters not directly relevant for mathematics. There were just about 1,000 of these. The additions were published as ISO/IEC 10646-2, and became part of Unicode 3.1.
Characters relevant to MathML that were before final ballot made up a long list of operators and symbols, including some special constants and non-marking characters (see Section 6.2.4 Non-Marking Characters and Section 6.3.1 Special Constants). They numbered about 590 in all. With some small technical improvements and compromises the proposed additions accepted were published as an amendment to [ISO/IEC 10646-1], and became part of Unicode 3.2.
Even with the good will shown to the mathematical community by the
Unicode process a small number of characters of special interest
to some may not yet have been included. The obvious solution of
avoiding their use may not satisfy all. For these characters the
Unicode mechanism involving Private Use Area codes could be deployed,
in spite of all the dangers of confusion and collisions of conventions
this brings with it. However, this is the situation for which
mglyph
was introduced. The use of mglyph
is recommended to refer to symbols not included in Unicode.