6. Entities, Characters and Fonts

• 6.1 Introduction
  • 6.1.1 The Intent of Entity Names
  • 6.1.2 The STIX Project
• 6.2 Entity Listings
  • 6.2.1 Non-Marking Entities
  • 6.2.2 Printing Entity List
  • 6.2.3 Special Constants
  • 6.2.4 Alphabetical Lists
  • 6.2.5 ISO Entity Set Groupings
    • 6.2.5.1 ISO Symbol Entity Sets
    • 6.2.5.2 ISO Math Font Entity Sets
    • 6.2.5.3 Other ISO Font Entity Sets
  • 6.2.6 Additional Entity Set Grouping

6.1 Introduction

6.1.1 The Intent of Entity Names

Notation has proved very important for mathematics. Mathematics has grown in part because of the succinctness and suggestiveness of its evolving notation. There have been many new signs evolved for use in mathematical notation, and mathematicians have not held back from making use of many symbols originally developed 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 in coding. It is difficult to read mathematics if glyphs are not available for presentation on specific display devices.

This situation poses a problem for the W3C Math Working Group. It does not fall naturally within the purview of a math for HTML specification and DTD production to worry about more than the entities allowed in the DTD. Moreover, as experience has shown, a long list of entities with no means to display them is of little use, and a cause of frequent frustrations in trying use a standard. On the other hand, a large collection of glyphs or characters without a standard way to refer to them is not of much use either.

The W3C Math Working Group has therefore taken on directly specification of part of the full mechanism of proceeding from notation to final presentation, and is collaborating with organizations undertaking specification of the rest.

For instance, we try to use entity names that are contained in ISO TR 9573, which supersedes the ISO TR 8879 annex as far as math is concerned. There are considerations of mathematical usage that do on occasion militate against this, and the TR 9573 lists need supplementing. We hope to be able to agree with the TR 9573 WG on suitable extensions, in the course of the revision of their document that they are presently undertaking.

The STIX project of the STIPUB group of scientific and technical publishers has also been working toward a common collection of mathematical symbols and names. The W3C Math Working Group expects to issue further updates on the matter of character entities as a consequence of this project's useful work. For the latest character tables and fonts information, see the W3C Math Working Group home page.

6.1.2 The STIX Project

The STIX project team leader, Nico Poppelier, is a member of the W3C Math Working Group. The STIX project, set up by the STIPUB group of publishers, aims to formulate a collection of characters needed in the course of scientific and technical publishing. A database of characters in common use is being produced by collaborating publishing organizations. The team will propose to the Unicode consortium the additions to the next revision of the Unicode character set that this process shows are needed, together with the appropriate character codes. Finally the STIX project will commission the production of a complete set of fonts covering those Unicode characters for science and technology, to be made available to the public under license, but free of charge. The STIPUB group recognizes that easy availability of the characters and fonts greatly facilitates communication and publication.

6.2 Entity Listings

This chapter of the MathML proposal contains a listing of entities for use in MathML.

To provide more background on the characters used by mathematics we have used a larger comparative database showing codes and meanings in other common math environments. The W3C Math Working Group is very grateful to Elsevier Science and to Wolfram Research (makers of Mathematica ®) for making available to us so much useful data.

6.2.1 Non-Marking Entities

Some character entities, although important for the quality of print rendering do not directly have glyph marks that correspond. They are called here non-marking entities. Below we have a table of those adopted for the purposes of MathML. Their roles are discussed in Chapters 3 and 4, respectively on Presentation and Content Markup. The values of the spaces given are recommendations. Some of these characters do not already have Unicode values. Arbitrary values up in the Private Zone E8 range have been assigned. The correspondence between the spacing values mentioned below and those in the Unicode descriptions are not exact, but are good matches.

6.2.2 Printing Entity Listings

Since the situation concerning availability of character codes from Unicode and under ISO 9573-13 is not yet fully clear at the time of writing, we have decided to proceed conservatively.

We have taken the ISO 9573-13 proposal, as conveyed to us from Anders Berglund, and have added a number of additional aliases based in the practice of the mathematical typesetting community. Thus the main influence outside ISO has been the names to be found in the TeX community.

To facilitate comprehension of a fairly large list of names, which totals over 2000 in this case, we offer the same information in more than one form.

We have entities listed by name and sample glyphs for all of them. Each entity 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 if there is a corresponding sample glyph to be found in ISO 10646. Those codes beginning with the hex digit E, e.g., E321, indicate assignments to the private zone of Unicode. This indicates that the character in question is not at present an official Unicode character. It is highly recommended that authors use entity names instead of Unicode values, especially for those characters in the Unicode private zone, as those values may change. It is hoped that most of these characters will become officially endorsed by Unicode and ISO under its 10646 standard in due course. In any case we expect fonts for these characters to become publicly available as the use of MathML develops. If the entity name is an alias then a reference back to the ISO form is given if there is one, and to a preferred form if not. The ISO or preferred forms have references to their alternates where they exist.

Newly Revised. The entity listings by alphabetical and Unicode order in section 6.2.4 have now been brought more into line with the corresponding ISO character sets, in that if some part of a set is included then the entire set is included. Also, ISOCHEM has been dropped. These changes have also been reflected in the entity declarations in the DTD in Appendix A.

The tables of character sets with glyphs given in section 6.2.5 have not been revised from the original tables. In cases where information section 6.2.4 and 6.2.5 conflict, the tables in 6.2.3 and the DTD should be considered normative.

6.2.3 Special Constants

To commence we list separately a few of the special characters which MathML has seen fit to be a little radical in introducing. There are two for special constants and one for calculus. They too must have private Unicode values.

6.2.4 Alphabetical Lists

The first table offered is a very large ASCII listing of printing entity names, ordered alphabetically, with upper-case preceding lower-case as in ASCII order. The Unicode numbers beginning with E are arbitrary assignments in the Private Area where there is presently no Unicode character available. When there is no Unicode offered at all it is because the characters listed can be thought of as font variations of common Roman alphabetic characters.

There is also an ASCII listing of printing entities ordered by Unicode number. Next we have collections of the entities in entity sets which are similar to the groupings in the corresponding ISO documents.

6.2.5 ISO Entity Set Groupings

In addition, we list the above material in the groupings used by ISO 9573-13 with an additional grouping of aliases introduced. This table makes explicit the entity groupings and provides links to ASCII listings of the groups and HTML tabular listings which display the glyphs, insofar as they are to be had, as well.

6.2.5.1 ISO Symbol Entity Sets

The symbols for mathematics that ISO have considered are organized, for both historical and mnemonic reasons into groupings with somewhat descriptive names. In the tables below we reproduce the newly proposed versions of these groups and give the corresponding Unicode sample glyphs. For each ISO 9573-13 group we give first an Extended version in ASCII listing which includes aliases, then a similar listing with sample glyphs, then the Basic ISO 9573-13 entity set and its version with included glyphs. The entries are organized alphabetically by entity name.

It should be noted that the sample glyphs given here are in GIF files intended for viewing on a monitor's screen at 72dpi. They are not suitable for printing, and in particular do not constitute a set of fonts covering the symbols of mathematics. In addition, it is important to note that the Unicode numbers assigned in the private zone, beginning with hex digits E2 and above, are arbitrary and only used here to ensure that sample glyphs are available for display. They do not constitute suggested assignments of codes. Such a set of fonts is under development in more than one context. The MathML Working Group is engaged in ensuring that fonts will be readily publicly available.

This first block of entity sets includes mostly non-letter symbols, along with a few letters loaded with mathematical semantics. At the end of the block we have included the table MMALIAS of the aliases introduced by MathML, which mostly come from the TeX community, and MMEXTRA with the additional character entities added by MathML. Note that some of the blocks are place-holders for a possible future expansion of the tables.

6.2.5.2 ISO Math Font Entity Sets

Mathematical literature displays the common use of particular font styles. Characters representing given letters which differ only in the glyph presentation are in principle not different for the purposes of a character registry such as Unicode, which is not supposed to take into account mere font differences. However usage has meant that both ISO and Unicode, like mathematics, recognize them as different entities. Therefore we include lists for Greek, script, open face (also known as double struck or blackboard bold), and fraktur (also known as gothic or German) fonts.

6.2.5.3 Other ISO Font Entity Sets

For reference we provide a list of the names of several other ISO font entity sets which are really normally used for text. ISOGRK4 is actually a collection of emboldened forms of the Greek letters.

6.2.6 Additional Entity Set Grouping

In addition to the above listed, for the sake of completeness, we provide a table of other entities not within the ISO lists which are referred to somewhere in this specification. It is not certain that all these characters, though of mathematical significance, will reach incorporation within Unicode. The W3C Math WG continues to wrestle with the problems of the characters of mathematics.

&LeftSkeleton;	E850	start of missing information
&RightSkeleton;	E851	end of missing information
&LeftBracketingBar;	F603	left vertical delimiter
&RightBracketingBar;	E604	right vertical delimiter
&LeftDoubleBracketingBar;	F605	left double vertical delimiter
&RightDoubleBracketingBar;	F606	right double vertical delimiter
─	E859	short horizontal line
|	E85A	short vertical line
≔	E85B	assignment operator
❘	E85C	vertical separating operator
⫤	E30F	alias for ⫤
⥰	F524	right double arrow with rounded head (looks like thin superset)
⊏̸	E604	negated set-like partial order operator
⊐̸	E615	negated set-like partial order operator
⊈	2288	alias of ⊈
⊉	2289	alias of ⊉
⥐	F50B	left-down-right-down harpoon
⥞	F50E	left-down harpoon from bar
⥖	F50C	left-down harpoon to bar
⥟	F50F	right-down harpoon from bar
⥗	F50D	right-down harpoon to bar
⇤	21E4	alias for ⇤
⥎	F505	left-up-right-up harpoon
↤	21A4	alias for ↤
⥚	F509	left-up harpoon from bar
⥒	F507	left-up harpoon to bar
⇥	21E5	alias for ⇥
⥛	F50A	right-up harpoon from bar
⥓	F508	up-right harpoon to bar
⩵	F431	two consecutive equal signs
⪢	E2F7	alias for ≫
⧏	F410	not left triangle, vertical bar
⪡	E2FB	alias for ≪
≭	226D	alias for &nasymp;
≂̸	E84E	alias for ≂̸
≎̸	E616	alias for ≎̸
≏̸	E84D	alias for ≏̸
⧏̸	F412	not left triangle, vertical bar
⪢̸	F428	not double greater-than sign
⪡̸	F423	not double less-than sign
&NotPrecedesTilde;	E5DC	alias for ⪯̸
⧐̸	E870	not vertical bar, right triangle
≿̸	E837	not succeeds or similar
⧐	F411	vertical bar, right triangle
∏	220F	alias for ∏
⋄	22C4	alias for ⋄
⨯	E619	cross or vector product
□	25A1	alias for □
⤓	F504	down arrow to bar
↧	21A7	alias for ↧
⥡	F519	down-left harpoon from bar
⥙	F517	down-left harpoon to bar
⥑	F515	up-left-down-left harpoon
⥠	F518	up-left harpoon from bar
⥘	F516	up-left harpoon to bar
⥝	F514	down-right harpoon from bar
⥕	F512	down-right harpoon to bar
⥏	F510	up-right-down-right harpoon
⥜	F513	up-right harpoon from bar
⥔	F511	up-right harpoon to bar
↓	E87F	short down arrow
↑	E880	sort up arrow
⤒	F503	up arrow to bar
↥	21A5	↥
̑	0311	breve, inverted (non-spacing)
‾	00AF	over bar
⏞	F612	over brace
⎴	F614	over bracket
⏜	F610	over parenthesis
_	0332	combining low line
⏟	F613	under brace
⎵	F615	under bracket
⏝	F611	under parenthesis
▫	F530	empty very small square
▪	F529	filled very small square
◻	F527	empty small square
◼	F528	filled small square
⧴	F51F	rule-delayed (colon right arrow)