% language=us runpath=texruns:manuals/math \def\unichar#1#2{#1 (U+#2: \char"#2)} \def\APL{\ss apl} % \useMPlibrary[dum] \startcomponent musings-unicode \environment musings-style % \usemodule[mathfun] \startchapter[title=Unicode] \startsection[title=Introduction] When working on a \TEX\ macro package for decades one can hardly avoid dealing with math; after all \TEX\ is pretty much about math. When this wonderful typesetting infrastructure was written it was all about quality and how to make your documents look nice. And for sure, Don Knuths documents looks nice, also because he pays a lot of attention to the \quotation {fine points of math typesetting}. The constraints of those time (like hardware, compilers, fonts, and for sure also time) made \TEX\ into what it is: eight bit character sets, eight bit fonts, eight bit hyphenation patterns, efficient memory usage and therefore carrying around as little as possible. It all makes sense. But one needs to pay attention. \footnote {And that is what Mikael Sundqvist and I have been doing a lot since we started upgrading math in \CONTEXT\ in combination with enhancing the math engine in \LUAMETATEX. The story here is a byproduct of our explorations and very much a combined effort.} Math typesetting is actually a sort of separated process in the engine: unprocessed lists go in and after some juggling a list of assembled boxes, glyphs, glues and penalties come out. I will not go into detail about that and only mention that in \LUAMETATEX\ we extended all this to be a bit more flexible and controllable, something that has been driven by the fact that we need to support \UNICODE\ fonts. This is all part of a related effort to move from eight bit \quote {everything} to \UNICODE\ \quote {everywhere}. Now, one can say a lot about \UNICODE\ but the main advantage is that it tries to cover \quote {all} characters ever encountered, including scripts (used in languages) that are long gone, as well as these little pictures that people like to see on the web: emojis. One can safely say that \UNICODE\ simplifies mixing languages and scripts, and thereby makes \TEX\ macro packages less complex. On the other hand, \UNICODE\ (or more precisely, related wide) fonts makes all kind of features possible and thereby add a complication. So, how about math? When Don Knuth gave us \TEX\ he also gave us fonts and there are plenty symbols in these fonts. But, as mathematicians seem to love variations on symbols soon more fonts arrived, most noticeably those from the \AMS\ that also added some more alphabets: mathematicians also love to render the shapes of letters differently. In order to access these glyphs names were invented that also sometimes suggested that there was some order in the matter. And, for some reason these names got aliases and soon we had a huge list of often obscure and inconsistent macro names. It didn't take long for a little mess and confusion to creep in. It has been said that the verbose \TEX\ math \ASCII\ input format is also a way for mathematicians to communicate, just because many use the same tool to render the formulas. Of course that gets obscured when one starts to add additional macros. It gets even more tricky once we start talking \quote {standard} as in \quotation {\LATEX\ is the standard}. That has for instance resulted in browsers interpreting \TEX\ like input without using \TEX\ (so how about expansion?). It has also sort of put \TEX\ into the range of possible word processing systems, which in turn leads to these \MSWORD\ versus Google docs versus \LATEX\ debates that can get rather nasty and unrealistic when it comes to discussing usage and quality. Interestingly, \MSWORD\ now has reasonable math, to some extent modelled after \TEX. It has some verbose \TEX\ like (but constrained) input and would do well for probably mostly people who occasionally have to inject some math. There were also attempts by the people at \MICROSOFT\ to normalize the input but we leave that aside now. However, because we now do have all these symbols and because source code editors make them accessible and show them there is a good chance that users will inject them, if only by cut and paste, so we do have to deal with that. This automatically puts us in the position that we need to deal with different meanings for the same symbol, which in turn might demand different spacing, penalties and such. In the end it is users that drive all this, not publishers; they don't really care and out|-|source typesetting anyway. We're not aware of any research and development being done and I suppose we would have noticed because after all we're involved in developing \LUATEX. It is one of the engines that does \OPENTYPE\ and \UNICODE\ math and no publisher or supplier ever took serious interest in it. From our perspective what users do is visible, everything else is hidden behind corporate curtains. And this is why nowadays we only need to care about users (mainly authors). Back to typesetting. For a long time all went well: one could typeset documents that looked good. Okay, not all looked good because not everyone paid attention to details, and the more the web evolved the more patching cut'n'paste of bad examples made its way into documents, but let's not start talking quality here. But then came \UNICODE\ and a while later people started talking about accessibility, cutting and pasting and more. In the meantime there had been developments like \MATHML\ and \OPENMATH\ that tried to structure and organize formulas in a more symbolic way. \footnote {It probably went unnoticed that \CONTEXT\ always supported rendering \MATHML, and as such had to deal with all the weird aspects (read: way it was used). Although one is not supposed to directly edit \MATHML\ we work with authors who are quite happy to do that simply because they code the documents in \XML\ because there is a need for high quality \PDF\ as well as \HTML\ output and a \CONTEXT\ based workflow can handle the \XML\ well. We're talking of large volumes here (mostly for basically free school math).} In the meantime the \TEX\ community had lost the edge on fonts, and \OPENTYPE\ math was invented by \MICROSOFT\ and implemented in \MSWORD\ before a substantial number of \TEX\ users understood what was happening. They had it coming. To a large extend one can say the same about math in \UNICODE. Where a Greek capital \quote {A} is seen as different from a Latin capital \quote {A}, even when they often have the same shape, a math italic variable \quote {h} was made synonym to \quote {Planck constant}, as if the letters used in math had no meaning at all. We'll see that a wide hat is an extensible character of zero width combining hat accent, which makes for curious handling of the initial character. There is more granularity in some symbols, especially popular symbols like slashes and bars, than in letters. It is as if the math community didn't care much about how the letters (variables) were communicated and perceived but were picky about the slope of slashes. It seems more of a visual world, which might actually be the reason structured input never really took of. Maybe \TEX ies just love the mix of characters, commands, spacing directives. Maybe they just love to reposition and space these glyphs to suit all kind of curious non|-|standard math rendering. All this makes it pretty hard to communicate meaning, and it is just one of the examples where the \TEX\ community, for as far involved, failed to make a strong case. Our personal opinion is that no one really cared because in the \TEX\ community it is all about rendering. The fact that we use math to communicate only gained attention when accessibility became hot and by then it was too late. Efforts like \OPENMATH\ started ambitious and in the end basically failed. Coding in \XML\ using \MATHML\ isn't much better and one always had to adapt to the latest fashion. Also, once plenty code shows up bugs become features. Browser support came and went and came back. Simplified input using for instance \ASCIIMATH\ started indeed simple but quickly became a (somewhat inconsistent) mess. What we see here is the same as everything web (and computer languages): we can do better, we start some project, then move on, and we end up with half|-|way abandoned results. The development cycles are short, results have to be achieved fast, there is no time (or interest) for iterating and refactoring. The word \quote {standard} and mantra \quote {everyone should use this} are quite popular. So where does that leave us with \TEX ? Well, with a mess. Decades of various efforts have not brought us a coherent system of organizing symbols and properties, made us end up with inconsistencies, made users revert to hacks, didn't make math easily transferable and complicates rendering. Personally we find it sort of strange that we spend time on for instance tagging and accessibility before we get these math alphabets and shared math specific symbols sorted out. If we cannot make good arguments for that (math being a script on its own with semantics and such) we waste energy and are pulling a dead horse. What puzzles us most is that one would expect mathematicians to be able to come up with strong arguments for a structured approach. But maybe it was simply the fact that \TEX\ math typesetting was pretty much driven by large commercial publishers and those providing services for them: the first category doesn't invest in these matters and even less today, and the second category makes money from sorting out the mess, so why get rid of it. Who knows. For us, it means that any complain about these matters deserves the same answer: the \TEX\ community created this mess, so it has to live with it. And the bad thing is: bugs and work|-|arounds eventually become features and then one is supposed to conform, even if deep down one knows better. It doesn't help that the community is proud of what it can render and has built itself a reputation that all is good. So why this criticism? Why not just abandon \TEX ? The answer is simple: \TEX\ is quite okay and cannot be blamed for where we are now. We need to think of solutions and in that respect the \CONTEXT\ users are lucky! They have always been told not to use this macro package for math because there are other standards and because publishers want \LATEX\ (even if they just let the manuscripts be recoded). That means that we don't really need to care much about the past. Those who use \CONTEXT\ can benefit from the compatibility we have anyway but also move forward to more structured and consistent math. It is in this perspective that we will discuss some more details next so that eventually we can draw some conclusions. The end goal is to have an additional layer of grouping math symbols that permits consistent high quality rendering in a mixed input environment. \stopsection \startsection[title=Molecules] Before we go into details about some characters, we spend some word on the rendering. The building blocks of a formula are atoms and internally the term nucleus is used for what we have without scripts. The simple sequence \type {1 + x} will result in a linked list of three atoms with three nuclei. In \type {x^2} the \type {x} is the nucleus. Atoms can have scripts: prescripts, postscripts and a prime. The majority of \UNICODE\ math characters become such atoms (nuclei and scripts) and they get a class property that determines their spacing, but that is not part of the \UNICODE\ specification. From the upcoming sections it will be clear that when we classify we don't get that much help from \MATHML\ or even the \TEX\ community either. In addition to these atoms the \LUAMETATEX\ engine (which builds upon \TEX) has what we can call molecules. There are several types: fractions, accents, fences, radicals. This distinction is to some extent present in \UNICODE: plenty of fraction related slashes, all kind of accents, vertical delimiters that can be made from snippets and act as fences, and a radical symbol. In \MATHML\ we see similar constructs but there in practice quite often operators need to be interpreted in a way that can distinguish between atoms and molecules. That is partly a side effect of applications that generate \MATHML. And as usual with standards pushed upon the world without years of exploration the confusion became part of the norm and will stay. In the \TEX\ engine over and under delimiters are implemented on top of radicals (using the same noad, the wrapper node for yet unprocessed math) but they have different code paths. Basically we have vertically fenced material and just like fractions have left and right fences as part of the concept (for binominals) the radical has a sort of left fence too. You can also wonder why we need accent noads while we support other delimiters with radicals. This organization mostly relates to subtypes and classes (and likely some limitations of the past) that have related spacing properties, but we can think if a generic structure noad and meaningful subtypes. However, that is not what we get so let's be more precise: {\bf Fractions:} these stack two atoms (or molecules) and separate them by a visible or phantom rule, or in \LUAMETATEX\ by a delimiter. They can have a left and right fence which originates in them also suitable for binominals. You may wonder why we don't use regular fences here. One reason we can think of is that when you fence something, you have an open and close class at the edges while with a fenced fraction the whole still is fraction. In \LUAMETATEX\ we can tweak classes at the edges but in regular \TEX\ there are fewer classes, so there constructs become ordinary or inner. {\bf Accents:} these put something on top of or below an atom (or molecule) and are driven by characters. The accent related commands take an integer (traditional) or three integers (extended) and it is this expected input that drives it. However, they are treated like delimiters. In traditional \TEX\ a delimiter is defined by two characters: the direct unscaled one, and when not found a second one drives the lookup from wider variants and eventually an extensible character. Accents just have the second one, which probably relates to the fact that the text ones that would be the starting point make no sense. It is this \quote {looking} for a single code point that makes that accents are not merged with the more general radical command space. Another reason is that accents deal a bit different with spacing and italic correction so even if we could merge, it would be more confusing in the end. {\bf Fences:} these come in pairs with optional middle ones. The reason for pairing is that they need to get the same size. That means that before we construct them the atom or molecule that they fence has to be analyzed. It also makes the result a construct of its own, although in \LUAMETATEX\ we can unpack that result so that it can be broken across lines. In practice that was never an issue because in a running text unscaled fences are used (just atoms with open and close classes assigned) but as soon as one goes to multi|-|line displays formulas things become more hairy. The related commands expect delimiters (the two part character definitions) but in the meantime are also happy with a single one because in the end \OPENTYPE\ math has all in one font. {\bf Radicals:} originally this only concerned roots but because they are basically wrappers we also use them for content that gets a delimiter above, below or both. In that sense the term radical can also be interpreted as \quote {extreme}, more than a carrot looking symbol. The related commands take one or more delimiters (or character) because we support left as well as right delimiters connected by a rule, so in the end radicals evolved into a construct with delimiters of all kind. So, the unique property of radicals is that the fences assume a cooperation between one or more glyphs and a rule. In \CONTEXT\ we support actuarian hooks as radicals that are used for annuity expressions, otherwise the \UNICODE\ symbols is useless and the \MATHML\ construct complex. So, where accents take numbers as delimiter specification, fences, fractions and radicals take specific math quantities or just letters. This makes that we will not merge these into one scanner and handler even if they all use the same (large) noad to store and carry around their properties. Also, it has some charm to keep the original \TEX\ distinctions. After all, it's not like \UNICODE, \MATHML\ or \OPENTYPE\ math fonts have brought some new insights: in the end they all draw from \TEX\ and they way it's done there. \stopsection \startsection[title=Symbols] There are plenty of symbols in \UNICODE. When we try to get an idea how we ended up with that set we're surprised that not much seems to be known about it. There are references to \ISO\ standards, usage by specific organizations (like those dealing with patents), there are references to lists of publishers. In personal communications with people involved it becomes clear that the criterion that some symbols really has to be used somewhere doesn't apply to these math symbols. There are bizarre specimens that we cannot locate anywhere. They are often assigned the \quote {relation} property which for \TEX\ is a safe bet because binary and relations get similar spacing, but binary makes an exception when it sits at the front. The fact that relation spacing is used can even obscure the fact that some characters have zero width properties; the results just look somewhat bad and one can always blame the font or renderer and adding some thin spacing is accepted behavior. So one can make the argument that because \TEX\ was the main renderer of math, a safe bet was better than a confusing and unproven|-|by|-|usage assignment to some category. In \TEX\ some symbols have multiple names, even when they have the same class. This indicates the wish for meaning at one end but shape at the other, and once a name has been assigned it sticks. It would be interesting to know how mathematicians see formulas: if one puts \type {\bar}s around a variable does one see \quotation {bar x bar} or \quotation {the modulus of x}, and how is translation to audio to be performed? One important aspect of using any symbol in \TEX, or basically any typesetting system that deals with math, is that the spacing depends on the meaning. Now, in the perspective of \UNICODE\ meaning is somewhat diffuse. A Latin capital \quote {A} related to \quote {a} is not the same as a Greek capital \quote {A} that relates to \quote {\alpha}. So, from the shape one cannot beforehand deduce what is meant, but when copying it the \UNICODE\ will expose the meaning. This is not the case in math: although many symbols have one meaning only, there are also plenty that can mean different things and the (\TEX) math community has not been able to make a strong case for providing different slots. Maybe the reason was that there already was a tradition of using commands that then relate a shape to a class that then results in appropriate spacing. Maybe it is also assumed that an article or book starts by explaining what a specific symbol means in that particular context. But that doesn't help much for copying. It also doesn't help with direct \UNICODE\ input. The way out for this last problem is that in \CONTEXT\ we will add additional properties to characters that then can communicate the class and thereby control the spacing. Although we initially did that at the \LUA\ end we now use the lightweight dictionary feature of the engine: a property, group, slot model. The main reason is that we foresee that at some point we might have to add property based rendering to the engine, and this opens up that possibility. Ever since we started with \LUATEX\ and \MKIV\ we have used the character database (in \LUA\ format) to store most properties so that we have all in one place. For figuring out the properties we can look at how traditionally symbols got multiple commands associated, how \MATHML\ looks at it, what \UNICODE\ reveals and what we find in fonts. It is a bit of jungle out there so for sure we have to make decisions ourselves. We next turn to that exploration. \stopsection \startsection[title=Slashes] The definition on the \WIKIPEDIA\ page [1] of slashes is as follows: \startquotation The slash is an oblique slanting line punctuation mark /. Once used to mark periods and commas, the slash is now used to represent exclusive or inclusive or, division and fractions, and as a date separator. It is called a solidus in \UNICODE, is also known as an oblique stroke, and has several other historical or technical names including oblique and virgule. \stopquotation The page then has a very detailed description on how slashes are used in text, mathematics, computing, currency, dates, numbering, linguistic transcriptions, line breaks, abbreviations, proofreading, fiction, libraries, addresses, poetry, music, sports, and text messages. It is a pretty good and detailed page which also gives a nice summary of usage in math. In mathematics, we use the slash (a forward leaning bar) for fractions, division, and quotient of set. Examples of fractions are $\vfrac {1} {2}$ but also $\percent$ sits in this category. \starttabulate[|T|l|l|] \NC U+0002F \NC \switchtobodyfont[stixtwo]$\utfchar{"0002F}$ \NC this is the official solidus \NC \NR % / \NC U+02044 \NC \switchtobodyfont[stixtwo]$\utfchar{"02044}$ \NC the mathematical fraction slash \NC \NR % ⁄ \NC U+02215 \NC \switchtobodyfont[stixtwo]$\utfchar{"02215}$ \NC the mathematical division slash \NC \NR % ∕ \NC U+02571 \NC \switchtobodyfont[stixtwo]$\utfchar{"02571}$ \NC a diagonal box drawing line \NC \NR % ╱ \NC U+029F8 \NC \switchtobodyfont[stixtwo]$\utfchar{"029F8}$ \NC the mathematical big solidus \NC \NR % ⧸ \NC U+0FF0F \NC \switchtobodyfont[stixtwo]$\utfchar{"0FF0F}$ \NC a full width solidus \NC \NR % ／ \NC U+1F67C \NC \switchtobodyfont[stixtwo]$\utfchar{"1F67C}$ \NC the very heavy solidus \NC \NR % 🙼 \stoptabulate The \STIX\ fonts have the first five, the rest is not there, so we can safely assume that they are not used in math. That brings us to the question that, say that the other ones are used, how does the user access them? In the editor they often look pretty much the same. For \TEX ies the answer is easy: you use a command. But as we already mentioned, there we enter a real fuzzy area: these commands either describe a shape or they communicate a meaning, at least, in an ideal world. Sometimes wrapping in a macro helps, like \typ {$\vfrac {1} {2}$}. In the document that explains \UNICODE\ math there is a section \quotation {Fraction Slash and Other Diagonals}. Even if we limit ourselves to the forward leaning slashes it looks like we need to include exotic symbols, as the empty set symbol with an left arrow on top: \type {U+29B4} a circle with left pointing arrow on top, that doesn't show up in most math fonts but \STIX\ has it {\switchtobodyfont[stixtwo]{$⦴$}}. We quote: \startquotation \type {U+2044 ⁄} \typ {FRACTION SLASH} is typically used to build up simple skewed fractions in running text. It applies to immediately adjacent sequences of decimal digits, that is, to spans of characters with the General Category property value \type {Nd}. For example, \type {1⁄2} should be displayed as \type {½}. In ordinary plain text, any character other than a digit delimits the numerator or denominator. So \type {5 1⁄2} should be displayed as \type {5½} since a space follows the \type {5}. In general mathematical use, a more versatile method for layout of fractions is needed (see, for example, Section 2.1 of [UnicodeMath]), however parsers of mathematical texts should be prepared to handle \typ {FRACTION SLASH} when it is received from other sources. \type {U+27CB} \typ {MATHEMATICAL RISING DIAGONAL} and \type {U+27CD} \typ {MATHEMATICAL FALLING DIAGONAL} are mathematical symbols for specific uses, to be distinguished from the more widely used solidi and reverse solidi operators as well as from nonmathematical diagonals. \stopquotation In \TEX\ there is no parsing going on: we just get sequences of atoms and the inter atom spacing applies. Curly braced arguments are used to communicate units that needs to be treated a while. As side note: where for some scripts there are special characters that tell where something (state) starts and ends this is not available for math, which makes it impossible to mark a sequence of characters as being something math. The whole repertoire of pre|-|composed fractions and super- and subscripted \UNICODE\ symbols are not to be used in math. Most documents that somehow relate to or (partially) originate in \TEX\ can be rather fuzzy, so we can read here: \startquotation \type {U+27CB} corresponds to the \LATEX\ entity \type {\diagup} and \type {U+27CD} to \type {\diagdown}. Their glyphs are invariably drawn with 45° and 135° slopes, respectively, instead of the more upright slants typical for the solidi operators. The diagonals are also to be distinguished from the two box drawing characters \type {U+2571} and \type {U+2572}. While in some fonts those characters may be drawn with 45° and 135° slopes, respectively, they are not intended to be used as mathematical symbols. One usage recorded for \type {U+27CB} and \type {U+27CD} is in the notation for spaces of double cosets. \stopquotation So, it is the angles that math users should translate into meaning which I guess is natural for them. From the above we cannot deduce if we should take them into account in a macro package. The \MATHML\ specification [3] keeps it abstract and talks about division without mentioning the rendering. In content \MATHML\ we have: \starttyping divide = element divide { CommonAtt, DefEncAtt, empty} \stoptyping and the suggested rendering (from an example) is a slash. In the chapter \quotation {Characters, Entities and Fonts} there is mentioning of: \startquotation There is one more case where combining characters turn up naturally in mathematical markup. Some relations have associated negations, such as \type {U+226F} [\typ {NOT GREATER-THAN}] for the negation of U+003E [\typ {GREATER-THAN SIGN}]. The glyph for U+226F [NOT GREATER-THAN] is usually just that for U+003E [\typ {GREATER-THAN SIGN}] with a slash through it. Thus it could also be expressed by \type {U+003E}|-|\type {U+0338} making use of the combining slash \type {U+0338} [COMBINING LONG SOLIDUS OVERLAY]. That is true of 25 other characters in common enough mathematical use to merit their own \UNICODE\ code points. In the other direction there are 31 character entity names listed in [\typ {Entities}] which are to be expressed using \type {U+0338} [\typ {COMBINING LONG SOLIDUS OVERLAY}]. \stopquotation A curious note is this: \startquotation For special purposes, one may need a symbol which does not have a \UNICODE\ representation. In these cases one may use the \type {mglyph} element for direct access to a glyph as an image, or (in some systems) from a font that uses a non|-|\UNICODE\ encoding. All \MATHML\ token elements accept characters in their content and also accept an \type {mglyph} there. Beware, however, that use of \type {mglyph} to access a font is deprecated and the mechanism may not work in all systems. The \type {mglyph} element should always supply a useful alternative representation in its alt attribute. \stopquotation At some point we experimented with very precise positioned \HTML\ from \TEX\ (read: \CONTEXT) and that worked very well: the rendering was exactly the same as \PDF\ but then suddenly it was no longer possible to access glyphs from fonts. The assumption had become that one should feed text into the font rendering machinery and use \OPENTYPE\ features to access specific shapes, which of course is a fragile approach (the libraries and logic keep evolving, and the most robust access is simply by index, or by glyph name if present, assuming that one uses the font that was meant to be used). So, how the \MATHML\ glyph element is supposed to work out well is not clear. Anyway, as we want nicely typeset math we don't care that much if features present in \LUAMETATEX\ and \CONTEXT\ are unique and cannot be reproduced otherwise. In \type {mathclass.txt} [4] which is \quotation {{\em not} formally part of the \UNICODE\ Character Database at this time} we see a classification: \starttabulate[|T|l|] \NC U+0002F \NC binary \NC \NR \NC U+02044 \NC binary \NC \NR \NC U+02215 \NC binary \NC \NR \NC U+02571 \NC not mentioned \NC \NR \NC U+029F8 \NC n-ary or large operator, often takes limits \NC \NR \NC U+0FF0F \NC not mentioned \NC \NR \NC U+1F67C \NC not mentioned \NC \NR \stoptabulate So, in the end we can focus on the four that are mentioned, and we will do that with the above in mind as well as what is common in the \TEX\ world. We will look at usage, classification (groups) and classes. % modern % ok, both the same % cambria % different, no extensible / % bonum % ok, both the same % pagella % ok, both the same % stixtwo % only / extensible, 2044 useless % lucida % both extensible, 2044 looks bad and more slope Unfortunately this sort of mess also results in a mess in fonts. For instance when we checked out the difference between \type {U+002F} and \type {U+2044} we found that in the fonts produced by the \TEX Gyre project both have proper dimensions (and look the same), so they can be used stand alone, but also as delimiters. In Cambria the dimensions are okay but only \type {U+2044} has extensible characters. In \CONTEXT\ we have defined \type {\slash} to use that slot but when you test Lucida and \STIX2 the results are disappointing: In Lucida the width of \type {U+2044} makes it unusable (it looks bad anyway), and in \STIX2 it is a bit wider so in the end it even becomes fuzzy what to recommend as fix: quarter width, half width or full width. Defining \type {\slash} as any of them gives at some point an issue so in the end we just patch the font in the goodie file: we make them the same and make sure they have extensible characters. After all, chances are slim that this will ever be fixed. In that respect a newer engine doesn't change the problem: we need to handle it in the macro package, but at least that can be done a bit more natural. \footnote {In principle, we can support the goodies in the generic font handler, but we think it makes no sense because it also relates to the way math is handled in general and supporting a wide range of different applications can only cripple the code, let along that agreeing on matters can be hard.} % \ctxlua{table.tocontext(characters.data[0x002F],"[0x002F]")} % \ctxlua{table.tocontext(characters.data[0x2044],"[0x2044]")} % \ctxlua{table.tocontext(characters.data[0x2215],"[0x2215]")} % \ctxlua{table.tocontext(characters.data[0x2571],"[0x2571]")} % \ctxlua{table.tocontext(characters.data[0x29F8],"[0x29F8]")} \stopsection \startsection[title=Bars] Again we start with the \WIKIPEDIA\ page, this time the one dedicated to bars [5]. The page starts with mathematics so that suggests that the (initial) author is familiar with usage in that field: if we cut and paste the itemized list we even get \TEX\ math (sort of). Examples of usage are: absolute value, cardinality, conditional probability, determinant, distance, divisibility, function evaluation, length, norm, order, restriction, set|-|builder notation, the Sheffer stroke in logic, subtraction, but also \quotation {A vertical bar can be used to separate variables from fixed parameters in a function, or in the notation for elliptic integrals}. Among the objectives of our exploration are grouping symbols in sets that represent related meanings and usage. Within these groups we can fine tune with classes but that is more geared at rendering. Although currently users enter specific usage of symbols with the same shape (or even \UNICODE) with commands we can imagine them entering the \quote {real} characters and in that case we need some automatic class assignment based on a group (or set of groups). The \WIKIPEDIA\ page mentions that in physics \quotation {The vertical bar is used in bra|–|ket notation in quantum physics}. It then goes on about usage in computing, phonetics and literature. This ordering is different from the slashes, but okay. The page then makes a distinction between solid and broken bars and there is some interesting history behind that, which relates to typewriters, terminals and printers in the perspective of distinction and indeed we noticed that on our keyboard the broken bar is still used, even if the rendering is solid. The page ends with the \UNICODE\ bars and entities. We mention most: \starttabulate[|T|l|l|] \NC U+007C \NC \switchtobodyfont[stixtwo]$\utfchar{"007C}$ \NC a single vertical line \NC \NR % | \NC U+00A6 \NC \switchtobodyfont[stixtwo]$\utfchar{"00A6}$ \NC a single broken line \NC \NR % ¦ \NC U+2016 \NC \switchtobodyfont[stixtwo]$\utfchar{"2016}$ \NC a double vertical line (norms) \NC \NR % ‖ \NC U+2223 \NC \switchtobodyfont[stixtwo]$\utfchar{"2223}$ \NC divides \NC \NR % ∣ \NC U+2225 \NC \switchtobodyfont[stixtwo]$\utfchar{"2225}$ \NC parallel lines \NC \NR % ∥ \NC U+2502 \NC \switchtobodyfont[stixtwo]$\utfchar{"2502}$ \NC a vertical box drawing line \NC \NR % │ \NC U+FF5C \NC \switchtobodyfont[stixtwo]$\utfchar{"FF5C}$ \NC a fullwidth vertical line \NC \NR % ｜ \stoptabulate Given the mentioned wide range of usage it will be clear bars that can be confusing and are pretty overloaded. We're not aware of broken bars being used in math, so we ignore these. The \UNICODE\ math draft talks of \quote {vertical lines} and distinguishes two series, delimiters: \starttabulate[|T|l|l|] \NC U+007C \NC \switchtobodyfont[stixtwo]$\utfchar{"007C}$ \NC single vertical lines \NC \NR \NC U+2016 \NC \switchtobodyfont[stixtwo]$\utfchar{"2016}$ \NC double vertical lines \NC \NR \NC U+2980 \NC \switchtobodyfont[stixtwo]$\utfchar{"2980}$ \NC triple vertical lines \NC \NR \stoptabulate and operators: \starttabulate[|T|l|l|] \NC U+2223 \NC \switchtobodyfont[stixtwo]$\utfchar{"2223}$ \NC divides (single line) \NC \NR \NC U+2225 \NC \switchtobodyfont[stixtwo]$\utfchar{"2225}$ \NC parallel (double lines) \NC \NR \NC U+2AF4 \NC \switchtobodyfont[stixtwo]$\utfchar{"2AF4}$ \NC binary relation (tripple lines) \NC \NR \NC U+2AFC \NC \switchtobodyfont[stixtwo]$\utfchar{"2AFC}$ \NC s large triplle operator \NC \NR \stoptabulate Watch the triples: these are not (yet) in the \WIKIPEDIA\ summary. Rightfully there is a remark that the official \UNICODE\ descriptions use \typ {BAR} and \typ {LINE} but \TEX ies can't complain about that, can they? After all, they also use these terms mixed. The delimiters sit at the edges but sometimes also in the middle. The operators are between other elements and the document states that they also should grow. And is it mentioned that spacing depends on usage. The large triple is an n-ary operator but as usual with math symbols the user (reader) has to guess what that actually means. It is actually unfortunate that the fences have no left, middle and right variant. Even if these render the same it would make life easier and consistency with other fences is also worth something. One wonders how it would have looked if accessibility demands had kicked in earlier. The \UNICODE\ \type {mathclass.txt} [4] provides: \starttabulate[|T|l|] \NC U+007C \NC fence (unpaired delimiter) \NC \NR \NC U+2016 \NC fence (unpaired delimiter) \NC \NR \NC U+2980 \NC fence (unpaired delimiter) \NC \NR \stoptabulate We assume that the unpaired qualification is actually an indication that usage as what in \TEX\ is called \quote {middle} is okay. The operators are classified as: \starttabulate[|T|l|] \NC U+2223 \NC relation \NC \NR \NC U+2225 \NC relation \NC \NR \NC U+2AF4 \NC binary \NC \NR \NC U+2AFC \NC large n-ary \NC \NR \stoptabulate % \ctxlua{table.tocontext(characters.data[0x007C],"[0x007C]")} % \ctxlua{table.tocontext(characters.data[0x00A6],"[0x00A6]")} % \ctxlua{table.tocontext(characters.data[0x2016],"[0x2016]")} % \ctxlua{table.tocontext(characters.data[0x2980],"[0x2980]")} % \ctxlua{table.tocontext(characters.data[0x2223],"[0x2223]")} % \ctxlua{table.tocontext(characters.data[0x2225],"[0x2225]")} % \ctxlua{table.tocontext(characters.data[0x2AF4],"[0x2AF4]")} % \ctxlua{table.tocontext(characters.data[0x2AFC],"[0x2AFC]")} The main problem with bars in \TEX\ is that there is no distinction between a left and right bar which makes it impossible to use them directly as fences. On can consider this to be an omission to \UNICODE\ math because shape rules over meaning. So anyway, this is something that a macro package has to deal with. If needed these can get a class on their own in which case we can define atom spacing rules that deal with them ending up left or right. In \UNICODE\ there are signals that deal with bidirectional text, so we see no reason why there shouldn't be similar provisions for math. \stopsection \startsection[title=Hyphens and Dashes] This section applies to text and math as both are riddled with horizontal lines: easy to scratch in wood, chisel in stone or draw on paper symbols. We limit ourselves to the straight ones, but similar observations can be made for curved ones. \WIKIPEDIA\ distinguishes hyphens, minus, and dashes so there are multiple pages dedicated to this. The page about minus mentions that there are three usages (somewhat rephrased): \startitemize[packed] \startitem It is used as subtraction operator and therefore a binary operator that indicates the operation of subtraction. \stopitem \startitem It can be function whose value for any real or complex argument is the additive inverse of that argument. \stopitem \startitem It can serve as a prefix of a numeric constant. When it is placed immediately before an unsigned numeral, the combination names a negative number, the additive inverse of the positive number that the numeral would otherwise name. \stopitem \stopitemize The functional variant is how content \MATHML\ sees it: you apply a minus operator to something, singular of multiple. We were surprised to see that there is a distinctive rendering suggested, something we have argued for at several occasions (mostly \TEX\ meetings): \startquotation In many contexts, it does not matter whether the second or the third of these usages is intended: \type {−5} is the same number. When it is important to distinguish them, a raised minus sign \type {¯} is sometimes used for negative constants, as in elementary education, the programming language \APL, and some early graphing calculators. \stopquotation Unfortunately that distinction was not recognized by the \TEX\ community at large which (we guess) is why we don't see it in \UNICODE, which on the other hand has plenty dashes as we will see soon. The page mentions usage in indicating blood types and music, which is a nice detail. It also mentions usage in computing, including regular expressions and in physics and chemistry indicating charge. It lists these codes for minus symbols: \starttabulate[|Tl|l|] \NC U+002D \NC hyphen minus \NC \NR \NC U+2212 \NC minus \NC \NR \NC U+FE63 \NC small hyphen minus \NC \NR \NC U+FF0D \NC full width hyphen minus \NC \NR \stoptabulate The page also mentions the commercial minus \type {⁒} (see also [7]) and division sign \type {÷} (see also [8]) and we think these should be supported in math mode simply because they can be part of (even simple text style) formulas. The fact that we use the hyphen as minus and expect it to render as a wider dash like shape is something that related to math mode in \TEX\ speak. In text mode we expect it to be seen as hyphenation related indicator. We won't go into details about automated hyphenation and explicit hyphens in text mode but here are the hyphens as mentioned on the hyphen specific \WIKIPEDIA\ page: \starttabulate[|Tl|l|] \NC U+002D \NC hyphen minus \NC \NR \NC U+00AD \NC soft hyphen \NC \NR \NC U+2010 \NC hyphen \NC \NR \NC U+2011 \NC non breaking hyphen \NC \NR \stoptabulate You might wonder why we mention text variants here and one reason is that we actually might need to provide a catch for the last two: maybe when a user copies these from a document (when rendered at all) we need to treat them as the simple hyphen minus and just remap them to the math minus when in math mode. Below, we will discuss dashes, and although these are also meant for text, a reason for exploring these can be found in the fact that \TEX\ users like to decorate the content in unexpected ways and lines (or rules) fit into that. The \WIKIPEDIA\ pages go into some details about the hyphens being used in compounds and there can be some confusion about whether to use endashes or hyphens for that. We're pretty sure that typesetting wars have been fought over that. Usage as pre- and suffixes definitely is worth noting (and we use them as such in this sentence). We leave out all the other usages and see what there is to tell about related symbols. The \WIKIPEDIA\ page about dashes is an extensive one. It starts out with the distinction between \unichar {figure dash} {2012}, \unichar {endash} {2013}, \unichar {emdash} {2014} and \unichar {horizontal bar} {2015}. Of these a \TEX ie will for sure recognize the endash and emdash. The hyphen is not a dash but if you look at \TEX\ input that double or triple hyphens get ligatured into en- and emdashes! The only certainty one has is that the endash is often half the width of an emdash. Also, the width of the emdash is often the same as the font size. One reason why a language subsystem of a \TEX\ macro package is complex is that it has to deal with cultural aspects and the usage as well as spacing around all these dashes can differ. When trying to support that a macro writer soon finds out that one user of language~X can tell you the rules are done this way, and a while later you get a mail from another user who claims that in language~X the rules are done that way. Word processing and dominance of English probably adds to the confusion. The same is true for quotes, but math doesn't need these, so we skip them. Now wait, you will say: does math use these dashes? Users probably will mix them in but more important is that the width of these dashes also has associated skips: \type {\enspace} and \type {\emspace} or \type {\quad} and these one definitely see users mix into math. The figure dash has the same width as digits which makes them useful in tables. In the fonts that come with \TEX\ it is the reverse: the digits have the same width and that width matches the endash. There is no habit of using the figuredash, but we might need to change that. After all, we now have the fonts! We do need to deal with the figure dash because users might mix math and text in tables, and although you can find plenty of badly typeset by \TEX\ tables, this is no excuse for using a mix of minus and figure dash in inconsistent ways. The \WIKIPEDIA\ page mentions the usage of the endash: as connector, as compound hyphen, and as sentence interrupter. Now the one that needs some attention is the second one. In Dutch, we can combine words in many ways and for educational purposes adding a compound dash makes sense. However, because the weight of the hyphen and endash in \TEX\ fonts is rather incompatible, in \CONTEXT\ we use(d) fakes: two overlapping hyphens. Another complication is that one has to wrap that in a discretionary node in order to make the hyphenator happy, but that is now delegated to the engine that can be configured to see certain characters as valid hyphenation points. Although we support discretionaries in math this doesn't relate to dashes but to pluses and minuses and such. The engine supports explicit discretionaries but can also automatically repeat symbols that are set up as repeatable across lines. We're not sure if users actually use en- and emdashes in math mode, but one can occasionally run into examples (on the web) where special effects are achieved in curious ways. \footnote {The math stream doesn't go through the font handler although embedded \type {\hbox}es get that treatment. This means that two hyphens in a row are just two atoms and not get collapsed to an endash.} It is worth pointing out that \WIKIPEDIA\ discusses \quotation {Ranges of values} and this is something we need to investigate in the perspective of math! Strictly spoken that is a text thing, but \unknown\ Among the many observed and suggested patterns we note that among \TEX ies using the endash as itemize symbols is also popular. Usage of the emdash is related to the use of parenthesis or colons, so it is more a kind of punctuation. It can also be used as an interrupt and again it is a candidate for an itemize symbol. There is of course a \TEX\ thing there: lack of text symbols made for a rather mixed usage of math and text symbols in itemizations. For instance a dotted one uses the well visible math dot instead of the often hardly visible text dot that simply was not present in \TEX\ fonts, so our eyes got accustomed to the bolder ones. It is one of the reasons why a \TEX\ macro package load a math font even when no math is used. Over the years in \TEX\ math and text symbols have been mixed in various ways, also a side effect if the limited amount of characters in text fonts and the abundance of them in math mode, even if most are only accessible by name. We need to deal with that historic mix. The page rightfully mentions that \TEX\ has no horizontal bar, also known as \quote {quotation dash}, used for dialogues in some languages. We should make a note then that it might be good to see if we have to reconfigure the sub|-|sentence presets to match that expectation. The proposed hack {\red MPS: where?} for a missing symbol is somewhat curious: \starttyping x \hbox{---}\kern-.5em--- x \stoptyping \startbuffer[dash-example] \uleaders \hbox to 1.5em {---\hskip 0pt minus .5em---} \hskip.125em minus .125em \relax \stopbuffer Why not \type {\hbox {---\kern-.5em---}} or just \type {---\kern-.5em---} to get the same effect? This also assumes that the font collapses these three hyphens into a dash, then it backtracks the symbol width and does a second one. \footnote {Here is some food for thought: for this kind of usage one can argue that such a dash should have some stretch. In \LUAMETATEX\ and therefore \CONTEXT\ we can do this: \typeinlinebuffer [dash-example] and get: \dorecurse {30} {x \getbuffer [dash-example] x}. Boxed material can be stretched and be taken into account when creating paragraphs. It is no big deal to wrap that in a macro, say \type {\figuredashed}.} Anyway, where figure dashes are related to minuses we can probably ignore this super minus resembling horizontal bar. \footnote {We can actually issue a warning when it is used in math mode.} The \WIKIPEDIA\ page ends with a summary of all kind of dashes, including underscores, script specific symbols, accents (like macron), modifiers and curly ones. Here we only mention the ones that can end up in some source when one cuts and pastes. Doing that can result in missing characters (because not all fonts provides them) or a change in meaning (for as far as the symbols relates to an intention). We show some that fit into this discussion and also mention the \UNICODE\ description: \starttabulate[|T|lb{\ttx}|p|] \NC U+002D \NC HYPHEN-MINUS \NC the usual hyphen but also used as minus \NC \NR \NC U+005F \NC LOW LINE \NC aka underscore \NC \NR \NC U+00AD \NC SOFT HYPHEN \NC valid hyphenation point (invisible) \NC \NR \NC U+2010 \NC HYPHEN \NC the real hyphen but more work on a keyboard \NC \NR \NC U+2011 \NC NON-BREAKING HYPHEN \NC a hard hyphen, disables following hyphenation \NC \NR \NC U+2012 \NC FIGURE DASH \NC see discussion above \NC \NR \NC U+2013 \NC EN DASH \NC see discussion above \NC \NR \NC U+2014 \NC EM DASH \NC see discussion above \NC \NR \NC U+2015 \NC HORIZONTAL BAR \NC see discussion above \NC \NR \NC U+2043 \NC HYPHEN BULLET \NC used in itemized lists \NC \NR \NC U+207B \NC SUPERSCRIPT MINUS \NC combined with pre-superscripted characters \NC \NR \NC U+208B \NC SUBSCRIPT MINUS \NC combined with pre-subscripted characters \NC \NR \NC U+2212 \NC MINUS SIGN \NC the math minus (rendering of hyphen) \NC \NR \NC U+23AF \NC HORIZONTAL LINE EXTENSION \NC build long connected horizontal lines \NC \NR \NC U+23E4 \NC STRAIGHTNESS \NC represents line straightness in technical context \NC \NR \NC U+2500 \NC BOX DRAWINGS LIGHT HORIZONTAL \NC part of the box-drawing repertoire \NC \NR \NC U+2796 \NC HEAVY MINUS SIGN \NC a visual variant with no meaning \NC \NR \NC U+2E3A \NC TWO-EM DASH \NC a visual variant with no meaning \NC \NR \NC U+2E3B \NC THREE-EM DASH \NC a visual variant with no meaning \NC \NR \NC U+FE58 \NC SMALL EM DASH \NC a visual variant with no meaning \NC \NR \NC U+FE63 \NC SMALL HYPHEN-MINUS \NC a visual variant with no meaning \NC \NR \NC U+FF0D \NC FULLWIDTH HYPHEN-MINUS \NC a visual variant with no meaning \NC \NR \stoptabulate The \UNICODE\ math draft only mentions the hyphen: \footnote {When I copy this snippet into the document source there are \typ {START OF TEXT} symbols at the places where a hyphenation occurs, which is probably a side effect of a bad \type {TOUNICODE} entry in the \PDF\ file, but it is kind of interesting in this perspective as definitely a hyphen is rendered.} \startquotation Minus sign. \type {U+2212} [or] \type{−} [known as] \typ {MINUS SIGN} is the preferred representation of the unary and binary minus sign rather than the \ASCII|-|derived \type {U+002D} [or] \type {-} [known as] \typ {HYPHEN-MINUS}, because minus sign is unambiguous and because it is rendered with a more desirable length, usually longer than a hyphen. \stopquotation and elsewhere we can read: \startquotation The \ASCII\ hyphen minus \type {U+002D} [or] \type {-} is a weakly mathematical character that may be used for the subtraction operator, but \type {U+2212} [or] \type {−} [known as] \typ {MINUS SIGN} is preferred for this purpose and looks better. \stopquotation We are not aware of the concept of weak mathematical characters, so we will not take that property too serious when we try to improve the rendering. This is basically it. There is no mentioning of classes (after all, traditional \TEX\ has no unary class) so it is assumed that the renderer does the right thing: interpreting the sequence of characters and apply spacing accordingly. There are users who like to see a unary minus being rendered differently, just as the minus that a student is supposed to key in a calculator and while the \WIKIPEDIA\ page mentions this explicitly, it is ignored here. Yes, having two distinctive slots for this would have been great. Maybe it is not seen as relevant enough by the community that would benefit most, but who knows what had happened it the \WIKIPEDIA\ page had been there before! The minus is mentioned in the somewhat curious section about how shapes should be positioned relative to the baseline, where the position of the minus relates to what in \TEX\ speak is the math axis. There is also some mentioning of non-mathematical use, like: \startquotation The concept of mathematical use is deliberately kept broad; therefore the Math property is also given to characters that are used as operators, but are not part of standard mathematical notation, such as \type {U+2052} \typ {COMMERCIAL MINUS}. \stopquotation There should be no confusion with the \typ {SET MINUS} which renders as a backslash, a \typ {(NEG\-ATED) MINUS TILDE} or \typ {(NEG\-ATED) SIMILAR MINUS SIMILAR} that look more like relations. {\red MPS: overfull hbox, and do you intend to hyphenate?} The \MATHML\ document recognizes the minus as being unary or binary. In content \MATHML\ it is easy: when applied to a single atom it is a unary. In presentation \MATHML\ minus is an operator that sits at the front of a row (unary) or in the middle (binary). Keep in mind that we are limited to \type {mn} for numbers, \type {mi} for alphabetic symbols and \type {mo} for operators, not to be confused with \TEX's math operators, because in \MATHML\ relations are also operators. One can wonder about a minus in \type {mn} elements. So to summarize: we definitely need to make sure that (whatever renders as) hyphens is dealt with in math as minus. We can wonder what to do with (especially) en- and emdashes and the other horizontal lines that actually might show up as (what we call) middle delimiters in mathematical constructs: if it's there, \TEX ies will use it! The lack of specific symbols for unary minus has to be compensated at the macro package level. % \ctxlua{table.tocontext(characters.data[0x002D],"[0x002D]")} % \ctxlua{table.tocontext(characters.data[0x2010],"[0x2010]")} % \ctxlua{table.tocontext(characters.data[0x2011],"[0x2011]")} % \ctxlua{table.tocontext(characters.data[0x2212],"[0x2212]")} % \ctxlua{table.tocontext(characters.data[0x2212],"[0x2213]")} % \ctxlua{table.tocontext(characters.data[0x2212],"[0x2214]")} % \ctxlua{table.tocontext(characters.data[0x2212],"[0x2215]")} % \ctxlua{table.tocontext(characters.data[0xFE63],"[0xFE63]")} % \ctxlua{table.tocontext(characters.data[0xFF0D],"[0xFF0D]")} % U+2043 HYPHEN BULLET % U+207B SUPERSCRIPT MINUS % U+208B SUBSCRIPT MINUS \stopsection \startsection[title=Pieces] In \UNICODE\ one can find all kind of constructors, for instance characters that find their origin in those character sets that had lines and corners for drawing on a terminal. It is therefore no surprise that there are also some constructors that relate to math. An example demonstrates this: \startbuffer[definition] \def\makeweird#1#2#3#4% {\vcenter\bgroup \offinterlineskip \hbox{$\scriptscriptstyle\char"#1$}\par \hbox{$\scriptscriptstyle\char"#2$}\par \hbox{$\scriptscriptstyle\char"#3$}\par \hbox{$\scriptscriptstyle\char"#4$}% \egroup} \def\lwA{\mathopen {\makeweird{23A7}{23A8}{23A8}{23A9}}} \def\rwA{\mathclose{\makeweird{23AB}{23AC}{23AC}{23AD}}} \def\lwB{\mathopen {\makeweird{23A7}{23AC}{23AC}{23A9}}} \def\rwB{\mathclose{\makeweird{23AB}{23A8}{23A8}{23AD}}} \def\lwC{\mathopen {\makeweird{23A7}{23AC}{23A8}{23A9}}} \def\rwC{\mathclose{\makeweird{23AB}{23A8}{23AC}{23AD}}} \stopbuffer \startbuffer[demo] $\lwA x + 4 + \lwB x^2 + 4^2 + \lwC x^3 + 4^3 \rwC \rwB \rwA$ \stopbuffer \typebuffer[definition,demo] This renders as: \startlinecorrection \getbuffer[definition] \scale[width=\textwidth]{\getbuffer[demo]} \stoplinecorrection So, we have official \UNICODE\ characters for constructing large fences. In the \UNICODE\ math documents there is some mentioning of this and interesting is that there are suggested compositions expressed in 2, 3, 5 etc. stacked \quote {lines} which makes one wonder how math is perceived (or supposed to be rendered). But what is really weird is that there are plenty of arrows but no snippets defined that can be used to create extended ones. Why vertical snippets and no horizontal ones? This is clearly an omission and the \TEX\ community did take care of this need. So, for horizontal arrows and alike one expects the font to handle it and for fences not? It is not only fences that have snippets, we also find them for integrals. But for vertical arrows they are lacking: that is completely up to the font. Now, for us that is fine, but again, for consistency they could have been there. It would make it possible to filter bits and pieces from fonts using official slots instead of private ones. So, to some extent we can best assume there is nothing like that and ignore whatever pieces are in \UNICODE\ anyway (like the braces in the example). One can even argue that because of this inconsistency a font designed can as well only use private slots and not provide snippets at all. So, how do we get out of this situation? Because no one cared getting it in \UNICODE, we can do as we like. Of course, we can define arrow fillers as has always been done in \TEX, but because in \LUAMETATEX\ we have a bit more in our toolkit, and because we want to support stretch fractions (where the rule is replaced by a horizontal delimiter) it was decided to define a tweak that deals with this: when the basic arrows have no horizontal parts defined, we just assemble them. For those arrows that have a hook or so at the other end, we use the space as extender. \footnote {Actually we no longer do that because the engine will center the arrow anyway when it's too short.} If we ever end up with proper snippets un \UNICODE\ then we also need adapted fonts, and then we can get rid of these hacks. That said: because all decent math fonts do have the three pairs or fences (brace, parenthesis, bracket) the vertical snippets are rather useless, unless one wants to construct assembled weird ones. This would be different for horizontal assemblies, because there is more variety in them. The official name for all related to characters that can stretch is \quote {delimiter}. In traditional \TEX\ one can define a command that becomes a character. In that case a family, class and slot is assigned. You can also directly access a character in which case one will assign these properties otherwise (no command is defined). The same is true for these delimiters. However, in traditional \TEX\ the larger character usually comes from a so called extension font and uses family~3). In \OPENTYPE\ fonts we have all in one font so there the large family, class, and slot are not used. An interesting side effect of the updated math machinery in \LUAMETATEX\ is that we no longer really need delimiter specifications when we use \OPENTYPE\ fonts. This is because in practice the only two classes that really matter are the open and close ones. There are basically two kinds of delimiters: fences and singulars. Fences need open and close and only bars have a dual character. So, when we don't define it as delimiter, the engine can still use that character and take its assigned class when used stand|-|alone, while in the case of fences these themselves are of class open and close. And, for instance a left brace can get class open because when used stand alone it is an unscaled left fence. In the rare case that one really need a different class we are using commands: some characters can be binary, ordinary or whatever so then commands relate a name to a class|-|character combination. Actually, in \CONTEXT\ we will switch to using dictionaries and field specific rendering instead, but that is a different story. We can illustrate the arrows with an example: \startbuffer $ x + \left\downarrow a \uparrow \frac{1}{b} \downarrow c \right\uparrow = y $ \stopbuffer \typebuffer The stand alone arrows are defines with class relation but when used as fences their spacing is driven by the fences themselves. \startlinecorrection \scale[width=\textwidth]{\showmakeup[mathglue]\mathspacingmode1\showglyphs\getbuffer} \stoplinecorrection This means that in \CONTEXT\ \LMTX\ we no longer have delimiter code definitions. Of course the engine has to be able to use math characters of any kind (by commands, direct or as \UTF) as delimiters, but that was not that hard to provide. It also simplifies the code we use for fencing as it can be less selective. Another interesting side effect of once again looking into these stretched characters is that the fraction mechanism that already was extended with skewed fractions, now supports any stretchable character as alternative for a fraction rule. \startbuffer $ p \leftarrowtext {a + b + c + d}{x + y} q \quad p \frac {a + b + c + d}{x + y} q $ \stopbuffer \typebuffer Watch the difference in spacing: here the class of the used delimiter determines the spacing around the (pseudo) fraction: \startlinecorrection \scale[width=\textwidth]{\showmakeup[mathglue]\mathspacingmode1\showglyphs\getbuffer} \stoplinecorrection Again this simplifies some code because normally one ends up with stacking stuff using leaders in between. \stopsection \startsection[title=Accents] When we talk about accents, we refer to tiny symbols that anchor themselves onto base characters. We limit ourselves to the ones common in Latin scripts because they are the ones used in math. Accents in \UNICODE\ are somewhat special. In the past, when encoding vectors were limited, accents were entered as part of an input sequence and then anchored by the renderer. Nowadays often pre|-|composed characters are used. A very cheap way of anchoring is to have accents that just overlay, and in practice centering an accent over a base character works sort of okay. As an example of an accent we will use the hat: \starttabulate[|T|c|l|c|] \NC U+005E \NC x\char"005E x m\char"005E m\NC \tex {Hat} \NC \im{x \char"005E x + m\char"005E m} \NC \NR % 94 \NC U+02C6 \NC x\char"02C6 x m\char"02C6 m\NC \tex {hat} \NC \im{x \char"02C6 x + m\char"02C6 m} \NC \NR % 710 \NC U+0302 \NC x\char"0302 x m\char"0302 m\NC \tex {widehat} \NC \im{x \char"0302 x + m\char"0302 m} \NC \NR % 770 \stoptabulate Normally the font handler will take care of anchoring \type {U+0302}, but it can only be done properly when there are anchors defined for what are called \quote {marks}: the official feature description is mark|-|to|-|base (or simply \type {mark}). The last column in the above table shows math and as we input a raw character we don't get proper anchoring: the zero width makes it overlap. % till here Now wait, you will say, but why does it actually overlap? The reason is that zero width is not actually zero width here! The glyph has a bounding box that goes into the negative horizontal direction and therefore, when such a shape gets injected into the output, the rendering in the viewer will move the left edge to the left. But because the \TEX\ engine only handles positive widths and because the width is explicitly part of a character specification anyway\footnote {The height and depth are not: these we derive from the bounding box.} we don't progress (advance) which is why the overlapping sort of works for the $x$ but less so for the $m$: in math mode we need to use these \type {\hat} and \type {\widehat} commands. The hat and widehat assignments were those of August 2022. In plain \TEX\ we see these definitions: \starttyping \def\hat {\mathaccent"705E } \def\widehat{\mathaccent"0362 } \stoptyping The \type {\mathaccent} primitive takes an integer that encodes the class, family, and slot in the 8 bit font encoding. Here we see that the hat comes from family 0, the upright math font. The widehat comes from extensible family 3. These two are independently defined. When you want a hat that spans the nucleus, you need to use the widehat. In the math engine spanning actually means that we have a delimiter and normally that means: start with a basic shape, when that is too narrow, go to the extensible font and follow the chain with increasing sizes and when you run out of those apply an extensible recipe. The sequence and extensible are both optional and the important part is that we first look at what is called the small character and then to the large one(s). However, the \type {\mathaccent} primitives doesn't take a delimiter! It directly starts following a chain if the given character has it (and then the character itself is of course the first in that chain). And this is where the problems start when we move to \OPENTYPE\ and \UNICODE\ math. \starttabulate[|T|l|l|] \NC U+005E \NC Hat \NC some useless, often ugly large glyph \NC \NR % 94 \NC U+02C6 \NC hat \NC it has width but no extensibles \NC \NR % 710 \NC U+0302 \NC widehat \NC it has zero width and extensibles \NC \NR % 770 \stoptabulate Now, if we define \type {\hat} as \type {U+02C6} we don't get the extensibles, and it basically is what was always done in \TEX\ macro packages following the plain suggestions. If we define \type {\widehat} we start out with a glyph that has likely zero width\footnote {Over the many years that \LUATEX\ evolved this was not guaranteed, for instance when wide (\UNICODE) fonts were constructed from traditional eight bit (\TEX\ encoded) fonts.} And, because \OPENTYPE\ starts with the base glyph and {\em then} uses a set of variants of eventually a recipe of parts, we suddenly have a different situation with \type {\mathaccent} than we normally have, where these are decoupled. Therefore, the definition of \type {\hat} and \type {\widehat} determines what an \OPENTYPE\ math engine will do, just as in regular \TEX, but we might need them to be defined differently. A solution would be to let \type {\mathaccent} (or \type {\Umathaccent}) directly go to the variants, but that is sort of weird. Because a zero width glyph doesn't match the criteria to span a nucleus it is likely to be skipped anyway, although there can be a case where the next in size overruns the width of the nucleus in which case the zero width one is used which itself is not that nice. We could actually derive the width from the boundingbox, but that would be a bit abnormal, and it makes no sense to burden the font machinery with that exception. Another approach we can follow is to just copy the extensibles from \type {U+0302} to \type {02C6} and use that one for \type {\hat} as well as \type {\widehat} and then make \type {\widehat} an alias to \type {\hat}. After, all, the main reason why we have two commands comes from the fact that \type {\mathaccent} doesn't take a delimiter but single character reference (encoded in an integer). Here is the whole list of accents: \starttabulate[||T||T|] \NC \tex{grave} \NC U+0060 \NC \tex{widegrave} \NC U+0300 \NC \NR \NC \tex{ddot} \NC U+00A8 \NC \tex{wideddot} \NC U+0308 \NC \NR \NC \tex{bar} \NC U+00AF \NC \tex{widebar} \NC U+0304 \NC \NR \NC \tex{acute} \NC U+00B4 \NC \tex{wideacute} \NC U+0301 \NC \NR \NC \tex{hat} \NC U+02C6 \NC \tex{widehat} \NC U+0302 \NC \NR \NC \tex{check} \NC U+02C7 \NC \tex{widecheck} \NC U+030C \NC \NR \NC \tex{breve} \NC U+02D8 \NC \tex{widebreve} \NC U+0306 \NC \NR \NC \tex{dot} \NC U+02D9 \NC \tex{widedot} \NC U+0307 \NC \NR \NC \tex{ring} \NC U+02DA \NC \tex{widering} \NC U+030A \NC \NR \NC \tex{tilde} \NC U+02DC \NC \tex{widetilde} \NC U+0303 \NC \NR \NC \tex{dddot} \NC U+20DB \NC \tex{widedddot} \NC U+20DB \NC \NR \stoptabulate The only accent that is an exception is the last one but is it really used? It anyway makes no real sense to assume that users will ever directly input the \UTF\ characters conforming the last column, so we can just go for the first one and use the extensibles from the second and see where we end up. Neither \MATHML\ nor \TEX\ related specifications seem to cover this well, so we can just do what suits us best. \startbuffer \showglyphs \im {\widehat{a} + \widehat {aa}} = \im {\hat {a} + \hat {aa}} = \im {\hat {a} + \hat[stretch=yes]{aa}} = \setupmathaccent[top][stretch=yes] \im {\hat {a} + \hat {aa}} \stopbuffer Because all has to fit into the \CONTEXT\ user interface and because we also want to be backward compatible (command wise), we end up with something: \typebuffer that gives us: \startpacked \glyphscale = \numexpr2*\glyphscale\relax \getbuffer \stoppacked Now, one problem, is of course that users can enter these modifiers as \UTF\ sequence in the input, just like they do with delimiters. Therefore we do support the following feature (which is under class control so disabled by default): \startbuffer \Umathcode "02C6 \mathaccentcode 0 "02C6 \edef \HiHatA {\Uchar"02C6} \Umathchardef \HiHatB \mathaccentcode 0 "02C6 $ \Uchar"02C6{x} + \HiHatA{xx} + \HiHatB{xx} = \widehat {xxxx} $ \stopbuffer \typebuffer You get this: \start \pushoverloadmode \getbuffer \popoverloadmode \stop The only cheat here is that normally accents come after the accentee, but we can live with that. After all, it's all about convenience. There is another aspect of accents that we need to mention here. The hat, tilde, and check are often used over not only single letters but also small expressions. So how come that fonts have only very few variants defined? We can imagine that in eight bit fonts the number of available slots plays a role but in \OPENTYPE\ fonts that is not the case. It therefore can be considered an oversight that usage of these wide accents has not be communicated well to the font designers. \def\CrappyHack#1{\im{ #1{a} + #1{a+b} + #1{a+b+c} + #1{a+b+c+d} + #1{a+b+c+d+e} + #1{a+b+c+d+e+f} }\par} \startpacked \CrappyHack\widehat \CrappyHack\widetilde \CrappyHack\widecheck \stoppacked The previous lines demonstrate that we can actually cheat a little for these three top accents: we can just scale the last variant horizontally. It was a few lines patch to \LUAMETATEX\ to make this automatic and triggered by setting the \type {extensible} field in a character table to \type {true} instead of a recipe. The ingredients to get this working were already there, and it works out quite well. The only complication was that the \type {flac} feature (that provides flat accents for cases where the nucleus is rather high) could interfere, but that was trivial to deal with in the code that does the goodies. \footnote {When we were testing fonts this got us by surprise when we tested Cambria that has these flat overloads for the tilde and check. Because supports this automatic (hidden from the user) one doesn't look into that direction when testing something.} When it comes to these delimiters that have no real solution in the font, we can consider delegating coming up with a glyph to the macro package at the time it is needed, and we can actually do that. However, this is mostly interesting for educational usage, where the amount of delimiters is predictable and limited. About a decade ago some mechanism was added to the \MKIV\ math machinery that support plugins so that we could use \METAFUN\ to generate (most noticeably) square root symbols the way we liked. \footnote {This was a fun project of Alan and Hans.} The main drawback is that mixing this in means matching to a font, and that is not always trivial. But it is this kind of trickery that makes working with \TEX\ fun. That said: what we are discussing here is more fundamental in the sense that we try to come up with generic engine solutions that just rely on the fonts. That way complex math with all reasonable symbols is also served. \footnote {These \METAFUN\ plugins are still possible, but we need to adapt some to \LMTX\ which will happen as we go.} Interestingly there are some arrows that act like accents. There are over- and under ones as well as combining (often zero width) accents. Fonts are not always consistent in how these extends (the wide ones). Often the combining accents are smaller and closer to the running text. Traditionally in \TEX\ fonts there are no extensible arrows: they are constructed from arrow heads, minus and equal signs with some negative spacing in between. One can therefore wonder is the smaller combining ones are appreciated by those who want stable math. It definitely means that we have to make choices. Even more interesting is that while \UNICODE\ has some means to construct braces from predictable \UNICODE\ slots. there is no way to do the same with arrows and (indeed) there are fonts out there with shaped arrows that demand different middle and end pieces. In fact, the same is true for rules that are not simple rectangles and radical extensions that are not flat rules either. In all these cases the usage patterns of accents and similar constructs has not really been fed back into the way \UNICODE\ and \OPENTYPE\ fonts support math. \footnote {One can argue that this is not what \UNICODE\ is for but if so, then some other bits and pieces also make little sense.} \stopsection \startsection[title=Bullets] In \TEX\ usage bullets are a it special. Because fonts had a limited number of slots available, bullets in for instance itemized lists traditionally were taken from a math font. The bullet in Computer Modern has a comfortable size and is quite useful for that. Bullets in text fonts often were (are) relatively small so even when they were available they were not really used. The official \UNICODE\ slot for bullet is \type {U+2022} and in this font it shows up as \quote {•}. The \WIKIPEDIA\ page on bullets (typography) mentions: \startquotation A variant, the bullet operator (\type {U+2219} ∙ \typ {BULLET OPERATOR}) is used as a math symbol, akin to the dot operator. Specifically, in logic, $x • y$ means logical conjunction. It is the same as saying \quotation {x and y} \stopquotation The page also mentions that \quotation {glyphs such as {\switchtobodyfont [stixtwo]$•$} and {\switchtobodyfont [stixtwo]$◦$}} have \quotation {reversed variants {\switchtobodyfont [stixtwo]$◘$} and {\switchtobodyfont [stixtwo]$◙$}} although we haven't see the reverse once in \TEX\ documents (yet), like these (we use \STIX2\ to show them): \starttabulate[|Tl|l|l|] \NC U+2022 \NC \switchtobodyfont[stixtwo]$•$ \NC BULLET \NC \NR \NC U+2023 \NC \switchtobodyfont[stixtwo]$‣$ \NC TRIANGULAR BULLET \NC \NR \NC U+2043 \NC \switchtobodyfont[stixtwo]$⁃$\NC HYPHEN BULLET \NC \NR \NC U+204C \NC \switchtobodyfont[stixtwo]$⁌$\NC LACK LEFTWARDS BULLET \NC \NR \NC U+204D \NC \switchtobodyfont[stixtwo]$⁍$\NC LACK RIGHTWARDS BULLET \NC \NR \NC U+2219 \NC \switchtobodyfont[stixtwo]$∙$ \NC BULLET OPERATOR (math) \NC \NR \NC U+25CB \NC \switchtobodyfont[stixtwo]$○$ \NC WHITE CIRCLE \NC \NR \NC U+25CF \NC \switchtobodyfont[stixtwo]$●$ \NC BLACK CIRCLE \NC \NR \NC U+25D8 \NC \switchtobodyfont[stixtwo]$◘$ \NC INVERSE BULLET \NC \NR \NC U+25E6 \NC \switchtobodyfont[stixtwo]$◦$ \NC WHITE BULLET \NC \NR \NC U+29BE \NC \switchtobodyfont[stixtwo]$⦾$ \NC CIRCLED WHITE BULLET \NC \NR \NC U+29BF \NC \switchtobodyfont[stixtwo]$⦿$ \NC CIRCLED BULLET \NC \NR \stoptabulate The reverse ones are not really reverse in \STIX2\ as they have bigger circles. There are a few more bullets mentioned but probably only because they have the word bullet in their description and they don't really look like bullets. Given the already discussed lack of granularity in some math symbols with multiple usage it is somewhat surprising that we have a math bullet. The weird looking left- and rightward bullets are kind of hard to distinguish. Let's hope that mathematicians don't discover these! This brings us to the more general way of looking at these bullets because among the popular math symbols used in text are also the triangles and (\TEX) math fonts came with. When we have a few commands for circular shapes like \typ {$\bullet \bigcirc \circ$} giving $\bullet \bigcirc \circ$ we have plenty of (black) triangles. For instance, we have \type {\triangledown} and \type {\bigtriangledown} and these have corresponding \UNICODE\ slots \type {U+25BD} and \type {U+25BF} but when you try these in for instance \STIX2, Pagella and Cambria you got: ▽ + ▿, ▽ + ? and ? + ?, where the question mark indicates a missing character. It is for that reason that \type {\triangledown} and \type {\bigtriangledown} are both defined as using the large one. This test also demonstrated us that we didn't have to waste time looking up what \MATHML\ had to tell about it. A typeset version of that specification was never a visual highlight and missing glyphs only makes that worse. And, when fonts lack shapes no one uses them anyway. However, it makes sense to think a bit about how to deal with this properly, and we will likely add some checking to the goodie files for it, so that when we do have them, we use them. \footnote {Most practical is to add this information to the character database which is a bit of work}. But even then, most troublesome is that the size (and even positioning) of these symbols is rather inconsistent across math fonts, but because they are seldom used it doesn't make much sense to compensate for that (read: we just wait till users ask for it). % {\switchtobodyfont[stixtwo]$\char"25BD+\char"25BF$}% +\triangledown+\bigtriangledown$ % {\switchtobodyfont[pagella]$\char"25BD+\char"25BF$}% +\triangledown+\bigtriangledown$ % {\switchtobodyfont[cambria]$\char"25BD+\char"25BF$}% +\triangledown+\bigtriangledown$ \stopsection \startsection[title=Punctuation] There are quite some punctuation symbols in \UNICODE\ but not for math where the main troublemakers are the period, comma, colon and semicolon. The first two can be used as separator in numbers, in which case we don't want any spacing, or they can be part of a (pseudo) sentence in a formula, or they can separate entries in a list (take coordinates). \starttyping 1.1 + 1.2 (1.1, 1.2) x + 1.1, x + 1.2 \stoptyping When used as separator in a sentence, which is more likely in display math than in inline math, the spacing after it can be either regular (as in text) or wide. And the symbol can come from the math font or text (and these can actually look different). In \CONTEXT\ (also pre \LMTX) we have some special trickery at work for spacing comma's and periods but we leave that aside now. What should be noted is that out|-|of|-|the|-|box spaces are ignored when math is scanned so we cannot take that surrounding into account when dealing with spacing in the engine. Although the \UNICODE\ specification provides a classification of characters that includes punctuation in practice we need to deal with it ourselves. For instance, by default a period is not considered punctuation but a command and semi colon are, while a colon is a relation! Take for instance $f.$ (math italic f followed by a period). Italic correction and math glyphs have this special relationship and it also shows up in punctuation. Imagine that we have a sequence of characters, say $fx$. These are actually two ordinary atoms but in $f,$ we have an ordinary atom followed by a punctuation atom so here spacing is determined by how these classes are set up. But, given the shape if the $f$ we actually don't want italic correction here. \startbuffer $fx + f. +f, + f: + f; + a. +a, + a: + a; + x, +x, + x: + x;$% \stopbuffer \startlinecorrection \scale[width=\textwidth]{% \getbuffer } \blank[halfline] \scale[width=\textwidth]{% \showmakeup[mathglue]% \mathspacingmode\plusone \showfontitalics \showfontkerns \showglyphs \getbuffer } \stoplinecorrection When you zoom in you can see the subtle spacing differences. We can compensate for the semi colon being a bit higher than the period by applying some kern, something that we can set up in the goodie file. Actually, if we assume that periods only occur in numbers we can make it punctuation and set it up for digit spacing but then commas etc also get done that way. A variant is to have two punctuation classes (or cheat and put the period in the digit class). No matter what we do, no help can be expected from documents mentioned: it's mostly a visual thing anyway. Let's end with the visual aspect: in most fonts the two colons \type {0x003A} and \type {0x2236} are different: one has more distance between the periods. Which one? Well, that depends on the font! Latin Modern has a cramped \type {0x2236} while \STIX2 has a cramped \type {0x003A}. Cambria has square dots for the {0x003A} and round ones slightly more cramped for \type {0x2236}. Lucida goes extreme: it has smaller dots far apart for \type {0x2236}. If the idea is that a reader should get from the shape what it's about one can wonder if texts get read the way the author intended. Of maybe shapes don't matter. Of course a macro package can obscure these inconsistencies by setting the math character code of \type {0x003A} to \type {0x2236} but that only obscures the fact that little attention has been paid: what one can consider bugs became features. \stopsection \startsection[title=Special ones] There are quite some characters that really depend on a math renderer. Examples are wide accents, fences, and arrows. Some constructs, like fractions use rules and these don't come from \UNICODE\ nor fonts. A mixed case is radicals: there is a \UNICODE\ point and fonts can provide larger variants. Normally one steps up a slightly slanted version but when things get large the radical becomes an extensible and therefore gets an upright shape. The engine is supposed to add a horizontal rule at the right location. Interesting is that there is no provision for a right end cap. The reason probably is that \TEX, being the major renderer, has no combined horizontal and vertical extenders and \OPENTYPE\ doesn't have that either. Some properties are driven by the fonts' math parameters which sort of makes the radical rendering a very restricted adventure: it is supposed to be used for roots only, either of not with a degree anchored in the right top area. It looks like that degree is not really to extend much beyond the left edge of the symbol. In \UNICODE\ there is an actuarian character \type {U+20E7} and support in fonts is not that good. We do support it because we ran into in \MATHML. However, it is a hack. The symbol as provided by fonts is rather useless. \startbuffer $ \sqrt {x + 1} + \annuity{x + 1} $ \stopbuffer \typebuffer Let's see how it renders: \startlinecorrection \scale[width=.5\textwidth]{\getbuffer} \stoplinecorrection We take the dimensions of a radical as template and when we look at the bare glyphs we see this: \startlinecorrection \scale[height=2\lineheight]{$\char"221A \enspace \char"20E7$} \stoplinecorrection Basically we have a right actuarian character like we have a left radical. But In this case the rule will go left instead of right. This is implemented on top of radicals so and driven by \type {\Udelimited} that takes two delimiters and doesn't scan for a degree. For two-sided roots (with degree) we have \type {\Urooted}. And like normal radicals the delimited one adapts itself to the content: \startbuffer $ \sqrt {x + \frac{1}{x}} + \annuity {x + \frac{1}{x}} $ \stopbuffer \typebuffer So we get: \startlinecorrection \scale[width=.5\textwidth]{\showstruts \getbuffer} \stoplinecorrection For the record: in \CONTEXT\ spacing is also driven by the struts and because we use the radicals renderer the gap and distance parameters also apply. It might look spacy, but keep in mind that we want radicals to look similar when we have more of them in line, and we can configure all. We have also enabled the feature that radicals at the same level are normalized in height and depth. Here are some variants: \startbuffer $ \lannuity {x + \frac{1}{x}} + \rannuity {x + \frac{1}{x}} + \lrannuity {x + \frac{1}{x}} $ \stopbuffer \typebuffer This gives: \startlinecorrection \scale[width=.75\textwidth]{\getbuffer} \stoplinecorrection So we can have a mix of left, right and both end radical like symbols that encompass the nucleus. We're not aware of more such characters in \UNICODE\ but when they show up we are prepared. Only real usage can result in some parameters being fine|-|tuned. \stopsection % \startsection[title=Summary] % % Here we give a summary of some of the things that added on top of \UNICODE\ and % \OPENTYPE\ math in order to be able to properly render these more complex atoms % and molecules. % % \stopsection \startsection[title=Final words] This text was written in 2022 when we were working on math, extending the goodie files with new tweaks, checking support in fonts and updating manuals. But, as we moved forward, for instance with adapting \TYPEONE\ support of Antykwa and Iwona to the new possibilities again we had to go back in time and figure out why actually things were done in certain ways. And I have to admit that we had some good laughs and quite some fun on seeing how strange and inconsistent the assumed structured and logical \TEX\ ecosystem deals with math. A wrapup like is is never complete and we can keep adding to it so just consider it to be a momentary impression. Personally I have to admit that I've always overestimated what happened outside the \CONTEXT\ bubble, especially given the claims made. Consistency in \UNICODE\ math is probably not as good as is could have been and the same is true for \OPENTYPE\ math support, but maybe I'm naive in expecting consistency and logic in math related work. The mere fact that Donald Knuth pays a lot of attention to the math in his writing doesn't automatically translate in all \TEX ies doing the same. I don't claim that \CONTEXT\ is doing better but I do hope that its users keep going for the best outcome. One final note. In \CONTEXT\ we always tried to keep up with developments and \UNICODE\ input as well as using \OPENTYPE\ math fonts are part of that. However, because we're not part of the \quote {gremia of \TEX\ math and related coding} it hardly matters what our opinions are with respect to these issues. The best we can do is adapt to whatever shows up, it being bad or good. It is however kind of funny to see (by now rusty) problems that have been noticed already long ago being presented as kind of new. Hopefully staying ahead and|/|or adapting with specific solutions doesn't'backfire to hard on the \CONTEXT\ users. If so, we're sorry for that. As long as they can render their documents well, it doesn't matter that much anyway. After all, we can always just blame \quote {the others involved}. \stopsection \startsection[title=Resources] \starttyping [1] en.wikipedia.org/wiki/Slash_(punctuation) [2] www.unicode.org/reports/tr25 [3] www.w3.org/TR/MathML3 [4] www.unicode.org/Public/math/revision-15/MathClass-15.txt [5] en.wikipedia.org/wiki/Vertical_bar [6] en.wikipedia.org/wiki/Dash [7] en.wikipedia.org/wiki/Commercial_minus_sign [8] en.wikipedia.org/wiki/Division_sign [9] en.wikipedia.org/wiki/Bullet_(typography) \stoptyping \stopsection % After reading the \UNICODE\ report about math I don't feel too guilty when people % complain about the \CONTEXT\ manuals. It is a curious mix of discussing % organization of symbols, rendering, usage, structure, exchange, parsing, % confusion, etc. and it is clearly a mix of experiences with the web, word % processing and \TEX\ and as such not that useable because it is just not how % \TEX\ works with input and fonts and how users perceive matters. But it % definitely helps to get an idea why we ended up with the current situation: the % unification of math was more a combination of what was there and not a fresh % start. Maybe that is not really possible anyway. If we flash forward a couple of % pages it will all look the same to us as stone age chiseling in stone. \stopchapter \stopcomponent