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Unicode Support for Mathematics Murray Sargent III Microsoft

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Overview Unicode math characters Semantics of math characters Unicode and markup Multiple ways of encoding math characters Not yet standardized math characters Inputting math symbols

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Unicode Math Characters 340 math chars exist in ASCII, U+2200 – U+22FF, arrows, combining marks of Unicode 3.0 996 math alphanumeric characters are in Unicode 3.1’s Plane 1 591 new math symbols and operators are in Unicode 3.2’s BMP One math variant selector One new combining character (reverse solidus).

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Basic Set of Alphanumeric Characters u Latin digits (0 - 9) u Upper- & lowercase Latin letters (a - z, A - Z) u Uppercase Greek letters Α - Ω plus the nabla ∇ and the variant of theta Θ given by U+03F4 u Lowercase Greek letters α - ω plus the partial differential sign ∂ and glyph variants of ε, θ, κ, φ, ρ, and π u Only unaccented forms of letters are used

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Math Alphanumeric Characters Math needs various Latin and Greek alphabets like normal, bold, italic, script, Fraktur, and open-face May appear to be font variations, but have distinct semantics Without these distinctions, you get gibberish, violating Unicode rule: plain text must contain enough info to permit the text to be rendered legibly, and nothing more Plain-text searches should distinguish between alphabets, e.g., search for script H shouldn’t match H, etc. Reduces markup verbosity

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Legibility Loss Legibility Loss Without math alphabets, the Hamiltonian formula H = dτ [εE 2 + μH 2 ] becomes an integral equation H = dτ [εE 2 + μH 2 ]

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Math Alphanumeric Chars (cont) Plaina-z, A-Z, 0-9, - , -Ω Bolda-z, A-Z, 0-9, - , -Ω Italica-z, A-Z, - , -Ω Bold italica-z, A-Z, - , -Ω Scripta-z, A-Z Bold scripta-z, A-Z Fraktura-z, A-Z Bold Fraktur a-z, A-Z Double strucka-z, A-Z, 0-9 Sans-serifa-z, A-Z, 0-9 Sans-serif bolda-z, A-Z, 0-9, - , -Ω Sans-serif italica-z, A-Z Sans-serif bold italica-z, A-Z, - , -Ω Monospacea-z, A-Z, 0-9

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How Display Math Alphabets? u Can use Unicode surrogate pair mechanisms available on OS u Alternatively, bind to standard fonts and use corresponding BMP characters u Second approach probably faster and to display Unicode one needs font binding in any event. But most traditional fonts are not suited to math alphabetic characters u A single math font may look more consistent

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Math Alphabetics via Glyph Variants u One approach to the math alphanumerics would be to use a set of math glyph variant selectors u Such a tag would follow a base character imparting a math style u Approach was dropped since it seemed likely to be abused u One math variant selector does exist to offer a different line slant for some composite symbols u Other variant selectors are being defined for nonmath purposes, e.g., Han variants

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Multiple Character Encodings u As with nonmath characters, math symbols can often be encoded in multiple ways, composed and decomposed u E.g., ≠ can be U+003D, U+0338 or U+2260 u Recommendation: use the fully composed symbol, e.g., U+2260 for ≠ u For alphabetic characters, use combining-mark sequences to get consistent typography u Some representations use markup for the alphabetic cases. This allows multicharacter combining marks.

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Compatibility Holes u Compatibility holes (reserved positions) exist in some Unicode sequences to avoid duplicate encodings (ugh!) u E.g., U+2071-U+2073 are holes for ¹²³, which are U+00B9, U+00B2, and U+00B3, respectively u Math alphanumerics have holes corresponding to Letterlike symbols. u Recommendation: you can use the hole codes internally, but must import and export the standard codes.

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Nonstandard Characters u People will always invent new math characters that aren’t yet standardized. u Use private use area for these with a higher-level marking that these are for math. u This approach can lead to collisions in the math community (unless a standard is maintained) u Cut/copy in plain text can have collisions with other uses of the private use area

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Unicode and Markup Unicode was never intended to represent all aspects of text Language attribute: sort order, word breaks Rich (fancy) text formatting: built-up fractions Content tags: headings, abstract, author, figure Glyph variants: Poetica font: 58 ampersands; Mantinia font: novel ligatures (TT, TE, etc.) MathML adds XML tags for math constructs, but seems awfully wordy

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Unicode Plain Text Can do a lot with plain text, e.g., BiDi Grey zone: use of embedded codes Unicode ascribes semantics to characters, e.g., paragraph mark, right-to-left mark Lots of interesting punctuation characters in range U+2000 to U+204F Extensive character semantics/properties tables, including mathematical, numerical

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Unicode Character Semantics u Math characters have math property u Math characters are numeric, variable, or operator, but not a combination u Properties are useful in parsing math plain text u MathML doesn’t use these properties: every quantity is explicitly tagged u Properties still can be useful for inputting text for MathML (noone wants to type all those tags!) u Sometimes default properties need to be overruled u Would be useful to have more math properties

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Plain Text Encoding T E X fraction numerator is what follows a { up to keyword \over Denominator is what follows the \over up to the matching } { } are not printed Simple rules give unambiguous “plain text”, but results don’t look like math How to make a plain text that looks like math?

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Simple plain text encoding Simple operand is a span of alphanumeric characters E.g., simple numerator or denominator is terminated by any operator Operators include arithmetic operators, most whitespace characters, all U+22xx, an argument “break” operator (displayed as small raised dot), sub/superscript operators Fraction operator is given by the Unicode fraction slash operator U+2044

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Fractions abc/d gives More complicated operands use parentheses ( ), brackets [ ], or { } Outermost parens aren’t displayed in built-up form E.g., plain text (a + c)/d displays as Easier to read than T E X’s, e.g., { a + c \over d } MathML: a + c d Neat feature: plain text looks like math

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Subscripts and Superscripts u Unicode has numeric subscripts and superscripts along with some operators (U+2070-U+208E) Others need some kind of markup like … u With special subscript and superscript operators (not yet in Unicode), these scripts can be encoded nestibly u Use parentheses as for fractions to overrule built-in precedence order

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Presentation markup E = m c 2 u Presentation markup directs how the math should be rendered.

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Content markup E m c 2 u Content markup describes the meaning of the expression, not the format.

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Unicode T E X Example

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Symbol Entry GUI PCs can display a myriad glyphs, mathematics symbols, and international characters Hard to input special symbols. Menu methods are slow. Hot keys are great but hard to learn Reexamine and improve symbol-input and storage methods With left/right Ctrl/Alt keys, PC keyboard gives direct access to 600 symbols. Maximum possible = 2 100 = 10 30 Use on-screen, customizable, keyboards and symbol boxes Drag & drop any symbol into apps or onto keyboards

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Hex to Unicode Input Method u Type Unicode character hexadecimal code u Make corrections as need be u Type Alt+x to convert to character u Type Alt+x to convert back to hex (useful especially for “missing glyph” character) u Resolve ambiguities by selection u Input higher-plane chars using 5 or 6-digit code u New MS Word standard

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Built-Up Formula Heuristics Math characters identify themselves and neighbors as math E.g., fraction (U2044), ASCII operators, U2200– U22FF, and U20D0–U20FF identify neighbors as mathematical Math characters include various English and Greek alphabets When heuristics fail, user can select math mode: WYSIWYG instead of visible math on/off codes

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Operator Precedence Everyone knows that multiply takes precedence over add, e.g., 3+5×3 = 18, not 24 C-language precedence is too intricate for most programmers to use extensively T E X doesn’t use precedence; relies on { } to define operator scope In general, ( ) can be used to clarify or overrule precedence Precedence reduces clutter, so some precedence is desirable (else things look like LISP!) But keep it simple enough to remember easily

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Layout Operator Precedence Subscript, superscript Integral, sum Functions Times, divide/ * × · Other operatorsSpace "., = - + Tab Right brackets)]}| Left brackets([{ End of paragraphFF EOP

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Mathematics as a Programming Language u Fortran made great steps in getting computers to understand mathematics u Java and C# accept Unicode variable names u C++ has preprocessor and operator overloading, but needs extensions to be really powerful u Use Unicode characters including math alphanumerics u Use plain-text encoding of mathematical expressions u Can’t use all mathematical expressions as code, but can go much further than current languages go u When to to multiply? In abstract, multiplication is infinitely fast and precise, but not on a computer

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void IHBMWM(void) { gammap = gamma*sqrt(1 + I2); upsilon = cmplx(gamma+gamma1, Delta); alphainc = alpha0*(1-(gamma*gamma*I2/gammap)/(gammap + upsilon)); if (!gamma1 && fabs(Delta*T1) < 0.01) alphacoh = -half*alpha0*I2*pow(gamma/gammap, 3); else { Gamma = 1/T1 + gamma1; I2sF = (I2/T1)/cmplx(Gamma, Delta); betap2 = upsilon*(upsilon + gamma*I2sF); beta = sqrt(betap2); alphacoh = 0.5*gamma*alpha0*(I2sF*(gamma + upsilon) /(gammap*gammap - betap2)) *((1+gamma/beta)*(beta - upsilon)/(beta + upsilon) - (1+gamma/gammap)*(gammap - upsilon)/ (gammap + upsilon)); } alpha1 = alphainc + alphacoh; }

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Conclusions Unicode provides great support for math in both marked up and plain text Unicode character properties facilitate plain-text encoding of mathematics but aren’t used in MathML Heuristics allow plain text to be built up Need two more Unicode assignments: subscript and superscript operators On-screen keyboards and symbol boxes aid formula entry Unicode math characters could be useful for programming languages

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