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Unicode Support for Mathematics. Murray Sargent III Microsoft. 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. Unicode Math Characters.

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unicode support for mathematics

Unicode Support for Mathematics

Murray Sargent III


  • Unicode math characters
  • Semantics of math characters
  • Unicode and markup
  • Multiple ways of encoding math characters
  • Not yet standardized math characters
  • Inputting math symbols
unicode math characters
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).
basic set of alphanumeric characters
Basic Set of Alphanumeric Characters
  • Latin digits (0 - 9)
  • Upper- & lowercase Latin letters (a - z, A - Z)
  • Uppercase Greek letters Α - Ω plus the nabla ∇ and the variant of theta Θ given by U+03F4
  • Lowercase Greek letters α - ω plus the partial differential sign ∂ and glyph variants of ε, θ, κ, φ, ρ, and π
  • Only unaccented forms of letters are used
math alphanumeric characters
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
legibility loss
Legibility Loss

Without math alphabets, the Hamiltonian formula 

H =  dτ[εE2 + μH2]

 becomes an integral equation

H =  dτ[εE2 + μH2]

math alphanumeric chars cont
Math Alphanumeric Chars (cont)

Plain a-z, A-Z, 0-9, -, -Ω

Bold a-z, A-Z, 0-9, -, -Ω

Italic a-z, A-Z, -, -Ω

Bold italic a-z, A-Z, -, -Ω

Script a-z, A-Z

Bold script a-z, A-Z

Fraktur a-z, A-Z

Bold Fraktur a-z, A-Z

Double struck a-z, A-Z, 0-9

Sans-serif a-z, A-Z, 0-9

Sans-serif bold a-z, A-Z, 0-9, -, -Ω

Sans-serif italic a-z, A-Z

Sans-serif bold italic a-z, A-Z, -, -Ω

Monospace a-z, A-Z, 0-9

how display math alphabets
How Display Math Alphabets?
  • Can use Unicode surrogate pair mechanisms available on OS
  • Alternatively, bind to standard fonts and use corresponding BMP characters
  • 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
  • A single math font may look more consistent
math alphabetics via glyph variants
Math Alphabetics via Glyph Variants
  • One approach to the math alphanumerics would be to use a set of math glyph variant selectors
  • Such a tag would follow a base character imparting a math style
  • Approach was dropped since it seemed likely to be abused
  • One math variant selector does exist to offer a different line slant for some composite symbols
  • Other variant selectors are being defined for nonmath purposes, e.g., Han variants
multiple character encodings
Multiple Character Encodings
  • As with nonmath characters, math symbols can often be encoded in multiple ways, composed and decomposed
  • E.g., ≠ can be U+003D, U+0338 or U+2260
  • Recommendation: use the fully composed symbol, e.g., U+2260 for ≠
  • For alphabetic characters, use combining-mark sequences to get consistent typography
  • Some representations use markup for the alphabetic cases. This allows multicharacter combining marks.
compatibility holes
Compatibility Holes
  • Compatibility holes (reserved positions) exist in some Unicode sequences to avoid duplicate encodings (ugh!)
  • E.g., U+2071-U+2073 are holes for ¹²³, which are U+00B9, U+00B2, and U+00B3, respectively
  • Math alphanumerics have holes corresponding to Letterlike symbols.
  • Recommendation: you can use the hole codes internally, but must import and export the standard codes.
nonstandard characters
Nonstandard Characters
  • People will always invent new math characters that aren’t yet standardized.
  • Use private use area for these with a higher-level marking that these are for math.
  • This approach can lead to collisions in the math community (unless a standard is maintained)
  • Cut/copy in plain text can have collisions with other uses of the private use area
unicode and markup
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
unicode plain text
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
unicode character semantics
Unicode Character Semantics
  • Math characters have math property
  • Math characters are numeric, variable, or operator, but not a combination
  • Properties are useful in parsing math plain text
  • MathML doesn’t use these properties: every quantity is explicitly tagged
  • Properties still can be useful for inputting text for MathML (noone wants to type all those tags!)
  • Sometimes default properties need to be overruled
  • Would be useful to have more math properties
plain text encoding
Plain Text Encoding
  • TEX 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?
simple plain text encoding
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
  • 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 TEX’s, e.g., {a + c \over d}
  • MathML: <mfrac><mrow><mi>a</mi><mo>+</mo> <mi>c</mi></mrow><mrow><mi>d</mi> </mrow></mfrac>
  • Neat feature: plain text looks like math
subscripts and superscripts
Subscripts and Superscripts
  • Unicode has numeric subscripts and superscripts along with some operators (U+2070-U+208E)
  • Others need some kind of markup like <msup>…</msup>
  • With special subscript and superscript operators (not yet in Unicode), these scripts can be encoded nestibly
  • Use parentheses as for fractions to overrule built-in precedence order
presentation markup
Presentation markup
  • Presentation markup directs how the math should be rendered.

<mrow> <mi>E</mi>











content markup
Content markup
  • Content markup describes the meaning of the expression, not the format.















symbol entry
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 = 2100 = 1030
  • Use on-screen, customizable, keyboards and symbol boxes
  • Drag & drop any symbol into apps or onto keyboards
hex to unicode input method
Hex to Unicode Input Method
  • Type Unicode character hexadecimal code
  • Make corrections as need be
  • Type Alt+x to convert to character
  • Type Alt+x to convert back to hex (useful especially for “missing glyph” character)
  • Resolve ambiguities by selection
  • Input higher-plane chars using 5 or 6-digit code
  • New MS Word standard
built up formula heuristics
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
operator precedence
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
  • TEX 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
layout operator precedence
Layout Operator Precedence

Subscript, superscript ¯ ­

Integral, sum ò S P

Functions Ö

Times, divide / * × · •

Other operators Space ". , = - + Tab

Right brackets )]}|

Left brackets ([{

End of paragraph FF EOP

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

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);



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;


  • 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