Plurimath

UnicodeMath support in Plurimath

Author’s picture Suleman Uzair Author’s picture Ronald Tse on 04 Apr 2024

Introduction

Plurimath has long been a versatile tool for handling various mathematical notation systems, including AsciiMath, MathML, LaTeX, and OMML.

Plurimath now includes support for UnicodeMath, opening up new possibilities for mathematical expression and representation.

UnicodeMath

Expressing math in Unicode

UnicodeMath is an innovative approach to encoding mathematical expressions using Unicode characters in a nearly plain-text format.

Invented by Murray Sargent III, long-time representative at the Unicode Consortium, retired physicist, and distinguished software engineer of Microsoft, UnicodeMath bridges the gap between human-readable mathematical notation and computer-encoded expressions.

UnicodeMath’s usefulness stems from the extensive availability of mathematical symbols in Unicode (ISO/IEC 10646). Unicode provides a vast array of mathematical symbols, making it possible to represent complex mathematical expressions using direct character input rather than relying on specialized markup commands.

Traditional mathematical typesetting systems often require users to memorize specific commands to represent mathematical symbols. For example, in LaTeX, one might need to use \sum for a summation symbol or \int for an integral. This approach burdens users with the task of memorizing these commands and understanding their visual characteristics.

In contrast, UnicodeMath allows users to directly input the actual mathematical symbols as they appear visually.

Table 1. Direct entry as Unicode instead of commands
UnicodeMath entry LaTeX command

Direct input of ∑ (U+2211 N-ARY SUMMATION)

\sum

Direct input of ∫ (U+222B INTEGRAL)

\int

Direct input of ∞ (U+221E INFINITY)

\infty

This direct symbol usage significantly reduces the cognitive load on users, especially those who are not professional mathematicians or frequent users of mathematical typesetting systems. It allows for a more intuitive and visual approach to inputting mathematical expressions, closely mimicking how one might write mathematics by hand.

Similar to other math expression languages, UnicodeMath also takes advantage of the spatial relationships between characters to infer mathematical meaning.

For example:

  • Superscripts can be denoted by using the ^ character: x^2 represents x².

  • Subscripts can be indicated with _: a_1 represents a₁.

  • Fractions can be written using /: a/b is interpreted as a fraction.

This approach combines the visual intuitiveness of handwritten mathematics with the precision and standardization offered by Unicode, creating a powerful and user-friendly system for mathematical notation in digital environments.

Name syntax string notation

While UnicodeMath primarily leverages the extensive set of mathematical symbols available in Unicode, there are instances where a desired symbol or operator may not have a direct Unicode representation.

To address this, UnicodeMath incorporates a "name syntax string notation" as an alternative input method.

The name syntax string notation allows users to input mathematical symbols or operators using descriptive names enclosed in backslashes.

There are actually two major reasons why the name syntax notation is necessary:

  1. Allows commonly-used symbols beyond those directly available in Unicode;

  2. Let’s face it: Unicode entry, especially for symbols on the Unicode math plane, is not always easiest on the keyboard. A named symbol provides an intuitive way to input complex or less common symbols.

Here are some examples of name syntax string notation in UnicodeMath:

  • \lceil represents the left ceiling bracket (⌈)

  • \rceil represents the right ceiling bracket (⌉)

  • \lfloor represents the left floor bracket (⌊)

  • \rfloor represents the right floor bracket (⌋)

  • \naryand represents the n-ary logical AND (⋀)

  • \naryor represents the n-ary logical OR (⋁)

In practice, this notation seamlessly integrates with direct Unicode input.

Example 1. UnicodeMath in name syntax string notation

This UnicodeMath expression:

\lceil x \rceil + \lfloor y \rfloor = z

Is interpreted and displayed as: ⌈x⌉ + ⌊y⌋ = z

The name syntax string notation also extends to more complex mathematical structures.

For instance:

  • \matrix{a,b;c,d} can be used to represent a 2x2 matrix

  • \eqarray{a=b;c=d} can create a system of equations

This notation system strikes a balance between the direct visual input of Unicode symbols and the expressive power of named commands, providing users with a flexible and comprehensive tool for mathematical expression.

It’s worth noting that while Plurimath supports both Unicode string input and name syntax string input for UnicodeMath, the output is always in Unicode representation. This ensures consistency and leverages the full visual capabilities of UnicodeMath across different platforms and applications.

Origins of UnicodeMath: Murray Sargent

Murray Sargent’s journey in developing UnicodeMath is rooted in his extensive experience with mathematical typesetting and software development. As a key figure in developing math support for Microsoft Office applications, Sargent recognized the need for a more intuitive, readable, and universally compatible mathematical notation system.

His work on UnicodeMath has revolutionized how users can input mathematical expressions directly into programs like Microsoft Word, PowerPoint, and OneNote. This innovation has significantly enhanced the user experience for students, educators, and professionals working with mathematical content in digital environments.

Implementations

The two prominent implementations of UnicodeMath are:

  • UnicodeMathML

  • Microsoft Equation Editor

UnicodeMathML (original) was originally created by Noah Doersing in 2019 as a JavaScript-based approach to allow easy entry of Unicode math symbols for UnicodeMath.

UnicodeMathML (new) is developed by Murray Sargent on top of Doersing’s UnicodeMathML, in order to keep the implementation updated with the latest specification.

The most popular implementation is likely the Microsoft Equation Editor, which is a component of Microsoft Office applications such as Word, PowerPoint, and OneNote offered on local installations and on Office 365.

This integration, spearheaded by Murray Sargent himself, allows users to input mathematical expressions using UnicodeMath directly into their documents.

The Microsoft Equation Editor interprets UnicodeMath in real-time, converting it into properly formatted mathematical expressions. This feature significantly enhances the user experience for anyone working with mathematical content in Microsoft Office, from students and educators to researchers and professionals.

For example, typing a^2 + b^2 = c^2 in the Equation Editor would automatically format it as a properly typeset mathematical equation, with superscripts and proper spacing.

Advantages of UnicodeMath

UnicodeMath offers compelling advantages over traditional mathematical notation systems:

Enhanced readability

UnicodeMath is encoded in a format that closely resembles displayed mathematics, making it intuitive to read and understand.

Reduced learning curve

For users new to mathematical typesetting, UnicodeMath presents a gentler learning curve compared to systems like LaTeX.

Preserved semantics

An equation encoded in UnicodeMath retains all semantics when it is copied and pasted across applications and platforms.

Here are some examples that illustrate these advantages.

In the first example, UnicodeMath and AsciiMath offer a more straightforward representation of simple fractions. While LaTeX provides more control over formatting, it requires learning specific commands, which can be a barrier for beginners.

Table 2. Fractions represented in different math expression languages
Notation system Expression Explanation

UnicodeMath

a/b

Simple and intuitive, resembling handwritten fractions.

LaTeX

\frac{a}{b}

Requires specific command and braces.

AsciiMath

a/b

Similar to UnicodeMath.

UnicodeMath shines in the following example by using the actual summation symbol (∑), making the expression more visually appealing and closer to traditional mathematical notation. LaTeX offers precise control but at the cost of readability in its raw form, while AsciiMath provides a middle ground.

Table 3. Summation represented in different math expression languages
Notation system Expression Explanation

UnicodeMath

∑_(i=1)^n i^2

Uses actual Unicode symbols, visually resembling handwritten math.

LaTeX

\sum_{i=1}^n i^2

Requires knowledge of specific commands and syntax.

AsciiMath

sum_(i=1)^n i^2

Uses plain text approximations of mathematical symbols.

In this complex expression involving integration, UnicodeMath’s use of actual symbols (∞, √) makes it more compact and visually similar to handwritten mathematics. LaTeX provides the most control for professional typesetting but is less intuitive to read in its raw form. AsciiMath offers a good balance but lacks the visual appeal of actual mathematical symbols.

Table 4. Complex expressions represented in different math expression languages
Notation system Expression Explanation

UnicodeMath

∫_0^∞ e(-x2) dx = √(π)/2

Compact and readable, using actual symbols for infinity and square root.

LaTeX

\int_0^\infty e{-x2} dx = \sqrt{\pi}/2

Precise but requires more specialized knowledge to interpret.

AsciiMath

int_0^oo e(-x2) dx = sqrt(pi)/2

Readable but uses text approximations for special symbols.

Leveraging UnicodeMath with Plurimath

General

With an understanding of the advantages of UnicodeMath, we explore how to utilize it in Plurimath.

Parsing

This code snippet demonstrates how to parse a UnicodeMath string into a Plurimath formula object.

string = '∑_(i=1)^n i^3'
formula = Plurimath::Math.parse(string, :unicode) (1)

  1. The :unicode parameter specifies that the input is in UnicodeMath format.

Generation and round-tripping

Here, we convert the formula object back to a UnicodeMath string. In other words, it normalizes an input UnicodeMath expression into a "cleaned" UnicodeMath expression.

formula.to_unicodemath # => '∑_(i = 1)^(n) i^(3)'

Notice how the Plurimath normalization process maintains the UnicodeMath format while potentially adjusting spacing for clarity.

While Plurimath supports both Unicode string input and name syntax string input for UnicodeMath, the output will always be in Unicode representation to maintain consistency and leverage the full visual capabilities of UnicodeMath.

Visualizing the parse tree

This parse tree visualization helps understand how Plurimath interprets the UnicodeMath expression, breaking it down into its constituent parts.

formula.to_display(:unicode) (1)
# |_ Math zone
#   |_ '∑_(i = 1)^(n) i^(3)'
#      |_ '∑' summation
#         |_ 'i = 1' lower limit
#         |_ 'n' upper limit
#         |_ 'i^(3)' expression

  1. The to_display(:unicode) method allows a Plurimath::Math::Formula object to be shown as a parse tree for UnicodeMath.

Converting UnicodeMath to MathML

This conversion to MathML showcases Plurimath’s ability to transform UnicodeMath into a more structured, XML-based format suitable for web applications and other digital platforms.

formula.to_mathml
# => "<math xmlns='http://www.w3.org/1998/Math/MathML'>
#       <mstyle displaystyle='true'>
#         <munderover>
#           <mo>∑</mo>
#           <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow>
#           <mi>n</mi>
#         </munderover>
#         <msup><mi>i</mi><mn>3</mn></msup>
#       </mstyle>
#     </math>"

Converting UnicodeMath to AsciiMath

This example demonstrates the conversion from UnicodeMath to AsciiMath, illustrating how Plurimath can bridge different mathematical notation systems.

formula.to_asciimath
# => "sum_(i=1)^n i^3"

Conclusion

The addition of UnicodeMath support to Plurimath represents a significant step forward in the realm of digital mathematical notation, and is a testament to Plurimath’s commitment to UnicodeMath as a math expression language.

Note
 Plurimath is the third major implementation of UnicodeMath and the second open-source implementation.

Support of UnicodeMath also demonstrates Plurimath’s dedication to handle major flavors of formal math representation languages for the user’s unhindered expressiveness:

  • Users can write math in the language of choice, and rely on Plurimath to automatically convert between representation languages with semantics preserved, purely according to their technical platform needs.

  • The mathematics expression models in Plurimath are demonstrably compliant to all supported math languages, including UnicodeMath and MathML.

Until then!