### Presenting Mathematics: LATEX and Nemeth Braille

#### Three Forms

A formula is shown as typeset in print. This is followed by two different linear forms: LATEX source and Nemeth braille. Print descriptors, which are revealed by hovering the mouse pointer, have been added for each braille cell in the Nemeth description. This is one way for sighted people and braille readers to read the same source.

##### Typeset Formula Displayed in Print

This is a picture of an important formula in college-level mathematics. It is used here to illustrate a number of features of the Nemeth Braille Code for Mathematics and Science Notation; it is not necessary to understand the formula in order to transcribe it to Nemeth although you do need to know the names of the special symbols. The print version is shown as it was displayed after the LATEX typesetting processor had been applied to the source. This picture was scanned in from a technical book and is typical of how mathematical formulas appear in print.

##### LATEX Source

Most print mathematics, including the formula just shown, is produced by automatic typesetting. This requires that the formula be entered into a computer. You may think of this as analogous to entering braille with the subsequent typesetting as analogous to actually embossing braille.

Just as experienced braillists often directly enter braille into the computer, persons who write mathematical formulas on a frequent basis often enter them directly in LATEX source. However, an alternative is user-friedly software like Scientific Notebook that automatically generates the LATEX source. With this type of math entry software, the user enters mathematics by using pull-down menus or another convenient method provided by the software's Graphical User Interface (GUI). It is not necessary for the user, including the braille transcriber, to know anything about LATEX in order to produce the required source.

However, just for completeness, here is the LATEX source that was actually used as the input to the typesetting processor for the previous formula. (Reading this directly is a bit like reading ASCII Braille, which is the format in which a computer stores braille that has been directly entered using six-key entry.)

$$e^x=\int_{-\infty}^x\sum_{n=0}^\infty {\lambda^n\over n!}d\lambda$$

This representation has 69 characters, including the two required spaces and the pair of  delimiters; this is about 50% more than the corresponding Nemeth transcription in the next section.

##### Same Formula in Nemeth

Here is the Nemeth braille transcription of the typeset formula show above. The MAVIS option in recent versions of Duxbury Braille Translator (DBT) can produce Nemeth automatically from LATEX source files exported from Scientific Notebook although the braille here was directly entered. If you hover your mouse over the braille cells, you can see how each group of cells corresponds to a portion of the LATEX source. [Note that the braille here is displayed as graphics for the benefit of persons who don't have a braille font; that is why it doesn't look quite as neat as it would otherwise. Also, this entire expression would almost fit on one standard 40-cell line.]

This representation uses 42 characters: 38 braille cells plus four required spaces to delimit the two equals marks.

Here is the Nemeth again. This time the print symbol corresponding to each braille cell is revealed by hovering the mouse pointer over the cell. If you aren't familiar with Nemeth, you might want to read more about it below before looking at the cells one-by-one.

#### An Introduction to Nemeth Braille

Important features of Nemeth are illustrated using the transcription shown above.

##### Indicators

Nemeth uses embedded indicators to show mathematically-significant composition (typesetting) and other information in a fashion similar to LATEX and other markup languages. The sample formula uses 11 different indicators as defined in the following list; the corresponding LATEX character is shown after the braille indicator where applicable.

1. (Single-letter) Capitalization indicator: (Not applicable.)
2. Greek letter indicator: (Not applicable.)
3. Subscript indicator:
_
4. Superscript indicator:
^
5. Baseline indicator: (Not needed in TEX for one-character subscripts and superscripts; expression must be enclosed in { … } otherwise.)
6. Multipurpose indicator (start modification): (Not applicable.)
7. Modifier directly-under indicator:
_
8. Modifier directly-over indicator:
^
9. Modified expression termination indicator: (Space.)
10. Simple fraction opening (start) indicator: ; TEX uses { … \over … } template for fractions.
11. Simple fraction closing (end) indicator:

##### Special Symbols

Nemeth has many one-cell and two-cell symbols for important mathematical symbols. The six used in the sample are listed here.

1. Factorial sign:
2. Horizontal fraction line (divide):
3. Integral sign:
4. Minus sign:
5. Equals mark:
6. Infinity:
##### Good Mnemonics

Nemeth uses the actual dot patterns of the braille cells to convey information by means of several simple conventions. (This is one reason why it is important to properly interface a braille display so that the reader gets the intended dot patterns.) These conventions include symmetries among the dot patterns of related symbols and similarities with corresponding print signs.

This is hard point for sighted persons to appreciate because when we first look at the braille dots, we may not see any patterns at all. Also, the blackprint braille cells shown here use shadow dots which tend to obscure the patterns produced by the dots. The compatible and easily recognized dot patterns of a well-designed braille code, including the patterns of several adjacent cells, may well be more important to the braille reader than a reduction in the number of cells.

• Indicators maintain the convention that cells used for this purpose only have dots in positions 4-6
• Subscript, baseline, and superscript are lower, "middle" and upper signs, respectively:
• Shapes of start and end fraction indicators are related and both resemble print parenthesis:
• Termination indicator resembles block:
• Fraction line resembles print slash used to show division:
• Equals mark resembles print equals mark: vs. =
• Minus sign resembles print minus sign vs. −
• Infinity somewhat resembles print infinity: vs. ∞
• Integral somewhat resembles print integral: vs. ∫

Another strategy that makes Nemeth easy to remember is the use of transliteration to represent letters in non-Latin alphabets. The relationship between the use of the same cell in the Nemeth code and in ordinary literary braille is also sometimes made use of.

• Greek letters are transliterated by corresponding Latin letters, e.g. l for lambda and s for sigma
• The factorial sign uses the same cell as the contraction for the word and, which is related to the meaning of factorial as "and multiply by all the numbers smaller than this", e.g.,
3! = 3 times 2 times 1
• Lower bounds and limits are always given before upper bounds and limits
• Expressions are presented in the same order as in print

The templates described in the next section are also an example of good mnemonics.

##### Powerful Templates

The Nemeth code uses templates for common arrangements. These templates use a prefix system so that the braille reader knows what to expect and doesn't have to read ahead to understand what type of expression is being presented. The example here makes use of both the modifier and fraction templates.

###### Modifier template

The modifier template is used to show subscript-like or superscript-like information that is typeset directly below or directly above a print symbol. Here we use the template for a symbol with information both below and above the same symbol; in the example given above, this template is used for a sum with both lower and upper limits.

1. Indicator for start of template:
2. Symbol (or expression) being modified
3. Directly-under indicator:
4. Expression that is directly under symbol being modified
5. Directly-over indicator:
6. Expression that is directly over symbol being modified
7. Modification termination indicator:
###### Simple-fraction template

The simple-fraction template is used for fractions which do not have additional fractions in either the numerator or denominator.

numerator denominator

### Summary

Nemeth braille is a linear math entry system especially designed for tactile reading of mathematics. As such, it is more appropriate for use by blind persons than linear systems designed for sighted persons or for computers. Current transcription software can produce Nemeth braille from LATEX source. LATEX source can be exported from a variety of math entry applications with user-friendly interfaces.

New software currently under development will also supply information about the braille cells in a form useable by sighted persons; this is illustrated here with the hand-entered information that appears when hovering the mouse over the braille cells.