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.
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.
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.
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.
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.
Important features of Nemeth are illustrated using the transcription shown above.
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.
(Not applicable.)
(Not applicable.)
_
^
(Not needed in TEX for one-character subscripts and superscripts; expression must be enclosed in { … } otherwise.)
(Not applicable.)
_
^
(Space.)
; TEX uses { … \over … } template for fractions.
Nemeth has many one-cell and two-cell symbols for important mathematical symbols. The six used in the sample are listed here.
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.
vs. =
vs. −
vs. ∞
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.
The templates described in the next section are also an example of good mnemonics.
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.
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.
… …
…
The simple-fraction template is used for fractions which do not have additional fractions in either the numerator or denominator.
numerator denominator
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.
This page was first posted December 9, 2002.
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