This is a copy of an article about reading Nemeth braille without using dots. The only difference between this copy and the original is that the braille cells are displayed using a braille font instead of a print font. If you don't see any dots, you will need to install one of Duxbury's braille fonts.
Contents
This article is intended for persons who are interested in learning how mathematics is represented in the Nemeth braille code used in the United States, Canada, and New Zealand. It might also be useful for sighted persons who are learning Nemeth math and need some hints for proofreading their work.
This page assumes no knowledge of braille and, in particular, does not show the braille cells as dots. (But click here for a version that does.) However, it does assume a basic familiarity with mathematical notation. If anything isn't clear, please send questions to the address at the bottom of the page.
After studying this material, you should be able to do a fairly good job of interpreting Nemeth braille at least up through algebra. Reading is, of course, much easier than writing. Familiarity with the information on this page is not intended to substitute for the thorough knowledge required of a professional transcriber.
Nemeth braille mathematics is a plain text linear system with some similarities to related systems such as LaTeX and MathML. [Braille math systems have to be linear because tactile readers can't easily sense the relative vertical positions of the braille cells. Thus, for example, something like x squared is written as x^2 (using the corresponding braille cells) rather than just x with the superscript 2 literally raised above the baseline.]
The main difference between Nemeth and the linear systems used with print math
is that Nemeth is much less verbose because it is designed for end use by tactile readers rather
than for machine processing.
Braille is represented electronically by ASCII characters which are simply the characters on a standard computer keyboard. ASCII Braille is not something that has been invented just for this article; this method for representing computer-generated braille has been in world-wide use for almost 40 years. There are several different conventions for associating the braille cells to the ASCII characters but the only one used in the United States, and, luckily, the most useful one for working with Nemeth math, is North American ASCII Braille or, more informally, ASCII Braille or computer braille.
With ASCII Braille, the braille cell for the letter a is associated with the ASCII character a, that for the letter b with the ASCII character b, that for the digit 1 with the ASCII character 1, etc.
Not all of the associations between the braille cells and the ASCII characters are useful for determining meaning; many are arbitrary. This is unavoidable since braille is not one-for-one with print:
From now on we will often refer to the braille cells by the common names of the ASCII characters used to transliterate them. In other words, we will write "comma character" as shorthand for "the braille cell represented in ASCII Braille by (the same character code as) the ASCII comma character." This way of referring to the braille cells ensures that you will generate the proper braille cell when entering braille from a standard keyboard.
If you are curious which braille cells are being referenced, you can either use this chart or
change to a braille font. (Note that the chart is based
on an older form of ASCII braille that uses print capital letter characters to represent
the braille small letters. They are treated as equivalent in modern six-dot ASCII
braille. This article uses the print small letter characters to show ASCII braille
although you may find that some .brf files uses capital letters.
If this is bothersome, you can always use the Change Case option in a wordprocessor to change from
uppercase to lowercase.)
You can, of course, generate braille math in a standard text editor using a standard keyboard. You simply need to enter the characters specified in this article. If you want to view the corresponding braille cells you will need to install a compatible braille font. You can download Duxbury's SimBraille and Braille fonts for free.
ASCII Braille files are often sent directly to an embosser or refreshable braille display but they can also be opened with a text editor for sighted viewing. Click here for information about obtaining braille files from braille-using students or braille software or here for information about viewing braille files with WordPad.
Let's start with some examples where Nemeth braille can be read off correctly using the ASCII transliteration to supply meaning. These examples includes two symbols that require some explanation.
First, numbers in Nemeth braille are sometimes written with a preceding hashmark character, #. The purpose of the hashmark character is to make the braille cells for numbers easier to read and can be safely ignored as far as interpreting.
The second unusual symbol is the equals mark which is period character, k character. (This looks funny in print but the braille dots actually resemble a print equals mark.) The equals mark and other comparison symbols in Nemeth braille must be both preceded and followed by a space. However, the majority of symbols should be left unspaced from the adjacent symbols.
#1+2 .k #3
x .k #0
#9.8-7.6 .k #2.2
#4*5 .k #20
#3(a+b) .k #3a+3b
f(x, y)
10's place
As the examples in the previous section illustrate, the ASCII transliteration of Nemeth braille supplies the correct meaning when the 26 letter characters or the 10 digit characters are used in math expressions. The ASCII transliteration also supplies the correct meaning for the ten special characters in the table below.
| Character | Interpretation |
|---|---|
| + | plus character represents cell for plus sign |
| - | hyphen-minus character represents cell for minus sign |
| * | asterisk character represents cell for times (dot, not cross) |
| / | slash character represents cell for horizontal fraction line of any length |
| . | period character represents cell for decimal point in a numerical item |
| ( | left parenthesis character represents cell for left parenthesis |
| ) | right parenthesis character represents cell for right parenthesis |
| # | hashmark character before numerical item represents numerical indicator |
| , | comma character after mathematical item, always followed by a space, represents cell for comma separator in math | ' | apostrophe character represents cell for apostrophe |
Nemeth, like any linear system with a limited character set, requires some standard method for representing characters that aren't included in the basic character set. The Nemeth code specifies braille representations for more than 400 mathematical symbols and other special characters as well as for the 230 or so letters of the Latin, Greek, Hebrew, Russian, and Fraktur alphabets.
The two most common methods for representing characters in other linear systems for mathematics are numerical character codes, as used in MathML, and mnemonic abbreviations, as used in LaTeX.
A few important mathematical symbols, e.g. the integral sign, are represented by single braille cells in the Nemeth code. However, most symbols are represented by two or more cells using a prefix-root representation with one or more leading cells and one final cell. The next section details the prefix-root system used for mathematical symbols and the following one does the same for alphabets.
| Character(s) in braille symbol | Interpretation | Standard print glyph |
|---|---|---|
| & | factorial | ! |
| ! | integral sign | ∫ |
| > | square root | √ |
| \ | vertical bar | | |
| .k | equals | = |
| @+ | vee | ∨ |
| .+ | union | ∪ |
| @( | left bracket | [ |
| @) | right bracket | ] |
| .( | left brace | { |
| .) | right brace | } |
| ,= | infinity | ∞ |
| @,p | paragraph symbol | ¶ |
| @,r | Crossed R, prescription | ℞ |
The table to the right shows the Nemeth representation of some common mathematical symbols. The first four symbols are represented with a single braille cell.
You might be wondering why the braille cell for the factorial sign is mapped to the ampersand character instead of the exclamation point character. There are two reasons. One reason is that, unlike print, Nemeth braille uses a different character for an exclamation point than for a factorial sign. The other reason is that the braille cell used for the factorial sign is the same braille cell used for the contraction for the word and in literary braille. (As for the braille exclamation point punctuation mark, ASCII braille maps the digit six character to that cell. See also section on punctuation marks.) You might be able to remember all this because factorial means multiply n and n-1 and so on.
The use of the exclamation point character for the cell that represents the integral sign is purely arbitrary. However, the cell itself resembles an integral sign, a useful item for the braille reader to know.
The Nemeth code makes considerable use of what might be called "visual mnemonics" where there is a relationship between the shape of the braille cells and the shape of the corresponding print glyph. This approach enhances communication between print users and braille users. The Nemeth code also uses symmetry in that pairs of symbols that are symmetric in print—e.g. a pair of parentheses or the the less-than and greater-than signs—have symmetric dot patterns.
The two-cell representation of the equals sign was explained earlier.
In examining the remaining symbols we note a sort of ad hoc mnemonic system that is based partly on grouping symbols into families according to their meaning. Thus the logical sum and union characters both use the braille plus sign as a root. Also, the brackets and braces use the corresponding braille parenthesis as a root.
The at-sign character prefix indicates a variant form as does the period character prefix. The at-sign character represents a braille cell with just one dot whereas the period character represents a braille cell with two dots; perhaps the former prefix implies less variation than the latter. The period character prefix is also used before letters to indicate italic style and its use here is something like "fancy style vaguely related to italics."
The comma character prefix means capital when immediately before a letter and, more generally, "big" or "enlarged."
To be continued....
This page was first posted on 4/21/2005.
Send questions to: info@dotlessbraille.org