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]]>Section 2.2 of Unicode Technical Report #25 discusses the math alphanumerics and why they’re important for math. OfficeMath uses math alphabetics for most variables and supports the math digit sets as well. Accordingly, we need mappings between Unicode and Nemeth braille math alphanumerics. No attempt is made to give UEB sequences for the Unicode math alphanumerics.

For the most part, the mappings are straightforward as illustrated in the table below. But due to its generative use of type-form and alphabetic indicators, Nemeth braille encodes some math alphabets not in Unicode, e.g., Greek Script and Russian Script. Meanwhile, Unicode has math double-struck and monospace English alphanumerics, which don’t exist in Nemeth braille. Unicode also has six alphabets that aren’t mentioned in the Nemeth specification but that can be defined unambiguously with Nemeth indicators, namely bold Fraktur (Nemeth calls Fraktur “German”), bold Script, and Sans Serif bold and/or italic. The table below includes unambiguous prefixes for these alphabets chosen such that the Nemeth bold indicator precedes the italic or script indicators, and the Sans Serif indicator precedes the bold indicator. These choices correspond to the orders in which the Unicode math alphabets are named. Changes in this ordering result in alternative prefixes that are also unambiguous, but it seems simpler for implementations and users to standardize on the Unicode name ordering. Note that for simplicity English letters in math expressions often aren’t preceded by the braille italic indicator even though they are understood to be math-italic characters. Both LaTeX and MathML also use this convention.

Letter/style |
Nemeth |
E.g. |
Unicode |
Nemeth |

English letter | ⠰ | A | U+0041 | ⠰⠠⠁ |

Fraktur letter | ⠸ | 𝔄 | U+1D504 | ⠸⠠⠁ |

Greek letter | ⠨ | Γ | U+0393 | ⠨⠠⠛ |

Greek letter alternatives | ⠨⠈ | 𝜀 | U+03B5 | ⠨⠈⠑ |

Russian letter | ⠈⠈ | Д | U+0414 | ⠈⠈⠠⠙ |

Hebrew letter | ⠠⠠ | ℵ | U+2135 | ⠠⠠⠁ |

Bold | ⠸ | 𝐀 | U+1D400 | ⠸⠰⠠⠁ |

Bold Fraktur | ⠸⠸ | 𝕬 | U+1D56C | ⠸⠸⠠⠁ |

Bold Italic | ⠸⠨ | 𝑨 | U+1D468 | ⠸⠨⠰⠠⠁ |

Bold Script | ⠸⠈ | 𝓐 | U+1D4D0 | ⠸⠈⠰⠠⠁ |

Italic | ⠨ | 𝐴 | U+1D434 | ⠨⠰⠠⠁ |

Italic Greek | ⠨⠨ | 𝛾 | U+1D6FE | ⠨⠨⠛ |

Script | ⠈ | 𝒜 | U+1D49C | ⠈⠰⠠⠁ |

Sans Serif | ⠠⠨ | 𝖠 | U+1D5A0 | ⠠⠨⠰⠠⠁ |

Sans Serif Bold | ⠠⠨⠸ | 𝗔 | U+1D5D4 | ⠠⠨⠸⠰⠠⠁ |

Sans Serif Bold Italic | ⠠⠨⠸⠨ | 𝘼 | U+1D63C | ⠠⠨⠸⠨⠰⠠⠁ |

Sans Serif Italic | ⠠⠨⠨ | 𝘈 | U+1D608 | ⠠⠨⠨⠰⠠⠁ |

Upper case | ⠠ | A | U+0041 | ⠠⠁ |

The Nemeth specification has Script Greek (in §22) as well as “alternative” Greek letters (in §23). Some of the latter may be referred to as “script”. Specifically, the Unicode math Greek italic letters 𝜃𝜙𝜖𝜌𝜋𝜅 have the alternative counterparts 𝜗𝜑𝜀𝜚𝜛𝜘, respectively. The symbol 𝜗 can be called “script theta”. Since Unicode doesn’t have a math script Greek alphabet, it makes sense to map Nemeth math script Greek letters to the alternative Greek letters, if they exist, on input and to use the Nemeth alternative notation on output. In addition, in Unicode the upper-case Θ has the alternative ϴ. In TeX and OfficeMath, the alternative letters are identified by control words with a “var” prefix, as in \varepsilon for 𝜀, while ϵ is given by \epsilon. Interestingly, modern Greek uses 𝜑 and 𝜀 instead of 𝜙 and 𝜖, but math notation considers the script versions to be the alternatives. The table below lists the Greek symbols explicitly.

Nemeth braille has several Russian alphabets (see §22 of the Nemeth spec). These alphabets map to characters in the Cyrillic range U+0410..U+044F. Unicode has no math Russian alphabets, but italic and bold Russian alphabets can be emulated using the appropriate Cyrillic characters along with the desired italic and bold formatting. The Unicode Technical Committee, which is responsible for the Unicode Standard, has not received any proposals for adding Russian math alphabets. At least in my experience, technical papers in Russian use English and Greek letters in math zones. In Russian technical documents, this has the nice advantage of easily distinguishing mathematical variables from normal text. Accordingly the OfficeMath math braille facility uses the Russian letter notation for Unicode math double-struck letters.

Unicode has four predefined Hebrew characters in the Letterlike Symbols range U+2135..U+2138: ℵ, ℶ, ℷ, ℸ, respectively. In math contexts, it makes sense to map those Hebrew letters in Nemeth braille to the Letterlike Symbols and to map the other Nemeth Hebrew letters to characters in the Unicode Hebrew range U+05D0..U+05EA. The Unicode Technical Committee has not received any proposals for adding more Hebrew math letters so they probably won’t appear in math zones, except, perhaps, as embedded normal text.

Most Unicode math digits can be represented by the appropriate type-form indicator sequences in the table above followed by the numeric indicator ⠼ (if necessary) and the corresponding ASCII digits. For example, a math bold 2 (𝟐—U+1D7D0) can be represented by ⠸⠼⠆or “_#2”. This works for the bold and/or sans-serif digits, but not for the double-struck and monospace digits, which have no Nemeth counterparts. Meanwhile Nemeth notation supports italic and bold italic digits, which aren’t in Unicode.

Digits in most math contexts don’t need a numeric indicator in Nemeth notation, e.g., most digits in fractions, subscripts, or superscripts. To optimize common numeric subscript expressions like a_{1}, the numeric indicator *and* the subscript indicator are omitted. In Nemeth ASCII braille, a_{1} is “A1” and in Nemeth braille it’s ⠁⠂. The ASCII braille representation is tantalizing since variables like A1, B2, etc., are used to index spreadsheets and it would be more natural if spreadsheet indices were a_{1}, b_{2}, etc., at least for people with a mathematical background.

Code |
Char |
Unicode Name |
Nemeth |
UEB |

0021 | ! | Exclamation mark | ⠸⠖ | ⠖ |

0021 | ! | Factorial | ⠯ | ⠖ |

0022 | “ | Quotation mark | ⠠⠶ | |

0023 | # | Number sign | ⠨⠼ | ⠸⠹ |

0024 | $ | Dollar sign | ⠈⠎ | ⠈⠎ |

0025 | % | Percent sign | ⠈⠴ | ⠨⠴ |

0026 | & | Ampersand | ⠸⠯ | ⠈⠯ |

0027 | ‘ | Apostrophe | ⠄ | ⠄ |

0028 | ( | Left parenthesis | ⠷ | ⠐⠣ |

0029 | ) | Right parenthesis | ⠾ | ⠐⠜ |

002A | * | Asterisk | ⠈⠼ | ⠐⠔ |

002B | + | Plus sign | ⠬ | ⠐⠖ |

002C | , | Comma | ⠠ | ⠂ |

002D | – | Hyphen | ⠸⠤ | ⠤ |

002E | . | Full stop | ⠸⠲ | ⠼⠲ |

002F | / | Solidus | ⠸⠌ | ⠸⠌ |

003A | : | Colon | ⠸⠒ | ⠒ |

003B | ; | Semicolon | ⠸⠆ | ⠆ |

003C | < | Less-than sign | ⠀⠐⠅⠀ | ⠈⠣ |

003D | = | Equals sign | ⠀⠨⠅⠀ | ⠐⠶ |

003E | > | Greater-than sign | ⠀⠨⠂⠀ | ⠈⠜ |

003F | ? | Question mark | ⠸⠦ | ⠦ |

0040 | @ | Commercial at | ⠈⠁ | ⠈⠁ |

005B | [ | Left square bracket | ⠈⠷ | ⠨⠣ |

005C | \ | Reverse solidus | ⠸⠡ | ⠸⠡ |

005D | ] | Right square bracket | ⠈⠾ | ⠨⠜ |

005E | ^ | Circumflex accent | ⠸⠣ | ⠈⠢ |

005F | _ | Low line | ⠨⠤ | |

007B | { | Left curly bracket | ⠨⠷ | ⠸⠣ |

007C | | | Vertical line | ⠡ | ⠸⠳ |

007D | } | Right curly bracket | ⠨⠾ | ⠸⠜ |

007E | ~ | Tilde | ⠈⠱ | ⠈⠔ |

00A2 | ¢ | Cent sign | ⠈⠉ | ⠈⠉ |

00A3 | £ | Pound sterling | ⠈⠇ | ⠈⠇ |

00A7 | § | Section sign | ⠈⠠⠎ | ⠘⠎ |

00A9 | © | Copyright sign | ⠷⠠⠉⠾ | ⠘⠉ |

00AC | ¬ | Not sign | ⠈⠹ | |

00AE | ® | Registered sign | ⠷⠠r⠾ | ⠘⠗ |

00B0 | ° | Degree sign | ⠘⠨⠡ | ⠘⠚ |

00B1 | ± | Plus-minus sign | ⠬⠤ | ⠸⠖ |

00B6 | ¶ | Pilcrow sign | ⠈⠠⠏ | ⠘⠏ |

00B7 | · | Middle dot | ⠡ | |

00D7 | × | Multiplication sign | ⠈⠡ | ⠐⠦ |

00F7 | ÷ | Division sign | ⠨⠌ | ⠐⠌ |

0391 | Α | Greek capital letter alpha | ⠨⠠⠁ | ⠠⠨⠁ |

0392 | Β | Greek capital letter beta | ⠨⠠⠃ | ⠠⠨⠃ |

0393 | Γ | Greek capital letter gamma | ⠨⠠⠛ | ⠠⠨⠛ |

0394 | Δ | Greek capital letter delta | ⠨⠠⠙ | ⠠⠨⠙ |

0395 | Ε | Greek capital letter epsilon | ⠨⠠⠑ | ⠠⠨⠑ |

0396 | Ζ | Greek capital letter zeta | ⠨⠠⠵ | ⠠⠨⠵ |

0397 | Η | Greek capital letter eta | ⠨⠠⠱ | ⠠⠨⠱ |

0398 | Θ | Greek capital letter theta | ⠨⠠⠹ | ⠠⠨⠹ |

0399 | Ι | Greek capital letter iota | ⠨⠠⠊ | ⠠⠨⠊ |

039A | Κ | Greek capital letter kappa | ⠨⠠⠅ | ⠠⠨⠅ |

039B | Λ | Greek capital letter lambda | ⠨⠠⠇ | ⠠⠨⠇ |

039C | Μ | Greek capital letter mu | ⠨⠠⠍ | ⠠⠨⠍ |

039D | Ν | Greek capital letter nu | ⠨⠠⠝ | ⠠⠨⠝ |

039E | Ξ | Greek capital letter xi | ⠨⠠⠭ | ⠠⠨⠭ |

039F | Ο | Greek capital letter omicron | ⠨⠠⠕ | ⠠⠨⠕ |

03A0 | Π | Greek capital letter pi | ⠨⠠⠏ | ⠠⠨⠏ |

03A1 | Ρ | Greek capital letter rho | ⠨⠠⠗ | ⠠⠨⠗ |

03A3 | Σ | Greek capital letter sigma | ⠨⠠⠎ | ⠠⠨⠎ |

03A4 | Τ | Greek capital letter tau | ⠨⠠⠞ | ⠠⠨⠞ |

03A5 | Υ | Greek capital letter upsilon | ⠨⠠⠥ | ⠠⠨⠥ |

03A6 | Φ | Greek capital letter phi | ⠨⠠⠋ | ⠠⠨⠋ |

03A7 | Χ | Greek capital letter chi | ⠨⠠⠯ | ⠠⠨⠯ |

03A8 | Ψ | Greek capital letter psi | ⠨⠠⠽ | ⠠⠨⠽ |

03A9 | Ω | Greek capital letter omega | ⠨⠠⠺ | ⠠⠨⠺ |

03B1 | α | Greek small letter alpha | ⠨⠁ | ⠨⠁ |

03B2 | β | Greek small letter beta | ⠨⠃ | ⠨⠃ |

03B3 | γ | Greek small letter gamma | ⠨⠛ | ⠨⠛ |

03B4 | δ | Greek small letter delta | ⠨⠙ | ⠨⠙ |

03B5 | ε | Greek small letter epsilon
(alternative epsilon) |
⠨⠈⠑ | ⠨⠑ |

03B6 | ζ | Greek small letter zeta | ⠨⠵ | ⠨⠵ |

03B7 | η | Greek small letter eta | ⠨⠱ | ⠨⠱ |

03B8 | θ | Greek small letter theta | ⠨⠹ | ⠨⠹ |

03B9 | ι | Greek small letter iota | ⠨⠊ | ⠨⠊ |

03BA | κ | Greek small letter kappa | ⠨⠅ | ⠨⠅ |

03BB | λ | Greek small letter lambda | ⠨⠇ | ⠨⠇ |

03BC | μ | Greek small letter mu | ⠨⠍ | ⠨⠍ |

03BD | ν | Greek small letter nu | ⠨⠝ | ⠨⠝ |

03BE | ξ | Greek small letter xi | ⠨⠭ | ⠨⠭ |

03BF | ο | Greek small letter omicron | ⠨⠕ | ⠨⠕ |

03C0 | π | Greek small letter pi | ⠨⠏ | ⠨⠏ |

03C1 | ρ | Greek small letter rho | ⠨⠗ | ⠨⠗ |

03C2 | ς | Greek small letter final sigma | ⠨⠒ | ⠨⠎ |

03C3 | σ | Greek small letter sigma | ⠨⠎ | ⠨⠎ |

03C4 | τ | Greek small letter tau | ⠨⠞ | ⠨⠞ |

03C5 | υ | Greek small letter upsilon | ⠨⠥ | ⠨⠥ |

03C6 | φ | Greek small letter phi
(alternative phi) |
⠨⠈⠋ | ⠨⠋ |

03C7 | χ | Greek small letter chi | ⠨⠯ | ⠨⠯ |

03C8 | ψ | Greek small letter psi | ⠨⠽ | ⠨⠽ |

03C9 | ω | Greek small letter omega | ⠨⠺ | ⠨⠺ |

03D1 | ϑ | Greek theta symbol | ⠨⠈⠹ | |

03D5 | ϕ | Greek phi symbol | ⠨⠋ | |

03D6 | ϖ | Greek pi symbol | ⠨⠈⠏ | |

03F0 | ϰ | Greek kappa symbol | ⠨⠈⠅ | |

03F5 | ϵ | Greek lunate epsilon symbol | ⠨⠑ | |

2014 | — | Em dash | ⠤⠤ | ⠠⠤ |

2015 | ― | Horizontal bar | ⠤⠤⠤⠤ | ⠐⠠⠤ |

2018 | ‘ | Left single quotation mark | ⠠⠦ | ⠠⠦ |

2019 | ’ | Right single quotation mark | ⠴⠠ | ⠠⠴ |

201C | “ | Left double quotation mark | ⠦ | ⠘⠦ |

201D | ” | Right double quotation mark | ⠴ | ⠘⠴ |

2020 | † | Dagger | ⠸⠻ | ⠈⠠⠹ |

2021 | ‡ | Double dagger | ⠸⠸⠻ | ⠈⠠⠻ |

2022 | • | Bullet | ⠔⠔ | ⠸⠲ |

2026 | … | Ellipsis | ⠄⠄⠄ | |

2030 | ‰ | Per mille sign | ⠈⠴⠴ | |

2031 | ‱ | Per ten thousand sign | ⠈⠴⠴⠴ | |

2032 | ′ | Prime | ⠄ | ⠶ |

2033 | ″ | Double prime | ⠄⠄ | ⠶⠶ |

210F | ℏ | Planck constant over two pi | ⠈⠓ | |

2113 | ℓ | Script small l
(differs from 1d4c1: 4-56-123) |
⠈⠇ | |

211D | ℝ | Double-struck capital R
(no nemeth double struck) |
||

212B | Å | Angstrom sign | ⠈⠠⠁ | |

2146 | ⅆ | Double-struck italic small d | ||

2147 | ⅇ | Double-struck italic small e | ||

2190 | ← | Leftwards arrow | ⠀⠫⠪⠒⠒⠀ | ⠳⠪ |

2191 | ↑ | Upwards arrow | ⠀⠫⠣⠒⠒⠕⠀ | ⠳⠬ |

2192 | → | Rightwards arrow | ⠀⠫⠕⠀ | ⠳⠕ |

2192 | → | Rightwards arrow | ⠀⠫⠒⠒⠕⠀ | |

2193 | ↓ | Downwards arrow | ⠀⠫⠩⠒⠒⠕⠀ | ⠳⠩ |

2194 | Left right arrow | ⠀⠫⠪⠒⠒⠕⠀ | ⠰⠳⠺⠗⠕ | |

2195 | Up down arrow | ⠀⠫⠣⠪⠒⠒⠕⠀ | ||

2196 | North west arrow | ⠀⠫⠘⠪⠒⠒⠀ | ⠳⠱ | |

2197 | North east arrow | ⠀⠫⠘⠒⠒⠕⠀ | ⠳⠎ | |

2198 | South east arrow | ⠀⠫⠰⠒⠒⠕⠀ | ⠳⠣ | |

2199 | South west arrow | ⠀⠫⠰⠪⠒⠒⠀ | ⠳⠜ | |

219A | ↚ | Leftwards arrow w stroke | ⠀⠳⠈⠫⠪⠒⠒⠻⠀ | |

219B | ↛ | Rightwards arrow w stroke | ⠀⠳⠈⠫⠒⠒⠕⠻⠀ | |

219C | ↜ | Leftwards wave arrow | ⠀⠫⠪⠔⠒⠢⠀ | |

219D | ↝ | Rightwards wave arrow | ⠀⠫⠔⠒⠢⠕⠀ | |

219E | ↞ | Leftwards two headed arrow | ⠀⠫⠪⠪⠒⠒⠀ | |

219F | ↟ | Upwards two headed arrow | ⠀⠫⠣⠒⠒⠕⠕⠀ | |

21A0 | ↠ | Rightwards two headed arrow | ⠀⠫⠒⠒⠕⠕⠀ | |

21A1 | ↡ | Downwards two headed arrow | ⠀⠫⠩⠒⠒⠕⠕⠀ | |

21A2 | ↢ | Leftwards arrow w tail | ⠀⠫⠪⠒⠒⠠⠽⠀ | |

21A3 | ↣ | Rightwards arrow w tail | ⠀⠫⠠⠯⠒⠒⠕⠀ | |

21A4 | ↤ | Leftwards arrow from bar | ⠀⠫⠪⠒⠒⠳⠀ | |

21A5 | ↥ | Upwards arrow from bar | ⠀⠫⠣⠳⠒⠒⠕⠀ | |

21A6 | ↦ | Rightwards arrow from bar | ⠀⠫⠳⠒⠒⠕⠀ | |

21A7 | ↧ | Downwards arrow from bar | ⠀⠫⠩⠳⠒⠒⠕⠀ | |

21A8 | ↨ | Up down arrow w base | ⠀⠫⠣⠪⠒⠒⠕⠳⠀ | |

21A9 | Leftwards arrow w hook | ⠀⠫⠪⠒⠒⠈⠽⠀ | ||

21AA | Rightwards arrow w hook | ⠀⠫⠈⠯⠒⠒⠕⠀ | ||

21AB | ↫ | Leftwards arrow w loop | ||

21AC | ↬ | Rightwards arrow w loop | ||

21AD | ↭ | Left right wave arrow | ⠀⠫⠪⠔⠒⠢⠕⠀ | |

21AE | ↮ | Left right arrow w stroke | ⠀⠳⠈⠫⠪⠒⠒⠕⠻⠀ | |

21AF | ↯ | Downwards zigzag arrow | ⠀⠫⠩⠔⠢⠔⠕⠀ | |

21B4 | ↴ | Rightwards arrow w corner downwards | ⠀⠫⠩⠠⠳⠒⠕⠀ | |

21B5 | ↵ | Downwards arrow w corner leftwards | ⠀⠫⠪⠒⠈⠳⠀ | |

21B6 | ↶ | Anticlockwise top semicircle arrow | ||

21B7 | ↷ | Clockwise top semicircle arrow | ||

21BA | ↺ | Anticlockwise open circle arrow | ⠀⠫⠢⠔⠕⠀ | |

21BB | ↻ | Clockwise open circle arrow | ⠀⠫⠪⠢⠔⠀ | |

21BC | ↼ | Leftwards harpoon w barb upwards | ⠀⠫⠈⠪⠒⠒⠀ | |

21BD | ↽ | Leftwards harpoon w barb downwards | ⠀⠫⠠⠪⠒⠒⠀ | |

21BE | ↾ | Upwards harpoon w barb rightwards | ⠀⠫⠣⠒⠒⠠⠕⠀ | |

21BF | ↿ | Upwards harpoon w barb leftwards | ⠀⠫⠣⠒⠒⠈⠕⠀ | |

21C0 | ⇀ | Rightwards harpoon w barb upwards | ⠀⠫⠒⠒⠈⠕⠀ | |

21C1 | ⇁ | Rightwards harpoon w barb downwards | ⠀⠫⠒⠒⠠⠕⠀ | |

21C2 | ⇂ | Downwards harpoon w barb rightwards | ⠀⠫⠩⠒⠒⠈⠕⠀ | |

21C3 | ⇃ | Downwards harpoon w barb leftwards | ⠀⠫⠩⠒⠒⠠⠕⠀ | |

21C4 | ⇄ | Rightwards arrow over leftwards arrow | ⠀⠫⠒⠒⠕⠫⠪⠒⠒⠀ | |

21C5 | ⇅ | Upwards arrow leftwards
of downwards arrow |
⠀⠫⠣⠒⠒⠕⠐⠫⠩⠒⠒⠕⠀ | |

21C6 | ⇆ | Leftwards arrow over rightwards arrow | ⠀⠫⠪⠒⠒⠫⠒⠒⠕⠀ | |

21C7 | ⇇ | Leftwards paired arrows | ⠀⠫⠪⠒⠒⠫⠪⠒⠒⠀ | |

21C8 | ⇈ | Upwards paired arrows | ⠀⠫⠣⠒⠒⠕⠐⠫⠣⠒⠒⠕⠀ | |

21C9 | ⇉ | Rightwards paired arrows | ⠀⠫⠒⠒⠕⠫⠒⠒⠕⠀ | |

21CA | ⇊ | Downwards paired arrows | ⠀⠫⠩⠒⠒⠕⠐⠫⠩⠒⠒⠕⠀ | |

21CB | ⇋ | Leftwards harpoon over rightwards
harpoon |
⠀⠫⠈⠪⠒⠒⠫⠒⠒⠠⠕⠀ | |

21CC | ⇌ | Rightwards harpoon over leftwards
harpoon |
⠀⠫⠒⠒⠈⠕⠫⠠⠪⠒⠒⠀ | |

21CD | ⇍ | Leftwards double arrow w stroke | ⠀⠳⠈⠫⠪⠶⠶⠻⠀ | |

21CE | ⇎ | Left right double arrow w stroke | ⠀⠳⠈⠫⠪⠶⠶⠕⠻⠀ | |

21CF | ⇏ | Rightwards double arrow w stroke | ⠀⠳⠈⠫⠶⠶⠕⠻⠀ | |

21D0 | ⇐ | Leftwards double arrow | ⠀⠫⠪⠶⠶⠀ | ⠰⠳⠶⠶⠪ |

21D1 | ⇑ | Upwards double arrow | ⠀⠫⠣⠶⠶⠕⠀ | ⠰⠳⠶⠶⠬ |

21D2 | ⇒ | Rightwards double arrow | ⠀⠫⠶⠶⠕⠀ | ⠰⠳⠶⠶⠕ |

21D3 | ⇓ | Downwards double arrow | ⠀⠫⠩⠶⠶⠕⠀ | ⠰⠳⠶⠶⠩ |

21D4 | ⇔ | Left right double arrow | ⠀⠫⠪⠶⠶⠕⠀ | |

21D5 | ⇕ | Up down double arrow | ⠀⠫⠣⠪⠶⠶⠕⠀ | |

21D6 | ⇖ | North west double arrow | ⠀⠫⠘⠪⠶⠶⠀ | |

21D7 | ⇗ | North east double arrow | ⠀⠫⠘⠶⠶⠕⠀ | |

21D8 | ⇘ | South east double arrow | ⠀⠫⠰⠶⠶⠕⠀ | |

21D9 | ⇙ | South west double arrow | ⠀⠫⠰⠪⠶⠶⠀ | |

21DA | ⇚ | Leftwards triple arrow | ⠀⠫⠪⠸⠸⠀ | |

21DB | ⇛ | Rightwards triple arrow | ⠀⠫⠸⠸⠕⠀ | |

21DC | ⇜ | Leftwards squiggle arrow | ⠀⠫⠪⠢⠤⠔⠒⠢⠀ | |

21DD | ⇝ | Rightwards squiggle arrow | ⠀⠫⠢⠤⠔⠒⠢⠕⠀ | |

21DE | ⇞ | Upwards arrow w double stroke | ⠀⠳⠳⠈⠫⠣⠒⠒⠕⠻⠀ | |

21DF | ⇟ | Downwards arrow w double stroke | ⠀⠳⠳⠈⠫⠩⠒⠒⠕⠻⠀ | |

21E0 | ⇠ | Leftwards dashed arrow | ⠀⠫⠪⠒⠀⠒⠀ | |

21E1 | ⇡ | Upwards dashed arrow | ⠀⠫⠣⠒⠀⠒⠕⠀ | |

21E2 | ⇢ | Rightwards dashed arrow | ⠀⠫⠒⠀⠒⠕⠀ | |

21E3 | ⇣ | Downwards dashed arrow | ⠀⠫⠩⠒⠀⠒⠕⠀ | |

21E4 | ⇤ | Leftwards arrow to bar | ⠀⠳⠫⠪⠒⠒⠀ | |

21E5 | ⇥ | Rightwards arrow to bar | ⠀⠫⠒⠒⠕⠳⠀ | |

21E6 | ⇦ | Leftwards white arrow | ||

21E7 | ⇧ | Upwards white arrow | ||

21E8 | ⇨ | Rightwards white arrow | ||

21E9 | ⇩ | Downwards white arrow | ||

21F3 | ⇳ | Up down white arrow | ||

21F4 | ⇴ | Right arrow w small circle | ⠀⠨⠡⠈⠫⠒⠒⠕⠻⠀ | |

21F5 | ⇵ | Downwards arrow leftwards
of upwards arrow |
⠀⠫⠩⠒⠒⠕⠐⠫⠣⠒⠒⠕⠀ | |

21F6 | ⇶ | Three rightwards arrows | ⠀⠫⠒⠒⠕⠫⠒⠒⠕⠫⠒⠒⠕⠀ | |

21F7 | ⇷ | Leftwards arrow w vertical stroke | ⠀⠳⠈⠫⠪⠒⠒⠻⠀ | |

21F8 | ⇸ | Rightwards arrow w vertical stroke | ⠀⠳⠈⠫⠒⠒⠕⠻⠀ | |

21F9 | ⇹ | Left right arrow w vertical stroke | ⠀⠳⠈⠫⠪⠒⠒⠕⠻⠀ | |

21FA | ⇺ | Leftwards arrow w double vertical stroke | ⠀⠳⠳⠈⠫⠪⠒⠒⠻⠀ | |

21FB | ⇻ | Rightwards arrow w double vertical stroke | ⠀⠳⠳⠈⠫⠒⠒⠕⠻⠀ | |

21FC | ⇼ | Left right arrow w double vertical stroke | ⠀⠳⠳⠈⠫⠪⠒⠒⠕⠻⠀ | |

21FD | ⇽ | Leftwards open-headed arrow | ||

21FE | ⇾ | Rightwards open-headed arrow | ||

21FF | ⇿ | Left right open-headed arrow | ||

2200 | ∀ | For all | ⠈⠯ | ⠘⠁ |

2202 | ∂ | Partial differential | ⠈⠙ | ⠈⠙ |

2203 | ∃ | There exists | ⠈⠿ | ⠘⠢ |

2204 | ∄ | There does not exist | ⠌⠈⠿ | |

2205 | ∅ | Empty set | ⠸⠴ | ⠈⠚ |

2207 | ∇ | Nabla | ⠨⠫ | ⠘⠙ |

2208 | ∈ | Element of | ⠀⠈⠑⠀ | ⠘⠑ |

2209 | ∉ | Not an element of | ⠀⠌⠈⠑⠀ | |

220B | ∋ | Contains as member | ⠀⠈⠢⠀ | ⠈⠘⠑ |

220C | ∌ | Does not contain as member | ⠀⠌⠈⠢⠀ | |

220E | ∎ | End of proof | ⠸⠳ | |

2211 | ∑ | N-ary summation | ⠨⠠⠎ | |

2212 | − | Minus sign | ⠤ | ⠐⠤ |

2213 | ∓ | Minus-or-plus sign | ⠤⠬ | ⠸⠤ |

2214 | ∔ | Dot plus | ⠐⠬⠣⠡⠻ | |

2215 | ∕ | Division slash | ⠸⠌ | |

2217 | ∗ | Asterisk operator | ⠈⠼ | |

2218 | ∘ | Ring operator | ⠨⠡ | ⠐⠴ |

221A | √ | Square root | ⠜ | ⠐⠩ |

221B | ∛ | Cube root | ⠣⠒⠜ | |

221C | ∜ | Fourth root | ⠣⠲⠜ | |

221D | ∝ | Proportional to | ⠀⠸⠿⠀ | ⠸⠐⠶ |

221E | ∞ | Infinity | ⠠⠿ | ⠼⠿ |

221F | ∟ | Right angle | ⠫⠪⠨⠗⠻ | |

2220 | ∠ | Angle | ⠫⠪ | ⠸⠪ |

2221 | ∡ | Measured angle | ⠫⠪⠈⠫⠁⠻ | ⠨⠸⠪ |

2223 | ∣ | Divides | ⠀⠳⠀ | |

2224 | ∤ | Does not divide | ⠀⠌⠳⠀ | |

2225 | ∥ | Parallel to | ⠀⠫⠇⠀ | ⠼⠇ |

2226 | ∦ | Not parallel to | ⠀⠌⠫⠇⠀ | |

2227 | ∧ | Logical AND | ⠈⠩ | ⠈⠦ |

2228 | ∨ | Logical OR | ⠈⠬ | ⠈⠖ |

2229 | ∩ | Intersection | ⠨⠩ | ⠨⠦ |

222A | ∪ | Union | ⠨⠬ | ⠨⠖ |

222B | ∫ | Integral | ⠮ | ⠮ |

222C | ∬ | Double integral | ⠮⠮ | ⠮⠮ |

222D | ∭ | Triple integral | ⠮⠮⠮ | ⠮⠮⠮ |

222E | ∮ | Contour integral | ⠮⠈⠫⠉⠻ | ⠈⠮ |

2232 | ∲ | Clockwise contour integral ???
§156 has half arcs |
⠮⠈⠫⠪⠢⠔⠻ | |

2233 | ∳ | Anticlockwise contour integral ??? | ⠮⠈⠫⠢⠔⠕⠻ | |

2234 | ∴ | Therefore | ⠀⠠⠡⠀ | ⠠⠡ |

2235 | ∵ | Because | ⠀⠈⠌⠀ | ⠈⠌ |

2236 | ∶ | Ratio | ⠀⠐⠂⠀ | ⠒ |

2237 | ∷ | Proportion | ⠀⠰⠆⠀ | ⠒⠒ |

2238 | ∸ | Dot minus | ⠨⠤ | |

2239 | ∹ | Excess | ⠀⠤⠐⠂⠀ | |

223A | ∺ | Geometric proportion | ⠀⠐⠤⠩⠡⠡⠣⠡⠡⠻⠀ | |

223B | ∻ | Homothetic | ⠀⠐⠈⠱⠩⠡⠣⠡⠻⠀ | |

223C | ∼ | Tilde operator | ⠀⠈⠱⠀ | |

223D | ∽ | Reversed tilde | ⠀⠠⠱⠀ | |

2242 | ≂ | Minus tilde | ⠀⠱⠈⠱⠀ | |

2243 | ≃ | Asymptotically equal to | ⠀⠈⠱⠱⠀ | ⠸⠔ |

2245 | ≅ | Approximately equal to | ⠀⠈⠱⠨⠅⠀ | ⠐⠸⠔ |

2248 | ≈ | Almost equal to | ⠀⠈⠱⠈⠱⠀ | ⠘⠔ |

224A | ≊ | Almost equal or equal to | ⠀⠈⠱⠈⠱⠱⠀ | |

224B | ≋ | Triple tilde | ⠀⠈⠱⠈⠱⠈⠱⠀ | |

224C | ≌ | All equal to | ⠀⠠⠱⠨⠅⠀ | |

224D | ≍ | Equivalent to | ⠀⠫⠠⠫⠈⠀ | |

224E | ≎ | Geometrically equivalent to | ⠀⠈⠣⠠⠣⠀ | |

224F | ≏ | Difference between | ⠀⠈⠣⠱⠀ | ⠘⠐⠶ |

2250 | ≐ | Approaches the limit | ⠀⠐⠨⠅⠣⠡⠻⠀ | |

2251 | ≑ | Geometrically equal to | ⠀⠐⠨⠅⠩⠡⠣⠡⠻⠀ | ⠨⠐⠶ |

2254 | ≔ | Colon equals | ⠀⠐⠂⠨⠅⠀ | |

2255 | ≕ | Equals colon | ⠀⠨⠅⠐⠂⠀ | |

2256 | ≖ | Ring in equal to | ⠀⠨⠡⠈⠨⠅⠻⠀ | |

2257 | ≗ | Ring equal to | ⠀⠐⠨⠅⠣⠨⠡⠻⠀ | |

2258 | ≘ | Corresponds to | ⠀⠐⠨⠅⠣⠫⠁⠻⠀ | |

2259 | ≙ | Estimates | ⠀⠐⠨⠅⠣⠸⠣⠻⠀ | |

225A | ≚ | Equiangular to | ⠀⠐⠨⠅⠣⠸⠩⠻⠀ | |

225B | ≛ | Star equals | ⠀⠐⠨⠅⠣⠫⠎⠻⠀ | |

225C | ≜ | Delta equal to | ⠀⠐⠨⠅⠣⠫⠞⠻⠀ | |

225D | ≝ | Equal to by definition | ⠀⠐⠨⠅⠣⠙⠑⠋⠻⠀ | |

225E | ≞ | Measured by | ⠀⠐⠨⠅⠣⠍⠻⠀ | |

225F | ≟ | Questioned equal to | ⠀⠐⠨⠅⠣⠸⠢⠻⠀ | |

2260 | ≠ | Not equal to | ⠀⠌⠨⠅⠀ | ⠐⠶⠈⠱ |

2261 | ≡ | Identical to | ⠀⠸⠇⠀ | ⠸⠿ |

2262 | ≢ | Not identical to | ⠀⠌⠸⠇⠀ | |

2264 | ≤ | Less-than or equal to | ⠀⠐⠅⠱⠀ | ⠸⠈⠣ |

2265 | ≥ | Greater-than or equal to | ⠀⠨⠂⠱⠀ | ⠸⠈⠜ |

2266 | ≦ | Less-than over equal to | ⠀⠐⠅⠨⠅⠀ | |

2267 | ≧ | Greater-than over equal to | ⠀⠨⠂⠨⠅⠀ | |

2268 | ≨ | Less-than but not equal to | ⠀⠐⠅⠌⠨⠅⠀ | |

2269 | ≩ | Greater-than but not equal to | ⠀⠨⠂⠌⠨⠅⠀ | |

226A | ≪ | Much less-than | ⠀⠐⠅⠈⠐⠅⠻⠀ | ⠨⠈⠣ |

226B | ≫ | Much greater-than | ⠀⠨⠂⠈⠨⠂⠻⠀ | ⠨⠈⠜ |

226D | ≭ | Not equivalent to | ⠀⠌⠈⠣⠠⠣⠀ | |

226E | ≮ | Not less-than | ⠀⠌⠐⠅⠀ | |

226F | ≯ | Not greater-than | ⠀⠌⠨⠂⠀ | |

2270 | ≰ | Neither less-than nor equal to | ⠀⠌⠐⠅⠱⠀ | |

2271 | ≱ | Neither greater-than nor equal to | ⠀⠌⠨⠂⠱⠀ | |

2272 | ≲ | Less-than or equivalent to | ⠀⠐⠅⠈⠱⠀ | |

2273 | ≳ | Greater-than or equivalent to | ⠀⠨⠂⠈⠱⠀ | |

2274 | ≴ | Neither less-than nor equivalent to | ⠀⠌⠐⠅⠈⠱⠀ | |

2275 | ≵ | Neither greater-than nor equivalent to | ⠀⠌⠨⠂⠈⠱⠀ | |

2276 | ≶ | Less-than or greater-than | ⠀⠐⠅⠨⠂⠀ | |

2277 | ≷ | Greater-than or less-than | ⠀⠨⠂⠐⠅⠀ | |

2278 | ≸ | Neither less-than nor greater-than | ⠀⠌⠐⠅⠨⠂⠀ | |

2279 | ≹ | Neither greater-than nor less-than | ⠀⠌⠨⠂⠐⠅⠀ | |

227A | ≺ | Precedes | ⠀⠨⠐⠅⠀ | |

227B | ≻ | Succeeds | ⠀⠨⠨⠂⠀ | |

227E | ≾ | Precedes or equivalent to | ⠀⠨⠐⠅⠈⠱⠀ | |

227F | ≿ | Succeeds or equivalent to | ⠀⠨⠨⠂⠈⠱⠀ | |

2282 | ⊂ | Subset of | ⠀⠸⠐⠅⠀ | ⠘⠣ |

2283 | ⊃ | Superset of | ⠀⠸⠨⠂⠀ | ⠘⠜ |

2284 | ⊄ | Not a subset of | ⠀⠌⠸⠐⠅⠀ | |

2285 | ⊅ | Not a superset of | ⠀⠌⠸⠨⠂⠀ | |

2286 | ⊆ | Subset of or equal to | ⠀⠸⠐⠅⠱⠀ | ⠸⠘⠣ |

2287 | ⊇ | Superset of or equal to | ⠀⠸⠨⠂⠱⠀ | ⠸⠘⠜ |

2288 | ⊈ | Neither a subset of nor equal to | ⠀⠌⠸⠐⠅⠱⠀ | |

2289 | ⊉ | Neither a superset of nor equal to | ⠀⠌⠸⠨⠂⠱⠀ | |

228A | ⊊ | Subset of w not equal to | ⠀⠸⠐⠅⠌⠱⠀ | ⠨⠘⠣ |

228B | ⊋ | Superset of w not equal to | ⠀⠸⠨⠂⠌⠱⠀ | ⠨⠘⠜ |

228C | ⊌ | Multiset | ⠨⠬⠈⠫⠪⠒⠻ | |

228D | ⊍ | Multiset multiplication | ⠡⠈⠨⠬⠻ | |

228E | ⊎ | Multiset union | ⠬⠈⠨⠬⠻ | |

2295 | ⊕ | Circled plus | ⠫⠉⠸⠫⠬⠻ | ⠰⠫⠿⠪⠐⠖⠱ |

2296 | ⊖ | Circled minus | ⠫⠉⠸⠫⠤⠻ | |

2297 | ⊗ | Circled times | ⠫⠉⠸⠫⠈⠡⠻ | |

2298 | ⊘ | Circled division slash | ⠫⠉⠸⠫⠸⠌⠻ | |

2299 | ⊙ | Circled dot operator | ⠫⠉⠸⠫⠡⠻ | |

229B | ⊛ | Circled asterisk operator | ⠫⠉⠸⠫⠈⠼⠻ | |

229C | ⊜ | Circled equals | ⠫⠉⠸⠫⠨⠅⠻ | |

229D | ⊝ | Circled dash | ⠫⠉⠸⠫⠤⠤⠻ | |

229E | ⊞ | Squared plus | ⠫⠲⠸⠫⠬⠻ | |

229F | ⊟ | Squared minus | ⠫⠲⠸⠫⠤⠻ | |

22A0 | ⊠ | Squared times | ⠫⠲⠸⠫⠈⠡⠻ | |

22A1 | ⊡ | Squared dot operator | ⠫⠲⠸⠫⠡⠻ | |

22A2 | ⊢ | Right tack | ⠀⠫⠳⠒⠀ | |

22A3 | ⊣ | Left tack | ⠀⠫⠒⠳⠀ | ⠈⠸⠒ |

22A4 | ⊤ | Down tack | ⠀⠫⠩⠳⠒⠀ | |

22A5 | ⊥ | Up tack (See also 27C2) | ⠀⠫⠏⠀ or ⠀⠫⠣⠳⠒⠀ | ⠼⠤ |

22A6 | ⊦ | Assertion | ⠸⠒ | |

22A8 | ⊨ | True | ⠘⠸⠒ | |

22B2 | ⊲ | Normal subgroup of | ⠈⠸⠣ | |

22B3 | ⊳ | Contains as normal subgroup | ⠈⠸⠜ | |

22B4 | ⊴ | Normal subgroup of or equal to | ⠸⠸⠣ | |

22B5 | ⊵ | Contains as normal subgroup or equal to | ⠸⠸⠜ | |

22B6 | ⊶ | Original of | ⠀⠫⠨⠡⠒⠡⠀ | |

22B7 | ⊷ | Image of | ⠀⠫⠡⠒⠨⠡⠀ | |

22B8 | ⊸ | Multimap | ⠀⠫⠒⠨⠡⠀ | |

22BE | ⊾ | Right angle w arc | ⠼⠸⠪ | |

22BF | ⊿ | Right triangle | ⠫⠞⠨⠗⠻ | |

22C5 | ⋅ | Dot operator | ⠡ | ⠐⠲ |

22C6 | ⋆ | Star operator | ⠫⠎ | |

22CD | ⋍ | Reversed tilde equals | ⠀⠠⠱⠱⠀ | |

22D6 | ⋖ | Less-than w dot | ⠀⠡⠈⠐⠅⠻⠀ | |

22D7 | ⋗ | Greater-than w dot | ⠀⠡⠈⠨⠂⠻⠀ | |

22D8 | ⋘ | Much less-than | ⠀⠐⠅⠈⠐⠅⠈⠐⠅⠻⠀ | |

22D9 | ⋙ | Much greater-than | ⠀⠨⠂⠈⠨⠂⠈⠨⠂⠻⠀ | |

22DC | ⋜ | Equal to or less-than | ⠀⠱⠐⠅⠀ | |

22DD | ⋝ | Equal to or greater-than | ⠀⠱⠨⠂⠀ | |

22EE | ⋮ | Vertical ellipsis | ⠀⠩⠒⠒⠒⠀ | |

22EF | ⋯ | Midline horizontal ellipsis | ⠀⠒⠒⠒ | |

22F0 | ⋰ | Up right diagonal ellipsis | ⠀⠘⠒⠒⠒⠀ | |

22F1 | ⋱ | Down right diagonal ellipsis | ⠀⠰⠒⠒⠒⠀ | |

22F2 | ⋲ | Element of w long horizontal stroke | ⠀⠱⠈⠈⠑⠻⠀ | |

22F5 | ⋵ | Element of w dot above | ⠀⠐⠈⠑⠣⠡⠻⠀ | |

22F6 | ⋶ | Element of w overbar | ⠀⠱⠈⠑⠀ | |

22FA | ⋺ | Contains w long horizontal stroke | ⠀⠱⠈⠈⠢⠻⠀ | |

22FD | ⋽ | Contains w overbar | ⠀⠱⠈⠢⠀ | |

2308 | ⌈ | Left ceiling | ⠈⠘⠷ | |

2309 | ⌉ | Right ceiling | ⠈⠘⠾ | |

230A | ⌊ | Left floor | ⠈⠰⠷ | |

230B | ⌋ | Right floor | ⠈⠰⠾ | |

2322 | ⌢ | Frown | ⠀⠫⠁⠀ | |

2323 | ⌣ | Smile | ⠀⠫⠄⠀ | |

25A0 | ■ | Filled square | ⠫⠸⠲ | |

25A1 | □ | Square | ⠫⠲ | |

25AC | ▬ | Filled rectangle | ⠫⠸⠗ | |

25AD | ▭ | Rectangle | ⠫⠗ | |

25B2 | ▲ | Black up-pointing triangle | ⠫⠸⠞ | |

25B3 | △ | White up-pointing triangle | ⠫⠞ | |

25BC | ▼ | Black down-pointing triangle | ⠸⠨⠫ | |

25CB | ○ | White circle | ⠫⠉ | ⠿ |

25CF | ● | Black circle | ⠫⠸⠉ | |

25EB | ◫ | White square w vertical bisecting line | ⠫⠲⠸⠫⠳⠻ | |

27C2 | ⊥ | Perpendicular | ⠀⠫⠏⠀ | |

27C3 | ⟃ | Open subset | ⠀⠨⠡⠈⠸⠐⠅⠻⠀ | |

27C4 | ⟄ | Open superset | ⠀⠨⠡⠈⠸⠨⠂⠻⠀ | |

27DC | ⟜ | Left multimap | ⠀⠫⠨⠡⠒⠒⠀ | |

27DD | ⟝ | Long right tack | ⠀⠫⠳⠒⠒⠀ | |

27DE | ⟞ | Long left tack | ⠀⠫⠒⠒⠳⠀ | |

27DF | ⟟ | Up tack w circle above | ⠫⠣⠳⠒⠒⠨⠡ | |

27E6 | ⟦ | Mathematical left white square bracket | ⠈⠸⠷ | |

27E7 | ⟧ | Mathematical right white square bracket | ⠈⠸⠾ | |

27E8 | ⟨ | Mathematical left angle bracket | ⠨⠨⠷ | |

27E9 | ⟩ | Mathematical right angle bracket | ⠨⠨⠾ | |

27F5 | ⟵ | Long leftwards arrow | ⠀⠫⠪⠒⠒⠒⠀ | |

27F6 | ⟶ | Long rightwards arrow | ⠀⠫⠒⠒⠒⠕⠀ | |

27F7 | ⟷ | Long left right arrow | ⠀⠫⠪⠒⠒⠒⠕⠀ | |

27F8 | ⟸ | Long leftwards arrow | ⠀⠫⠪⠶⠶⠶⠀ | |

27F9 | ⟹ | Long rightwards arrow | ⠀⠫⠶⠶⠶⠕⠀ | |

27FA | ⟺ | Long left right arrow | ⠀⠫⠪⠶⠶⠶⠕⠀ | |

27FB | ⟻ | Long leftwards arrow from bar | ⠀⠫⠪⠒⠒⠒⠳⠀ | |

27FC | ⟼ | Long rightwards arrow from bar | ⠀⠫⠳⠒⠒⠒⠕⠀ | |

27FD | ⟽ | Long leftwards double arrow from bar | ⠀⠫⠪⠶⠶⠶⠳⠀ | |

27FE | ⟾ | Long rightwards double arrow from bar | ⠀⠫⠳⠶⠶⠶⠕⠀ | |

27FF | ⟿ | Long rightwards squiggle arrow | ⠀⠫⠢⠤⠔⠒⠢⠤⠔⠒⠢⠕⠀ | |

2921 | ⤡ | North west and south east arrow | ⠀⠫⠘⠪⠒⠒⠕⠀ | |

2922 | ⤢ | North east and south west arrow | ⠀⠫⠰⠪⠒⠒⠕⠀ | |

2983 | ⦃ | Left white curly bracket | ⠨⠸⠷ | |

2984 | ⦄ | Right white curly bracket | ⠨⠸⠾ | |

2991 | ⦑ | Left angle bracket w dot | ⠡⠈⠨⠨⠷⠻ | |

2992 | ⦒ | Right angle bracket w dot | ⠡⠈⠨⠨⠾⠻ | |

29C4 | ⧄ | Square rising diagonal slash | ⠫⠲⠸⠫⠔⠻ | |

29C5 | ⧅ | Square falling diagonal slash | ⠫⠲⠸⠫⠢⠻ | |

29C6 | ⧆ | Squared asterisk | ⠫⠲⠸⠫⠈⠼⠻ | |

29CA | ⧊ | Triangle w dot above | ⠐⠫⠞⠣⠡⠻ | |

29CB | ⧋ | Triangle w underbar | ⠫⠞⠱ | |

29CC | ⧌ | s in triangle §114 | ⠫⠞⠎ | |

29E6 | ⧦ | Gleich stark | ⠫⠳⠶⠶⠳ | |

2A0C | ⨌ | Quadruple integral operator | ⠮⠮⠮⠮ | ⠮⠮⠮⠮ |

2A0D | ⨍ | Finite part integral | ⠮⠈⠱⠻ | |

2A0E | ⨎ | Integral w double stroke | ⠮⠈⠱⠱⠻ | |

2A16 | ⨖ | Quaternion integral operator | ⠮⠈⠫⠲⠻ | |

2A18 | ⨘ | Integral w times sign | ⠮⠈⠈⠡⠻ | |

2A19 | ⨙ | Integral w intersection | ⠮⠈⠨⠩⠻ | |

2A1A | ⨚ | Integral w union | ⠮⠈⠨⠬⠻ | |

2A1B | ⨛ | Integral w overbar (upper) | ⠣⠮ | |

2A1C | ⨜ | Integral w underbar (lower) | ⠩⠮ | |

2A22 | ⨢ | Plus sign w small circle above | ⠐⠬⠣⠨⠡⠻ | |

2A23 | ⨣ | Plus sign w circumflex accent above | ⠐⠬⠣⠸⠣⠻ | |

2A24 | ⨤ | Plus sign w tilde above | ⠐⠬⠣⠈⠱⠻ | |

2A25 | ⨥ | Plus sign w dot below | ⠐⠬⠩⠡⠻ | |

2A2A | ⨪ | Minus sign w dot below | ⠐⠱⠩⠡⠻ | |

2A30 | ⨰ | Multiplication sign w dot above | ⠐⠈⠡⠣⠡⠻ | |

2A31 | ⨱ | Multiplication sign w underbar | ⠈⠡⠱ | |

2A38 | ⨸ | Circled division sign | ⠫⠉⠸⠫⠨⠌⠻ | |

2A39 | ⨹ | Plus sign in triangle | ⠫⠞⠸⠫⠬⠻ | |

2A3A | ⨺ | Minus sign in triangle | ⠫⠞⠸⠫⠤⠻ | |

2A3B | ⨻ | Multiplication sign in triangle | ⠫⠞⠸⠫⠈⠡⠻ | |

2A40 | ⩀ | Intersection w dot | ⠡⠈⠨⠩⠻ | |

2A41 | ⩁ | Union w minus sign | ⠤⠈⠨⠬⠻ | |

2A42 | ⩂ | Union w overbar | ⠱⠨⠬ | |

2A43 | ⩃ | Intersection w overbar | ⠱⠨⠩ | |

2A51 | ⩑ | Logical AND w dot above | ⠐⠈⠩⠣⠡⠻ | |

2A52 | ⩒ | Logical OR w dot above | ⠐⠈⠬⠣⠡⠻ | |

2A5C | ⩜ | Logical AND with horizontal dash | ⠱⠈⠈⠩⠻ | |

2A5D | ⩝ | Logical OR with horizontal dash | ⠱⠈⠈⠬⠻ | |

2A5E | ⩞ | Logical AND w double overbar | ⠱⠱⠈⠩ | |

2A5F | ⩟ | Logical AND w underbar | ⠈⠩⠱ | |

2A60 | ⩠ | Logical AND w double underbar | ⠈⠩⠱⠱ | |

2A62 | ⩢ | Logical OR w double overbar | ⠱⠱⠈⠬ | |

2A63 | ⩣ | Logical OR w double underbar | ⠈⠬⠱⠱ | |

2A66 | ⩦ | Equals sign w dot below | ⠀⠐⠨⠅⠩⠡⠻⠀ | |

2A67 | ⩧ | Identical w dot above | ⠀⠐⠸⠇⠣⠡⠻⠀ | |

2A6A | ⩪ | Tilde operator w dot above | ⠀⠐⠈⠱⠣⠡⠻⠀ | |

2A6C | ⩬ | Similar minus similar | ⠀⠈⠱⠱⠈⠱⠀ | |

2A6D | ⩭ | Congruent w dot above | ⠀⠐⠈⠱⠨⠅⠣⠡⠻⠀ | |

2A6E | ⩮ | Equals w asterisk | ⠀⠐⠨⠅⠣⠈⠼⠻⠀ | |

2A6F | ⩯ | Almost equal to w circumflex | ⠀⠐⠈⠱⠈⠱⠣⠸⠣⠻⠀ | |

2A70 | ⩰ | Approximately equal or equal to | ⠀⠈⠱⠈⠱⠨⠅⠀ | |

2A71 | ⩱ | Equals sign above plus sign | ⠀⠨⠅⠬⠀ | |

2A72 | ⩲ | Plus sign above equals sign | ⠀⠬⠨⠅⠀ | |

2A73 | ⩳ | Equals sign above tilde operator | ⠀⠨⠅⠈⠱⠀ | |

2A74 | ⩴ | Double colon equal | ⠀⠐⠂⠐⠂⠨⠅⠀ | |

2A75 | ⩵ | Two consecutive equals signs | ⠀⠨⠅⠨⠅⠀ | |

2A76 | ⩶ | Three consecutive equals signs | ⠀⠨⠅⠨⠅⠨⠅⠀ | |

2A77 | ⩷ | Equals sign w two dots above and two dots below | ⠀⠐⠨⠅⠩⠡⠡⠣⠡⠡⠻⠀ | |

2A78 | ⩸ | Equivalent w four dots above | ⠀⠐⠸⠇⠣⠡⠡⠡⠡⠻⠀ | |

2A79 | ⩹ | Less than w circle inside | ⠀⠐⠅⠈⠨⠡⠻⠀ | |

2A7A | ⩺ | Greater than w circle inside | ⠀⠨⠂⠈⠨⠡⠻⠀ | |

2A7B | ⩻ | Less-than w question mark above | ⠀⠐⠐⠅⠣⠸⠦⠻⠀ | |

2A7C | ⩼ | Greater-than w question mark above | ⠀⠐⠨⠂⠣⠸⠦⠻⠀ | |

2A85 | ⪅ | Less-than or approximate | ⠀⠐⠅⠈⠱⠈⠱⠀ | |

2A86 | ⪆ | Greater-than or approximate | ⠀⠨⠂⠈⠱⠈⠱⠀ | |

2A8B | ⪋ | Less-than above double-line equal above greater-than | ⠀⠐⠅⠨⠅⠨⠂⠀ | |

2A8C | ⪌ | Greater-than above double-line equal above less-than | ⠀⠨⠂⠨⠅⠐⠅⠀ | |

2A8D | ⪍ | Less-than above similar or equal to | ⠀⠐⠅⠈⠱⠱⠀ | |

2A8E | ⪎ | Greater-than above similar or equal to | ⠀⠨⠂⠈⠱⠱⠀ | |

2A8F | ⪏ | Less-than above similar above greater-than | ⠀⠐⠅⠈⠱⠨⠂⠀ | |

2A90 | ⪐ | Greater-than above similar above less-than | ⠀⠨⠂⠈⠱⠐⠅⠀ | |

2A91 | ⪑ | Less-than above greater-than above double-line equal | ⠀⠐⠅⠨⠂⠨⠅⠀ | |

2A92 | ⪒ | Greater-than above less-than above double-line equal | ⠀⠨⠂⠐⠅⠨⠅⠀ | |

2A99 | ⪙ | Double-line equal to or less-than | ⠀⠨⠅⠐⠅⠀ | |

2A9A | ⪚ | Double-line equal to or greater-than | ⠀⠨⠅⠨⠂⠀ | |

2A9D | ⪝ | Similar or less-than | ⠀⠈⠱⠐⠅⠀ | |

2A9E | ⪞ | Similar or greater-than | ⠀⠈⠱⠨⠂⠀ | |

2A9F | ⪟ | Similar above less-than above equals sign | ⠀⠈⠱⠐⠅⠨⠅⠀ | |

2AA0 | ⪠ | Similar above greater-than above equals sign | ⠀⠈⠱⠨⠂⠨⠅⠀ | |

2AAE | ⪮ | Equals sign with bumpy above | ⠀⠈⠣⠨⠅⠀ | |

2AAF | ⪯ | Precedes above single-line equals sign | ⠀⠨⠐⠅⠱⠀ | |

2AB0 | ⪰ | Succeeds above single-line equals sign | ⠀⠨⠨⠂⠱⠀ | |

2AB1 | ⪱ | Precedes above single-line not equal to | ⠀⠨⠐⠅⠌⠱⠀ | |

2AB2 | ⪲ | Succeeds above single-line not equal to | ⠀⠨⠨⠂⠌⠱⠀ | |

2AB3 | ⪳ | Precedes above equals sign | ⠀⠨⠐⠅⠨⠅⠀ | |

2AB4 | ⪴ | Succeeds above equals sign | ⠀⠨⠨⠂⠨⠅⠀ | |

2AB5 | ⪵ | Precedes above not equal to | ⠀⠨⠐⠅⠌⠨⠅⠀ | |

2AB6 | ⪶ | Succeeds above not equal to | ⠀⠨⠨⠂⠌⠨⠅⠀ | |

2AB7 | ⪷ | Precedes above almost equals to | ⠀⠨⠐⠅⠈⠱⠈⠱⠀ | |

2AB8 | ⪸ | Succeeds above almost equals to | ⠀⠨⠨⠂⠈⠱⠈⠱⠀ | |

2AB9 | ⪹ | Precedes above not almost equals to | ⠀⠨⠐⠅⠌⠈⠱⠈⠱⠀ | |

2ABA | ⪺ | Succeeds above not almost equals to | ⠀⠨⠨⠂⠌⠈⠱⠈⠱⠀ | |

2ABB | ⪻ | Double precedes | ⠀⠨⠐⠅⠈⠨⠐⠅⠻⠀ | |

2ABC | ⪼ | Double succeeds | ⠀⠨⠨⠂⠈⠨⠨⠂⠻⠀ | |

2ABD | ⪽ | Subset w dot | ⠀⠡⠈⠸⠐⠅⠻⠀ | |

2ABE | ⪾ | Superset w dot | ⠀⠡⠈⠸⠨⠂⠻⠀ | |

2ABF | ⪿ | Subset with plus sign below | ⠀⠐⠸⠐⠅⠩⠬⠻⠀ | |

2AC0 | ⫀ | Superset with plus sign below | ⠀⠐⠸⠨⠂⠩⠬⠻⠀ | |

2AC1 | ⫁ | Subset with multiplication sign below | ⠀⠐⠸⠐⠅⠩⠈⠡⠻⠀ | |

2AC2 | ⫂ | Superset with multiplication sign below | ⠀⠐⠸⠨⠂⠩⠈⠡⠻⠀ | |

2AC5 | ⫅ | Subset of above equals sign | ⠀⠸⠐⠅⠨⠅⠀ | |

2AC6 | ⫆ | Superset of above equals sign | ⠀⠸⠨⠂⠨⠅⠀ | |

2AC7 | ⫇ | Subset of above tilde operator | ⠀⠸⠐⠅⠈⠱⠀ | |

2AC8 | ⫈ | Superset of above tilde operator | ⠀⠸⠨⠂⠈⠱⠀ | |

2AC9 | ⫉ | Subset of above almost equal to | ⠀⠸⠐⠅⠈⠱⠈⠱⠀ | |

2ACA | ⫊ | Superset of above almost equal to | ⠀⠸⠨⠂⠈⠱⠈⠱⠀ | |

2ACB | ⫋ | Subset of above not equals sign | ⠀⠸⠐⠅⠌⠨⠅⠀ | |

2ACC | ⫌ | Superset of above not equals sign | ⠀⠸⠨⠂⠌⠨⠅⠀ | |

2AD3 | ⫓ | Subset above superset | ⠀⠸⠐⠅⠸⠨⠂⠀ | |

2AD4 | ⫔ | Superset above subset | ⠀⠸⠨⠂⠸⠐⠅⠀ | |

2AD5 | ⫕ | Subset above subset | ⠀⠸⠐⠅⠸⠐⠅⠀ | |

2AD6 | ⫖ | Superset above superset | ⠀⠸⠨⠂⠸⠨⠂⠀ | |

2AD7 | ⫗ | Superset beside subset | ⠀⠸⠨⠂⠐⠸⠐⠅⠀ | |

2AEF | ⫯ | Vertical line w circle above | ⠀⠫⠩⠨⠡⠒⠒⠀ | |

2AF0 | ⫰ | Vertical line w circle below | ⠀⠫⠣⠨⠡⠒⠒⠀ | |

2AF1 | ⫱ | Down tack w circle below | ⠀⠫⠣⠨⠡⠒⠒⠳⠀ | |

2AF2 | ⫲ | Parallel w horizontal stroke | ⠀⠱⠈⠫⠇⠻⠀ | |

2AF3 | ⫳ | Parallel w tilde operator | ⠀⠈⠱⠈⠫⠇⠻⠀ | |

2AF4 | ⫴ | Triple vertical bar binary relation | ⠀⠳⠳⠳⠀ | |

2B1F | ⬟ | Black pentagon | ⠫⠸⠢ | |

2B20 | ⬠ | White pentagon | ⠫⠢ | |

2B21 | ⬡ | White hexagon | ⠫⠖ | |

2B22 | ⬢ | Black hexagon | ⠫⠸⠖ | |

2B2C | ⬬ | Black ellipse | ⠫⠸⠑ | |

2B2D | ⬭ | White ellipse | ⠫⠑ | |

2B30 | ⬰ | Left arrow w small circle | ⠀⠨⠡⠈⠫⠪⠒⠒⠻⠀ | |

2B31 | ⬱ | Three leftwards arrows | ⠀⠫⠪⠒⠒⠫⠪⠒⠒⠫⠪⠒⠒⠀ | |

2B32 | ⬲ | Left arrow w circled plus | ⠀⠫⠪⠒⠒⠈⠫⠉⠸⠫⠬⠻⠀ | |

2BC3 | ⯃ | Horizontal black octagon | ⠫⠸⠦ |

In general, Unicode’s math characters are simpler to work with than the braille sequences since they are assigned separate character codes instead of being composed as sequences of 64 braille codes. Unicode has about 2310 math characters (see Math property in DerivedCoreProperties.txt) and to distinguish all of those without indicators would require 12-dot braille! Such a system would be hard to learn. LaTeX describes characters using control words consisting of a backslash followed by combinations of the 64 ASCII letters. That approach has mnemonic value, but it’s not as concise as the Nemeth braille character code sequences. When you get a feel for the Nemeth approach, a character’s Nemeth sequence gives a good idea of what a character is even if you haven’t encountered it before. UnicodeMath and Nemeth braille are intended to be read by human beings, whereas LaTeX and MathML are intended to be read by computer programs, notwithstanding that some TeXies can read LaTeX fluently! Nemeth math alphabets not in Unicode probably don’t have to be considered unless they show up in published documents.

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]]>The post Cascadia Code Font appeared first on Math in Office.

]]>The Cascadia Code font is an open-source font developed to improve the readability of program text in plain-text editors such as in the Windows Terminal, Microsoft Visual Studio, and Visual Studio Code. A major restriction in these contexts is fixed-width character spacing. So, in displaying “<=” as ‘≤’, some extra leading and trailing space is added, and the glyph is widened. The font also works as designed in Notepad. It uses 104 ligatures as well as extendable glyph sequences to render character combinations such as shown in the table

characters |
display |
mechanism |

<= | ≤ | ligature |

>= | ≥ | ligature |

!= | ≠ | ligature |

==> | ⇒ | sequence |

<== | ⇐ | sequence |

===> | ⟹ | sequence |

<=== | ⟺ | sequence |

:= | ≔ | ligature |

When you concatenate many equal signs together, you see a continuous horizontal double bar, which is a great way to divide up sections of code. And you can insert ‘<’ and ‘>’ along the way to make super long arrows with or without tails.

Let’s examine how Cascadia Code displays “<=” as ‘≤’. Consider the string “a<b<=c=d”. It is rendered using the glyphs

a | < | b | ≤ | c | = | d | |

00e1 | 0809 | 00fc | 0a72 | 0c2d | 00fd | 0806 | 0103 |

Here the glyph 0a72 is named less_equal.liga and is the complete ≤ ligature. The glyph 0c2d is named LIG and has no ink. LIG is included to preserve the two-character advance width and the two-character count. Many other glyph pairs have LIG as the second “glyph”. The advance widths and left side bearings of the symbols are

glyph ID |
advance width |
left side bearing |
name |
comment |

0806 | 1200 | 100 | equal | |

0809 | 1200 | 96 | less | |

0a72 | 1200 | 506 | less_equal.liga | large left side bearing |

0c2d | 1200 | 0 | LIG | 0 left side bearing |

The sequence ==> is displayed as ⇒ by the glyph sequence 0a93 0a91 0a99, where 0a93 starts the horizontal double bar, 0a91 continues it and 0a93 finishes it with an arrowhead.

Cascadia Code PL includes italic, variable-font weights from 200 to 700, Hebrew, Arabic, braille cells, line drawing characters, and lots more cool transformations. If the font isn’t installed on your computer, you can download it from here. Happy programming!

The post Cascadia Code Font appeared first on Math in Office.

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