Unicode Math Braille Sequences

Murray

Unicode has a math symbol set of about 2310 symbols that’s described in Section 2 of Unicode Technical Report #25. The present post attempts to give Nemeth math braille sequences for the Unicode math symbol set. Many of the braille symbol sequences are taken directly from the Nemeth specification and the UEB specification. In addition, many sequences have been added following the rules in the Nemeth specification. Unfortunately, a significant number of symbols don’t have obvious braille sequences and the OfficeMath Nemeth braille facility uses the notation \x…. for them where ….. gives the hexadecimal Unicode codepoint. In braille this is ⠡⠭…. The UEB column is missing many more entries, since UEB doesn’t give rules for expressing undefined math symbols in braille. We start with the Unicode math alphanumerics and then give a table with the math symbols.

Unicode math alphanumerics

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.

Mappings

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 ⠠⠠ 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 ⠠⠁

 

Greek Alternative Letters

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.

Russian Letters

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.

Hebrew 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.

Math Digits

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 a1, the numeric indicator and the subscript indicator are omitted. In Nemeth ASCII braille, a1 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 a1, b2, etc., at least for people with a mathematical background.

Unicode math symbols

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 Planck constant over two pi ⠈⠓
2113 Script small l

(differs from 1d4c1: 4-56-123)

⠈⠇
211D Double-struck capital R

(no nemeth double struck)

212B Angstrom sign ⠈⠠⠁
2146 Double-struck italic small d
2147 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 ⠫⠸⠦

Conclusions

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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