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Angles and Arrows and Greek Letters

Angles

In braille, angles can be drawn with raised lines, or they can be represented by an angle sign. The angle sign can be used to describe the angle, just as it is in print. When transcribed into braille, it follows print spacing and formatting.

Angle ∠	⠸⠪
Measured angle sign ∡	⠨⠸⠪
Measured right angle sign ⦝	⠼⠸⠪
Degree °	⠘⠚

Examples:

$\angle 5$

⠸⠪⠀⠼⠑

$\text{Measure}\angle \mathrm{CDE}\text{.}$

⠠⠍⠂⠎⠥⠗⠑⠀⠸⠪⠠⠠⠉⠙⠑⠲

$\measuredangle A={60}^{°}$

⠨⠸⠪⠀⠠⠁⠀⠐⠶⠀⠼⠋⠚⠘⠚

$\measuredangle \mathrm{A}+\measuredangle \mathrm{B}=⦝\mathrm{C}$

⠨⠸⠪⠠⠁⠐⠖⠨⠸⠪⠠⠃⠀⠐⠶⠀⠼⠸⠪⠠⠉

$\angle \mathrm{A}=\angle \mathrm{B}$

⠸⠪⠠⠁⠀⠐⠶⠀⠸⠪⠠⠃

Angles are a circular measurement notated in degrees. When the dot in circle is used in print, a simple circle shape symbol can be used instead.

$⨀={360}^{°}$

⠰⠫⠿⠀⠐⠶⠀⠼⠉⠋⠚⠘⠚

$⨀P<⨀Q$

⠰⠫⠿⠱⠠⠏⠀⠈⠣⠀⠰⠫⠿⠱⠠⠟

$\angle \mathrm{B}\text{\&}\angle \mathrm{D}\text{are corresponding angles}$

⠸⠪⠠⠃⠀⠈⠯⠀⠸⠪⠠⠙⠀⠜⠑⠀⠉⠕⠗⠗⠑⠎⠏⠕⠝⠙⠬⠀⠁⠝⠛⠇⠑⠎

Arrows

In braille, arrows can be drawn with raised lines, or they can be represented by an arrow indicator followed by a directional arrow symbol. Although there are many types of arrows and many types of arrow shafts used for a variety of math expressions, the following are the most common simple arrows used.

Arrow Indicator	⠳
Right pointing arrow $\rightarrow$	⠳⠕
Left pointing arrow $\leftarrow$	⠳⠪
Bidirectional arrow $\leftrightarrow$	⠳⠺⠗⠕
Rightward harpoon arrow ⇀	⠳⠈⠗⠕

A simple arrow has a shaft and a barb (v-tip) on one end. Arrows are used in math to represent concepts such as an approaching value, conditional statements, limits, or in chemistry, transformations. They are comparison signs, so they are preceded and followed by a space. The arrow indicator requires a grade 1 indicator unless the expression is already in uncontracted mode.

Examples:

$A\to B$

⠠⠁⠀⠰⠳⠕⠀⠰⠠⠃

$\rightharpoonup x$

⠰⠳⠈⠗⠕⠀⠰⠭

$a\to b\leftrightarrow c\leftarrow d$

⠰⠰⠰⠁⠀⠳⠕⠀⠃⠀⠳⠺⠗⠕⠀⠉⠀⠳⠪⠀⠙⠰⠄

Note: The above expression uses the grade 1 passage indicator and terminator because it requires so many grade 1 indicators.

$(x^2-1)\leftrightarrow (x+1)(x+1)$

⠐⠣⠭⠰⠔⠼⠃⠐⠤⠼⠁⠐⠜⠀⠰⠳⠺⠗⠕⠀⠐⠣⠭⠐⠖⠼⠁⠐⠜⠐⠣⠭⠐⠖⠼⠁⠐⠜

Note: the grade 1 indicator is needed before the arrow indicator because there is a space directly in front of it, and the default mode is contracted.

Shafts

Straight, short	⠒
Straight, long	⠒⠒⠒
Double, short	⠶
Dotted shaft	⠂⠂⠂

All shafts can be made longer by repeating the shaft symbol. The shaft symbol is placed between the arrow indicator and the arrow tip symbol.

Examples:

⇢

⠰⠳⠂⠂⠕

⟹

⠰⠳⠶⠶⠶⠕

Note: The above example uses a double shaft.

⟶

⠰⠳⠒⠒⠒⠒⠒⠒⠕

Greek Letters

Greek letters are widely used in math and science. The Greek letter indicator is place in front of the corresponding English letter to indicate that a Greek letter is being used.

Greek letter indicator	⠨
Capital letter indicator	⠠

Example:

Letter a: ⠁

alpha: ⠨⠁

A capitalized Greek letter is made by placing the capital letter indicator before the Greek letter indicator.

Capital letter B: ⠠⠃

capital beta: ⠠⠨⠃

The effect of the Greek letter indicator only applies to the letter it precedes and, although print Greek letters are italicized, the italic typeform is omitted in braille.

{a, α, b, β}

⠸⠣⠁⠂⠀⠨⠁⠂⠀⠰⠃⠂⠀⠨⠃⠸⠜

Lowercase Greek Letter			Capital Greek Letter
α	⠨⠁	alpha	Α	⠠⠨⠁
β	⠨⠃	beta	Β	⠠⠨⠃
γ	⠨⠛	gamma	Γ	⠠⠨⠛
δ	⠨⠙	delta	Δ	⠠⠨⠙
ε	⠨⠑	epsilon	Ε	⠠⠨⠑
θ	⠨⠹	theta	Θ	⠠⠨⠹
λ	⠨⠇	lambda	Λ	⠠⠨⠇
μ	⠨⠍	mu	Μ	⠠⠨⠍
ν	⠨⠝	nu	Ν	⠠⠨⠝
π	⠨⠏	pi	Π	⠠⠨⠏
ρ	⠨⠗	rho	Ρ	⠠⠨⠗
σ	⠨⠎	sigma	Σ	⠠⠠⠨⠎
φ	⠨⠋	phi	Φ	⠠⠨⠋
ω	⠨⠺	omega	Ω	⠠⠨⠺

Examples:

$\pi \mathit{rad}={180}^{°}$

⠨⠏⠀⠗⠁⠙⠀⠐⠶⠀⠼⠁⠓⠚⠘⠚

Note: Greek letters are already preceded by an indicator and so a grade 1 indicator is not needed because there is no contracted meaning for the symbol.

$r={\displaystyle \sqrt{\frac{a}{\pi }}}$

⠰⠗⠀⠐⠶⠀⠰⠰⠩⠷⠁⠨⠌⠨⠏⠾⠬

$A=\pi r^2$

⠠⠁⠀⠐⠶⠀⠨⠏⠗⠰⠔⠼⠃

$\phi ={\displaystyle \frac{1+\sqrt{5}}{2}}$

⠨⠋⠀⠐⠶⠀⠰⠷⠼⠁⠐⠖⠩⠼⠑⠬⠨⠌⠼⠃⠾

$m={\displaystyle \frac{\Delta y}{\Delta x}}$

⠰⠍⠀⠐⠶⠀⠰⠷⠠⠨⠙⠽⠨⠌⠠⠨⠙⠭⠰⠾

Note: The capital indicator goes in front of the Greek letter indicator.