# Math Greek Letters

Greek letters often appear as variables in mathematical text. This post describes their support in [La]TeX and Unicode and how their math usage sometimes differs from modern Greek text. For example, the upper-case upsilon looks like a Latin Y in modern Greek, but a curvy Υ in math zones. Similarly, in math zones the two standard Greek letter glyphs φ and ε are variants of ϕ and ϵ, respectively. The Greek math letters appear in Appendix F (p. 434) of The TeXbook along with the TeX control words that define them. The control words can also be used for inputting the characters in other math applications, such as Microsoft Word, PowerPoint and OneNote (see Appendix B of UnicodeMath).

## Variant Greek letters

Several letters have variant glyphs and both the regular and variant characters can occur in the same document with different meanings. In my papers on laser theory, I’ve used θ for the polar angle in spherical coordinates and ϑ for a complex mode coupling coefficient. The following table shows the Greek letters that have variants

 Letter.. Unicode.. TeX.. Variant.. Unicode.. TeX ϵ 03F5 \epsilon ε 03B5 \varepsilon θ 03B8 \theta ϑ 03D1 \vartheta Θ 0398 \Theta ϴ 03F4 \varTheta π 03C0 \pi ϖ 03D6 \varpi ρ 03C1 \rho ϱ 03F1 \varrho σ 03C3 \sigma ς 03C2 \varsigma ϕ 03D5 \phi φ 03C6 \varphi

## Epsilon and phi

ε and φ are standard letters in the modern Greek alphabet, but they are variants in math text as shown in the table. This is the choice made by TeX back in the early 1980s and the math community depends on it. Up to Version 3.0, the Unicode Standard displayed ϕ for 03C6, but in 3.0 it was changed to conform with modern Greek usage as documented in Section 2.3.1 Representative Glyphs for Greek Phi of Unicode Technical Report #25 Unicode Support for Mathematics. Unicode 3.1 added ϵ (03F5).

## Upsilon

In The TeXbook Appendix F, the upper-case upsilon \Upsilon appears with curvy arms as in the Cambria Math Υ for 03A5. But in the STIX math 2 font, 03A5 appears as an upper-case Latin Y, which isn’t useful for math. Recognizing that fact, Unicode 1.1 encoded 03D2 for the curvy arm glyph ϒ (which unfortunately looks like a Y here; the WordPress blog infrastructure doesn't allow font changes). It has the odd name GREEK UPSILON WITH HOOK SYMBOL. One should not take Unicode names too literally except as being unique identifiers. Until recently, Cambria Math had no glyph at 03D2, but now it’s defined and preferred for math. Admittedly upper-case and lower-case upsilons aren’t commonly used in math, but I anyhow used υ extensively in laser theory to represent complex frequencies such as γ + iω. Here γ is a decay rate and ω is a (real) frequency.

## Unicode math Greek alphabets

TeX only defines upper-case Greek letters that have glyphs distinct from the Latin alphabet, that is, ΓΔΘΛΞΠΣΥΦΨΩ with the notation \Gamma for Γ. Unicode and Office apps support all upper-case Greek letters using the TeX convention that the first letter is upper case as in \Alpha for Α (upper-case alpha) and. For math, Greek letters that look identical to Latin letters aren’t useful, since they’d be interpreted as Latin letters. Lower-case Greek letters are italicized by default, while upper-case Greek letters are upright by default. This convention also applies to ∂ (\partial—italicized) and ∇ (\nabla—not italicized).

[La]TeX has control words to change the math style of math letters, such as \mathbf{…}, which changes the letters … to bold face. Ironically that construct doesn’t affect Greek letters, probably an early omission, but now unchangeable for archival reasons. Unicode chose to encode the various math alphabets explicitly instead of using variation selectors. Section 2.2 of Unicode Technical Report #25 describes the mathematical alphabets in the Unicode Standard, which are contained mostly in the plane-1 range U+1D400..U+1D7FF along with some characters in the Letterlike Symbols (U+2100..U+214F). Here is Table 2.1 listing the various math styles

 Math Style Characters from Basic Set.. Location plain (upright, serifed) Latin, Greek and digits BMP bold Latin, Greek and digits Plane 1 italic Latin and Greek Plane 1* bold italic Latin and Greek Plane 1 script (calligraphic) Latin Plane 1* bold script (calligraphic).. Latin Plane 1 Fraktur Latin Plane 1* bold Fraktur Latin Plane 1 double-struck Latin and digits Plane 1* sans-serif Latin and digits Plane 1 sans-serif bold Latin, Greek and digits Plane 1 sans-serif italic Latin Plane 1 sans-serif bold italic Latin and Greek Plane 1 monospace Latin and digits Plane 1

Here the * notes that some characters of the corresponding alphabet are located in the Unicode Letterlike Symbols block.

There’s no sans serif upright Greek alphabet, but LaTeX can display this math style. LaTeX can also display two script variations as described in the post Unicode Math Calligraphic Alphabets, while Unicode only has one. Accordingly, more Unicode math alphabets may be added in the future.

Nemeth braille can represent all the math Greek alphabets and variants as described in the post Nemeth Braille Alphanumerics and Unicode Math Alphanumerics.