As longtime readers of this blog have probably noticed, I like to poke around in Unicode occasionally. It’s an endless system of rabbit holes to explore.
This morning I was looking at the Miscellaneous Mathematical Symbols block. These are mostly obscure symbols, though I’m sure for each symbol that I think is obscure, there is someone out there who uses it routinely.
Perpendicular
The only common symbol in this block is ⟂ (U+27C2) for perpendicular. Even so, this symbol is a variation on ⊥ (U+22A5). The distinction is semantic rather than visual: U+22A5 is used for the Boolean value “false.”
In addition to using ⟂ to denote perpendicular lines, some (e.g. Donald Knuth) use the symbol to denote that two integers are relatively prime.
Geometric algebra
The block contains ⟑ (U+27D1) which is used in geometric algebra for the geometric product, a.k.a. the dot-wedge product. The block also contains the symbol for the dual operator ⟇ (U+27c7), the geometric antiproduct. Incidentally, Eric Lengyel’s Projective Geometric Algebra site officially sponsors these two Unicode symbols.
I’m sure these symbols predate Eric Lengyel’s use of them, but I can only recall seeing them used in his work.
Database joins
Unicode has four symbols for database joins. The bowtie symbol ⨝ (U+2A1D) is used for inner (natural) joins is in another block. The Miscellaneous Mathematical Symbols block has three other symbols for outer joins: left, right, and full. I posted a table of these on @CompSciFact this morning.
— Computer Science (@CompSciFact) April 14, 2025
Angle brackets
The Miscellaneous Mathematical Symbols block also has angle brackets: ⟨ (U+27E8) and ⟩ (U+27E9). These correspond to \langle and \rangle in LaTeX. I’ve used the LaTeX commands, but I wasn’t sure whether I’d ever used the Unicode characters. I searched this blog and found that I did indeed use the characters in my post on the Gram matrix.
A few of these can be entered using the TeX input-method in Emacs.
⟂ is \perp
⨝ is \Join or \bowtie
⟨ and ⟩ are \langle and \rangle
these are easier to remember if you often find yourself in Emacs.