Angle brackets are used in various sorts of mathematical expressions: ⟨x, y⟩ can denote an inner product or other such pairing, ⟨a, b | ab = ba2⟩ a presentation of a group, and k⟨X⟩ a free associative algebra. On the typewriter, they are rendered using the greater-than and less-than signs, < and >. Printed versions of angle bracket characters have angles ranging from nearly as sharp as < and > to much flatter, often compressing the symbols to the same widths as parentheses ( and ). Some examples from the literature showing this range of forms are this, this, and this. The LaTeX symbols \langle and \rangle are very flat (as are the html symbols ⟨ and ⟩ used above, at least as they appear on my browser).
Perhaps the tendency to use very flat symbols is based on the goal of emphatically distinguishing these from the inequality symbols. My experience is that occasions where angle brackets could be confused with inequality signs are rare, but that, on the other hand, in rapid reading, the eye can mistake very flat angle brackets for parentheses, leading to genuine confusion, e.g., between k⟨X⟩, a free associative algebra, and k(X), a field of rational functions.
Hence I have put together some not-so-flat alternatives to the LaTeX angle-bracket symbols \langle and \rangle. I give below two versions. One pair, denoted \langl and \rangl, gives angle brackets with an angle of 120°, the other, \lang and \rang, makes the angle 90°. (Other angles would be equally easy to produce.) Both use the graphics package.
To see how these look, with the standard LaTeX symbols given for comparison, see this PDF file. (I recommend clicking a few times to enlarge what you are looking at. I also like a little more space than LaTeX assigns between such symbols and what they enclose, so I give there, along with the default LaTeX output, an example using \langle\kern.5pt and \kern.5pt\rangle.) The definitions of the new glyphs (also shown on the pdf page) are:
\usepackage{graphics}
\newcommand{\langl}{\begin{picture}(4.5,7)
\put(1.1,2.5){\rotatebox{60}{\line(1,0){5.5}}}
\put(1.1,2.5){\rotatebox{300}{\line(1,0){5.5}}}
\end{picture}}
\newcommand{\rangl}{\begin{picture}(4.5,7)
\put(.9,2.5){\rotatebox{120}{\line(1,0){5.5}}}
\put(.9,2.5){\rotatebox{240}{\line(1,0){5.5}}}
\end{picture}}
\newcommand{\lang}{\begin{picture}(5,7)
\put(1.1,2.5){\rotatebox{45}{\line(1,0){6.0}}}
\put(1.1,2.5){\rotatebox{315}{\line(1,0){6.0}}}
\end{picture}}
\newcommand{\rang}{\begin{picture}(5,7)
\put(.1,2.5){\rotatebox{135}{\line(1,0){6.0}}}
\put(.1,2.5){\rotatebox{225}{\line(1,0){6.0}}}
\end{picture}}
I am not proficent enough with TeX to give these symbols the special properties of LaTeX delimiters, e.g., the ability to change size depending on what they enclose. Perhaps someone else can tell me how. I welcome comments of any sort.
Incidentally, I have found that if I use a formula containing the above glyphs in a section title, LaTeX gives error messages, and refuses to continue until I hit carriage-returns; but that if \newcommand is replaced by \DeclareRobustCommand in both the above definitions, the problem goes away. I don't know whether there are any drawbacks to using \DeclareRobustCommand . Apparently, this problem (and, I suppose, this solution) applies to any command which contains a new environment, as specified by a \begin...\end pair.
The above symbols are enough for most areas of mathematics that use angle brackets, but in noncommutative ring theory, a free associative algebra k⟨X⟩ has a universal skew field of fractions (an analog of the commutative construction of k(X) from k[X] ), and we like to denote this by parentheses superimposed on angle brackets, as is done here, and here. But often such superimposed symbols are not available, in which case one is forced to make do with the workaround k(⟨X⟩), as is done here. This unfortunately looks very much like k((X)) or k⟨⟨X⟩⟩, which represent two sorts of formal power series constructions. Below are LaTeX constructs – again in two versions – for the desired superimposed symbols. Here is how they look (with the above workaround for comparison). One has to use sharper angles in these symbols than in the angle bracket symbols shown above, to keep the brackets from running too close to the parentheses; I have made the angles 80° and 60°.
The code is:
\usepackage{graphics}
\newcommand{\langlC}{\begin{picture}(7,7)
\put(1.8,0){$($}
\put(1.1,2.5){\rotatebox{40}{\line(1,0){6.5}}}
\put(1.1,2.5){\rotatebox{320}{\line(1,0){6.5}}}
\end{picture}}
\newcommand{\ranglC}{\begin{picture}(7,7)
\put(1.2,0){$)$}
\put(.8,2.5){\rotatebox{140}{\line(1,0){6.5}}}
\put(.8,2.5){\rotatebox{220}{\line(1,0){6.5}}}
\end{picture}}
\newcommand{\langC}{\begin{picture}(9,7)
\put(3.0,0){$($}
\put(1.1,2.5){\rotatebox{30}{\line(1,0){8.0}}}
\put(1.1,2.5){\rotatebox{330}{\line(1,0){8.0}}}
\end{picture}}
\newcommand{\rangC}{\begin{picture}(9,7)
\put(2.0,0){$)$}
\put(.8,2.5){\rotatebox{150}{\line(1,0){8.0}}}
\put(.8,2.5){\rotatebox{210}{\line(1,0){8.0}}}
\end{picture}}