Angle brackets are used in various sorts of mathematical expressions:
⟨**x**, **y**⟩ can denote an inner
product or other such pairing,
⟨*a, b* | *ab* = *ba*^{2}⟩
a presentation of groups, 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}}
`