# How can I make angle brackets function like lists?

Is there a way to make <a,b,c> behave like {a,b,c}, but remain displayed as <a,b,c> (short of setting up new rules for every operation involving such a list that I care about)?

Note: I've typed "<" and ">" here for readability, but really mean \[LeftAngleBracket] and \[RightAngleBracket]. That, way, I could write expressions that look like the normal vector-of-components notation that my intro physics students see in their textbook, but Mathematica would treat as the three-element lists that are the normal way to represent vectors (e.g., to distribute scalar multiplication, calculate dot and cross products, etc.).

I don't know where to start. The kinds of rules I've learned to define seem designed to replace one symbol with its equivalent, rather than simply treating it as its equivalent. I have a feeling this is easy, but it's a side of Mathematica I haven't learned about yet. Can anyone point me in the right direction?

The Notation package is built for this.

Needs["Notation"];
Notation[ParsedBoxWrapper[RowBox[{"\[LeftAngleBracket]", "x___", "\[RightAngleBracket]"}]] \[DoubleLongLeftRightArrow]
ParsedBoxWrapper[RowBox[{"{", "x___", "}"}]]]


(It looks better when input via the Notation palette. Don't be frightened by the box manipulation - I used the palette to construct that expression.)

If you want to see exactly how it looks in the frontend, a cell which contains the expression above looks like:

Cell[BoxData[
RowBox[{"Notation", "[",
RowBox[{
TemplateBox[{RowBox[{"\[LeftAngleBracket]", "x___", "\[RightAngleBracket]"}]},
"NotationTemplateTag"], " ", "\[DoubleLongLeftRightArrow]", " ",
TemplateBox[{RowBox[{"{", "x___", "}"}]},
"NotationTemplateTag"]}], "]"}]], "Input"]

• Aah, okay. It seemed like a common enough thing to want to do. When I took my first look at the Notation Package doc page, though the immediate reference to "constructing explicit MakeExpression and MakeBoxes rules" scared me away. Aug 22, 2015 at 20:49
• Yeah, it's not a particularly intuitive package. The docs tutorial under Notation/tutorial/NotationSymbolizeAndInfixNotation is pretty good, though. Aug 22, 2015 at 20:50
• Wow, that solved my problem far more easily than I had expected. Thanks! Aug 22, 2015 at 20:59
• Unfortunately, this has an undesired side effect: It make Mathematica use angle brackets instead of curly-braces for ALL lists. I'd like it to leave curly-braces as curly-braces, and angle-brackets as angle-brackets, but evaluate expressions involving angle-brackets as if they were curly-braces. Is that possible? Nov 5, 2015 at 16:12

One hacky possibility is the following:

f_[a___, AngleBracket[b__], c___] ^:= With[{r = f[a, List[b], c]},
If[VectorQ[r],
AngleBracket @@ r,
r
]
]


Examples:

AngleBracket[1,2,3]^2
RandomInteger[1, {3, 3}] . AngleBracket[a, b, c]
`

\[LeftAngleBracket]1, 4, 9\[RightAngleBracket]

\[LeftAngleBracket]a + b, b, a + b + c\[RightAngleBracket]