[contravariant]

If that qualifies then surely symbolic notation itself is also a visualisation. It's not as if it's just a syntax tree, quite a few properties (associativity, distributivity, symmetry) are suggested purely by the way we denote sums and multiplication.

If you wants things to be even more visual you could use Penrose diagrams [0], though I don't find them particularly intuitive.

[0]:https://en.m.wikipedia.org/wiki/Penrose_graphical_notation

[Someone]

> quite a few properties (associativity, distributivity, symmetry) are suggested purely by the way we denote sums and multiplication.

I disagree. If “a + b + c” suggests “+” is associative, and “a × b × c” that “×” is associative, what about “a - b - c”, “a / b / c” and “a ^ b ^ c”?

Or are you suggesting Σaᵢ and Πaᵢ suggest associativity? If so, how?

(I also think you mean “commutativity”, not “symmetry” (https://en.wikipedia.org/wiki/Commutative_property) symmetry typically is about relations (https://en.wikipedia.org/wiki/Symmetric_relation)