Ratotopes

[ubersketch]

empty string = point
(a) = shape produced by changing a's size sinusoidally and using those as slices
ab = cartesian product of a and b

(empty string) = point
() = line
(()) = circle
()() = square
((())) = sphere
(())() = cylinder
(()()) = crind
()()() = cube
(((()))) = glome
((()))() = spherinder
((())()) = crilinder
(())()() = cubinder
((()())) = crond
(()())() = crindinder
(()()()) = cubind
()()()() = tesseract
(())(()) = duocylinder
Ratotope as a name was picked for being an anagram of Rotatope.