I'm actually not sure. I'd consider an ice cream sandwich to be a sandwich, but also a sub, and neither the hotdog nor the taco. I think I might consider myself to be a structural purist/ingredient rebel, and could go so far as to SN/IR, with the caveat that Structural Neutrality only counts as a sandwich if the food is principally designed to be eaten horizontally (with ingredients-sides along the edges) rather than vertically (with at least one ingredient-side along the top or bottom). Wraps are wraps, and calling anything a sandwich defeats the purpose of even having a word for sandwich (to be linguistically useful, the sign must signify something, and it cannot signify everything).
Pizza can satisfy all six cube rules.
Toast. A pizza pie with toppings on top. A slice of pizza. Bread on the bottom.
Sandwich. Stack two slices of pizza on top of each other. Bread on top and bottom.
Taco. Fold the slice of pizza horizontally. Bread surrounding three sides. Left, right, bottom.
Sushi. Roll the slice of pizza. Bread surrounds it with two openings at the end.
Quiche. Deep dish pizza.
Calzone. You could have a stromboli with pizza toppings. Bread outside, toppings inside.
Cake. See sandwich but add a slice of pizza. Boom, three layers.
A deep dish pizza is not a quiche. It is a toast with larger-than-average thickness. The only sense in which a deep dish pizza might qualify as a quiche is when the deep dish pizza is considered as a whole, with the entire perimeter crust intact; but if that counts, then a pizza cannot be a toast, and a regular pizza pie would count as a quiche as well.
Also, adding more pizza slices, one on top of the other, does not make a pizza slice a sandwich or a cake. It makes it one of two or three pizza slices, one on top of the other. (to illustrate my point, consider bread: two slices of bread are not "sandwich". Each slice of bread may function
as a constituent part of a sandwich, but each slice of bread is a slice of bread, not a sandwich in and of itself)
It doesn't really matter though, as I read through that Cube Rule site, and the entire theory is bunk. It's not a serious and well-reasoned attempt at gastronomical taxonomy; it's lazy clickbait.
