The Topology of Sandwiches
The debate over whether a hot dog qualifies as a sandwich has torn apart the
internet. Once you move past the triviality of the issue, the question is actually quite
complex. The ramifications of one’s stance on the debate impact the entire taxonomy
of food items. Merriam-Webster defines sandwich as two or more slices of bread or a split
roll having a filling in between or one slice of bread covered with food. At first glance, it
would seem this issue has been resolved. After all, a hot dog bun is a split roll and
the meat cylinder represents filling. However, as I will show, this definition is far too
vague to be operative in this discussion. For me, this question has always been a
topology problem.
Topology is a mathematical discipline concerned with the position of shapes.
Literally, it is the study of position or location. Topologists treat two objects as the
same if one can be deformed theoretically into the other. As such, the number of
topological shapes is limited as each can be represented in countless different ways
through manipulation. Shapes in topology are malleable and can self-intersect, but
Figure 1: A torus (donut) being deformed into a coffee mug. These objects are
topologically identical.
never crease, tear or glue together. It is an abstract discipline, oft used to solve
problems qualitatively. Here, we will examine the nature of sandwiches through a
topological lens and attempt to classify bread-related foods as sets of shapes.
We begin with some definitions and basic shapes to be used henceforth.
Firstly, let’s expand on the idea of topological shapes. Two objects share a shape if
you can manipulate one into the other using continuous deformations. This could
mean folding, stretching or compressing for example. Shapes are also topological
spaces. A space is a set X with a collection of T subsets of X. A set has an interior
and a boundary and two sets share an intersection value based on the sets of
intersections between their respective interiors and boundaries. Let A and B be two
sets with interior 𝑖 and boundary 𝜕, and 𝛼 ∩ 𝛽 denote the intersection of 𝛼 and 𝛽.
The intersection value of A and B is then:
𝐼!,! = (𝜕! ∩ 𝜕! , 𝑖! ∩ 𝑖! , 𝜕! ∩ 𝑖! , 𝑖! ∩ 𝜕! )
Where a value of 0 is returned for an empty set and 1 otherwise. Thus, a set B
contained entirely within the interior of a set A without the intersection of their
boundaries (Figure 2) would have an intersection value of (0,1,0,1). Basic topological
shapes are planes, spheres and tori, of which there are endless geometric
representations. Merriam-Webster’s definition raises concerns as two or more slices
of bread and a single slice or split roll represent two entirely different topological
spaces due to the restriction that one cannot obtain two slices from a single slice or
vice versa. This imposes that the class sandwich encompass both types of spaces.
Hence, as we will work through, the classic definition becomes far too inclusive.
Figure 2: A space containing two subsets A and B with intersection value (0,1,0,1).
One element I will borrow from the Merriam-Webster definition is the
required presence of bread in the sandwich space. Bread mustn’t necessarily be
leavened, but must be made from flour. An important distinction in my adapted
definition is that filling and not food will be the operative description of the two
provided my Merriam-Webster. In our topological analysis, it is not a requirement
that the filling be edible. In fact, the nature and shape of the sandwich filling is not of
concern. Any non-bread, non-intersecting subset in the sandwich space may be
oriented or re-arranged in space to become part of the filling. As a final guideline, we
think of air as empty space devoid of dust and molecules which could be thought of
as filling.
Two basic shapes will be considered: the sphere and the torus. All breads may
be represented as one of these two forms. They equate to a loaf and a bagel, in
Figure 3: Representations of bread types.
sandwich terms, respectively. It is worth noting now that a split loaf, as it is described
by Merriam-Webster, is nothing more than a sphere. We can immediately
understand that a hot dog and an open-faced sandwich are equivalent topological
spaces. Less intuitively, this category includes pizza, cheese and crackers, ice cream
cones, and pita with hummus. Indeed, conceding that this set belongs in the
sandwich family opens the door for a multitude of food items.
There are, at minimum, three classes of foods that may be constructed from
one or more breads of one type and one filling. The six classes yielded from breadfilling permutations of either bread type are summarized in figure 4. Hot dogs belong
to the first class, submarines. The topological space defining this class contains one
loaf and filling, with no intersections. A set containing two or more breads, such as a
classic sliced bread sandwich, must belong to a different class, as a loaf cannot be
split or otherwise divided to obtain two loaves. Hence, the notions that a hot dog is a
Figure 4: Classification and topological representation of bread foods.
sandwich and that a traditional ham-and-cheese is not a sandwich are mutually
exclusive. A space containing 𝑛 ≥ 2 loaves and filling, satisfying:
∀𝐴! , 𝐴
!!!
!!!
𝐴! = (0,0,0,0)
where 𝐴! is the 𝑖th set in the space, can be arranged to construct an 𝑛-1 story
sandwich. Thus, while a singular hot dog is not a sandwich, two or more hot dogs
occupying a neighborhood are a sandwich. Raviolis occur when intersections
between the filling and the loaf’s interior are present. These arrangements may not be
deformed into submarines or sandwiches and include calzones, tortellini and beef
wellington.
Three separate families of bread-related foods exist under the bagel umbrella.
Churros, donuts and bagels with cream cheese belong to the donut class.
Bagelwiches do not overlap with sandwiches in the same way bagels and loaves
represent two distinct shapes. Pretzels generally exist as triple tori and are beyond the
scope of this analysis.
Wraps such as burritos are submarines, as they contain only one loaf which
cannot and does not fuse with itself to form a bagel. Pastas such as spaghetti or
fusilli, interestingly, are multi-story sandwiches when paired with sauce. Tubular
varieties of noodle such as macaroni or penne, on the other hand form bagelwiches.
It is clear the dictionary definition of sandwiches encompasses far too many foods for
it to be taken seriously. From a topological perspective, any food space containing
bread would become a sandwich, a surely preposterous notion. It is in the interest of
preserving the sanctity and integrity of sandwiches that I conclude hot dogs and
other submarines inhabit an entirely separate group of foods, as do raviolis and all
bagel products.