Toward a Unified Theory of Sandwiches: Sandwich Topology

Toward A Unified Theory of Sandwiches: Sandwich Topology Joshua Eby∗ Weizmann Institute of Science Matteo Lotitoy and Brian Maddockz University of Cincinnati We present a unified picture of sandwich topologies, which considers the open-faced sandwich, dubbed S1, to be the fundamental unit of sandwichhood. All other sandwich topologies are shown to be equivalent to convolutions of many S1 units. This picture unites previous work in the field and is an important first step towards a Consistent Representation of a Unified Sandwich Theory (CRUST). The conditions on the nature of ingredients in a proper sandwich is left for future work. INTRODUCTION The sandwich is a staple of cuisine across continents, and can be found in the context of breakfast1, lunch, or dinner. It is said to have been invented by John Montagu fourth Earl of Sandwich; it is known that \The original sandwich was in fact a piece of salt beef between two slices of toasted bread" [2]. Today's notion of a sandwich is somewhat more broad; the three defining fea- tures of the original sandwich, \toasted", \beef", \between two slices of bread" can each be relaxed. Uncon- troversial sandwichhood is routinely granted to objects formed from untoasted ingredients; similarly, they may contain no meat at all, and they might even be formed from tortilla or pita rather than bread. FIG. 1: The space of possible beliefs about sandwichhood, as It is thus interesting to ask what constitutes sandwich- organized by topology and ingredients; reproduced from [8]. hood in the first place. A complete classification will answer such controversial questions as: is a hot dog on a bun a sandwich? is a lettuce wrap a sandwich [3]? if I sounds like an authority on the matter. put two pizzas on top of each other, have I made a pizza Upon further investigation, the NHDSC determination sandwich (or is an ordinary pizza an open-faced sandwich seems to be more about publicity than a consistent con- [4])? among numerous others. Indeed, the consideration sideration of the facts. On what grounds did they ex- of questions like these represents an important motiva- clude hot dogs from sandwichhood? Is it the choice of tion for this work. The ultimate goal is what we call a toppings? Is it the type or orientation of the bread? Consistent Representation of Unified Sandwich Theory None of these appear to have even been considered. They (CRUST), which will answer these questions and related assert: \Our verdict is... a hot dog is an exclamation ones by providing a framework for unifying disparate ar- of joy, a food, a verb describing one 'showing off' and eas of current sandwich research. In the end, a CRUST even an emoji. It is truly a category unto its own" [5]. would provide the natural boundary for the field, inside We find this answer altogether unsatisfying; it brings us which productive sandwich research can be conducted. no closer to understanding at a fundamental level what In lieu of a complete investigation, one might choose to makes something a sandwich. This classification is also blindly accept the proclaimations of certain \governing in direct tension with the definition of another regulatory bodies" on a case-by-case basis. We believe this approach body, the USDA, which indeed defines a hot dog as an is not acceptable, and is unlikely to end in a consistent open-faced sandwich [6]. classification. Take, for example, the 2015 announcement The hot dog is a somewhat complicated case, but by by the National Hot Dog and Sausage Council (NHDSC), no means unique. Other similar controversies and in- that a hot dog is not a sandwich [5]. Does this settle the consistencies in sandwich law have been pointed out [7]. debate? Surely this council, by its name and charge, It seems clear that an appeal to the authority of some sandwich regulatory body is not a clear path forward, as different authorities have different definitions, and none seem particularly concerned with consistency across 1 The most important meal [1] sandwich types. 2 To ground our work, we will use a certain set of work can be viewed as building on this general taxon- \common-sense" notions on which to build our classifi- omy of sandwich beliefs, towards a unified theory which cation. For example, certain objects (e.g. a hammer, the describes the space of possible sandwiches. Pacific Ocean, or a pile of grass clippings) cannot qualify as sandwiches, and a theory which grants such objects Because we focus only on questions of topology, this sandwichhood should be deemed a failure. Similarly, an work will present what we believe to be necessary (but edible object consisting of meat and cheese between two not sufficient) conditions for sandwichhood. The ques- pieces of bread is clearly a sandwich; indeed, in a sense it tion of ingredients is somewhat subtle, and deserving is the exemplar of what a sandwich should be. It is valu- of its own separate treatment. For example, is there a able to check an abstract sandwich theory against such minimum number of ingredients necessary to constitute common-sense notions. a sandwich?3 Need these ingredients fall into particular We will refer occasionally to this notion of an exam- classes (meat, cheese, vegetables) or can they be anything plar sandwich, one which cannot be excluded from the at all? Need the outside consist of bread, or do other in- collection of sandwiches by a consistent theory; we call gredients qualify? These questions are important, but we this a Sandwich of Unmistakeable Belonging (hereafter leave them for future work. SUB); this will be a common-sense check of sandwichhood. The SUB consists of two pieces of bread, contain- It will be adequate for our purposes here to accept only ing small slices or pieces of meat and cheese. One could very basic assumptions to constrain the ingredients. We imagine using a definitive SUB which included other op- will assume that any sandwich must consist of an outer tional ingredients, such as condiments (e.g. mayonnaise) layer (which we will call bread) and some toppings. This or vegetables (e.g. lettuce and tomato), but this is a mat- terminology should not be taken too literally; clearly the ter of taste.2 Crucially, we require that the SUB does outer layer of a sandwich might plausibly be made of not contain such controversial ingredients as, for exam- something other than actual bread, as in the case of a ple, egg, cream cheese, or additional slices of bread. Even wrap.4 In the original sandwich [2], the \bread" was more crucially, a SUB will also not contain hammers or literal bread, and the \toppings" consisted only of salted the Pacific Ocean. beef. For reasons that will become apparent shortly, we In the next section, we outline some basic terminology will not assume that toppings to be enclosed by bread, and assumptions of this work. We will then proceed to but rather assume that toppings live on a single side of classify the topology of sandwiches, an important step a given bread unit. towards the desired CRUST. We seek to classify all valid sandwich topologies. To avoid double-counting and to maintain consistent iden- FIRST STEPS tification of equivalent topologies, a few rules are necessary. First, as we have already stated, we assume toppings to live on a single side of a given slice of bread; As a starting point, it is crucial to distinguish orthog- bread with toppings on both sides seems clearly to vi- onal dimensions on which sandwichhood might depend. olate common-sense notions of sandwichhood, and they We identify two important broad classes of properties: also make a mess in the kitchen [1]. Second, we im- (1) Topology, the topic of this work, is determined the pose the requirement that bread not be torn, so that we organization of the ingredients in physical space; and (2) may count sandwich topologies consistently by number the nature of the ingredients, which we leave to a future of bread slices; one slice may not be torn into two, as work. There may indeed be further dimensions to specify this would constitute an inequivalent topology. Third, for a full classification of sandwichhood, but we will not the curving or deforming of a given piece of bread con- discuss this issue further here. It is important not to bite stitutes an equivalent topology (given that no new tears off more than one can chew. or holes are formed). In this sense, excluding ingredien- Our acknowledgement of these two important dimen- tial considerations, a tortilla wrap is equivalent to both sions of sandwichhood is not exactly novel. Indeed, a a pita wrap and a bread-based open-faced sandwich. classification of \Sandwich Alignments" has been pro- posed by [8]; the graphic from the original work is reproduced in Figure 1. It is important to note, however, that the author merely presents a classification of possi- 3 It has occasionally been suggested that a single piece of bread ble stances about sandwichhood, but does not argue in could be considered a sandwich; such inflammatory statements any way for which stance is best. With this in mind, our will not be commented on here, both due to the controversy surrounding this proposal and because of our focus on topology (rather than ingredients). 4 As we will make clear below, the distinction between an wrap and a bread-based open-face sandwich is ingrediential, not topo- 2 Pun intentional. logical, but should be investigated further in the near future. 3 FIG. 3: Compact arrow representations of the first few terms in the sandwich expansion.

Load more