Toward a Unified Theory of Sandwiches: Sandwich Topology

Toward A Uniﬁed Theory of Sandwiches: Sandwich Topology

Joshua Eby∗ Weizmann Institute of Science

Matteo Lotito† and Brian Maddock‡ 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. (a-h) correspond to the same physical objects as in Figure 2. In our notation the left- most arrow should be regarded as the “bottom” of the sandwich, but by construction the classification of sandwich type FIG. 2: Pictographic representations of the first few terms in is rotation-invariant. the sandwich expansion. a) The fundamental sandwich unit S1; b) A single S1 flipped vertically, defined as equivalent to (a); erally, we suggest that the topology of a given sandwich c) S2, defined as a normal + an inverted S1; should not be affected by its orientation compared to d) Two upright S1 units, or 2S1; the external world; a sandwich in the vacuum of space e) Three upright S1 units, or 3S1; is well-defined irrespective of any external influences. A f) An S with an S on top; 2 1 sandwich can, however, change topology through the in- g) Two upright S1 units and an inverted S1, defining an S3; ternal rotation of some of its parts, as illustrated by the h) A vertical inversion of (g), which is equivalently an S3. qualitative difference beween the open configuration 2S1 and the closed confiuguration S2. This will have con- sequences when we count possible topologies. Without TOPOLOGY OF A SANDWICH loss of generality, when we build higher-order sandwich contributions, we may regard the “bottom” S1 unit as Of crucial importance to the topological classification being “face-up” with respect to some fixed external ob- of sandwichhood is the distinction between “closed” and server. This naturally rules out also any sandwich can- “open-faced” sandwiches. These are topologically dis- didate whose toppings are fully external, but we have al- tinct configurations, in the sense that an open-faced ready suggested that such objects cannot be considered sandwich has a single bread boundary whereas a closed sandwiches. sandwich has two; one cannot deform an open-faced We may continue in this way, building up higher- sandwich continuously into a closed one without tear- order sandwiches in what we call the Sequential Ap- ing the bread manifold, an invalid procedure. We accept proach to N-order Determination With Inequivalent as intuitively obvious that both of these types are valid Configuration-Having (SANDWICH). In Figure 2 we sandwich configurations, provided other relevant criteria show all inequivalent configurations of N ≤ 3 S1 sand- (e.g. ingredients) are satisfied. wich units. Clearly (a) and (b) are equivalent, as the Once it is accepted that both closed and open-faced rotation is defined only with respect to some outside ob- sandwiches are acceptable sandwich topologies, the nat- server. The difference between (c) and (d) has already ural next question is the relation between the two. We been pointed out. Similarly, the 3S1 configuration (e) can posit that a closed sandwich is fundamentally the con- be understood as three separate S1 units stacked, com- junction of two open-faced sandwiches, making the open- pared with a single S1 on top of an S2 (f). Both of these faced sandwich the fundamental unit of sandwichhood. open configurations should be contrasted with the closed 5 We refer to this fundamental unit as S1. A typical closed S3 (g) and (h), which are equivalent because one can sandwich (e.g. a SUB) is an S2, which signifies both it’s be obtained by a complete vertical rotation of the other. closed-ness as well as the number of S1 units it contains. An equivalent, but more compact, representation of the On the other hand, two S1 units on top of each other SANDWICH up to N = 3 is given in Figure 3, where an is not an S2; we classify it instead as 2S1, two distinct upright S1 is given by an up-arrow, and an upside-down sandwiches stacked together rather than a single closed S1 is given by a down-arrow. sandwich. We may represent an N-order SANDWICH configura- It is also important to note certain symmetry considerations. Consider a single S1. If we flip it upside-down, we clearly have not generated a new topology; similarly with a rotation in any direction by any angle. More gen- 5 We may also refer to this as a “Big Mac” configuration [9]. 4

For N > 3, there is an additional ambiguity that must be acknowledged. Consider N = 4, as depicted in Figure 4. The labels on (j-p) indicate the ambiguity in how we define the sandwich decomposition. We propose a solution to this degeneracy, which we call the CLuster Untangling Breakdown (CLUB) procedure as an added requirement on N > 3 SANDWICH expan- sions. In the CLUB+SANDWICH procedure, we define the sandwich configuration unambiguously by breaking it into a natural selection of parts: starting at the bottom, if two breads are not connected by a layer of toppings, we CLUB it and define the split at that point. Consider configuration (j) as an example. Starting at the bottom (left, in the Figure), the first two S1 units form a united S2; on top of this, is a disconnected (face- FIG. 4: Arrow representations of the N = 4 SANDWICH up) S1, so we take this to be the natural splitting point. expansion. To resolve the ambiguity in the decomposition of Indeed, we must not forget that the goal is to classify in the sandwiches, we use the CLUB procedure; the result is the a useful way how sandwiches will appear in the physi- blue-colored options below each representation. cal world. The fact that no toppings exist between the second and third S1 units means that these should be regarded stacked, separated sandwiches. On the other tion, labeled DN , mathematically as hand, configuration (k) has the same set of possible de- N compositions, but under CLUB is unambiguously defined X DN = ni Si, (1) as an S1 S3; the top S1 is disconnected from the lower 3- i=1 sandwich. The final result for 4-sandwiches is where the coefficients ni are the number of clearly- E4 = {4S1,S1 S3, 2S2,S4}, (4) separated sandwiches in the SANDWICH that have con- as expected. figuration Si. By definition, We do not prove it rigorously here, but we believe the N CLUB+SANDWICH is a well-defined and unambiguous X i ni = N. (2) procedure for sorting arbitrary N-sandwiches into equiv- i=1 alence classes, and that the resulting space of sandwich equivalence classes EN then spans all possibilities. The set EN ≡ {DN } represents the collection of distinct equivalence classes of SANDWICH configurations. For the case of N ≤ 3, the configurations given in Figures 2 CONCLUSIONS and 3, we have

a) n1 = 1; b) n1 = 1; We have presented a rigorous analysis of the topology of sandwiches of arbitrary size. We have made very c) n1 = 0, n2 = 1; d) n1 = 2, n2 = 0; minimal assumptions, in particular that an open-faced sandwich is indeed a sandwich, and as a result we posit e) n1 = 3, n2 = 0, n3 = 0; that the open-faced sandwich S1 is indeed the funda-

f) n1 = 1, n2 = 1, n3 = 0; mental sandwich unit. All other sandwiches are formed from permutations of different numbers of S1 units. This g) n1 = 0, n2 = 0, n3 = 1; classification only leaves out one possible collection of permutations, namely, ones in which all external S con- h) n = 0, n = 0, n = 1. 1 1 2 3 stituents face outward. We have argued that such con- Observe that two configurations are equivalent if and figurations do not qualify as valid sandwich topologies. only if they have identical SANDWICH coefficients ni. We have further constructed a procedure to classify This implies any N-sized sandwich into its relevant equivalence class of sandwichhood, where N is defined as the number of E1 = {S1},E2 = {2S1,S2},E3 = {3S1,S1 S2,S3}. constituent S1 units. The expansion into equivalence (3) classes, dubbed SANDWICH, is a simple and direct pro- The cardinality of EN represents the total number of cedure for any sandwich up to N = 3 (the Big Mac con- topologically inequivalent configurations formed from N figuration). For N > 3, we supplement it by a CLUB se- open-faced sandwiches. lection rule which extends the validity of the procedure. 5

It is currenly only a well-founded conjecture that the [3] The Lunch Dude, “Which sandwich? CLUB+SANDWICH analysis works at arbitrarily large At Jimmy John’s, it’s the ’unwich’.” N, but certainly sandwiches beyond N = 6 or so are http://www.timesrecordnews.com/story/entertainment/food- mostly of only theoretical interest anyway. drink/2017/02/03/which-sandwich-jimmy-johns-s- unwich/97264366/ The question of ingredients is a complicated one, which [4] Drew Brown, “Pizza is a sandwich.” we have mostly ignored here. We discussed in the in- https://www.vice.com/en ca/article/nz8b88/pizza-is- troduction a few of the more pressing issues, but a full a-sandwich treatment will be delayed to a future work. [5] NHDSC, “National Hot Dog and Sausage Council Announces Official Policy On ‘Hot Dog as Sandwich’ Controversy.” http://www.hot-dog.org/press/national- Acknowledgements hot-dog-and-sausage-council-announces-official-policy- hot-dog-sandwich-controversy [6] NPR, “What Burritos And Sandwiches The authors thank E. Black, C. Doyle, R. Legman, S. Can Teach Us About Innovation.“ Linser, A. Malsom, P. Malsom, S. Palladino, I. Shojaei, https://www.npr.org/sections/alltechconsidered/2014/06 J. Stanton, J. Thompson (“The J”), and D. Tuckler for /26/325803580/what-burritos-and-sandwiches-can-teach- valuable contributions during “Eclipse Weekend”. us-about-innovation [7] Mental Floss, ”5 Ways to Define a Sandwich, According to the Law.“ http://mentalfloss.com/article/501011/5-ways- define-sandwich-according-law [8] @matttomic, ”The Sandwich Alignment Chart.“ ∗ Electronic address:[email protected] https://twitter.com/matttomic/status/859117370455060481 † Electronic address:[email protected] [9] McDonalds, ”The one and only. Big Mac.“ ‡ Electronic address:[email protected] https://www.mcdonalds.com/us/en-us/product/big- [1] Mom, private correspondence. mac.html [2] The History of Hinchingbrooke House. http://www.hinchhouse.org.uk/fourth/fourth.html