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You're familiar with a sandwich, food contained by two opposite and discontiguous pieces of bread.
Bake a baguette (or perhaps focaccia in some kind of special pan?) into a möbius strip, halve it, and fill it with your ingredients. The added pressure of ingredients will keep it inside as long as
you don't use too much mayo or take any tension off the ring by doing something like taking a bite. But mathematicians and theoretical gastronomers will flock to partake of this topological culinary masterpiece.
The technical explanation is quite straightforward. Sandwiches are known in FCT as a Category 2 food. Toast is 1, Hotdogs are 2B (Bordering), Sushi is 4, Dumplings are 6, and bowls of things are either soup or salad depending on wetness. This is intuitive and any armchair sophist can run it through the rigors during a cup of tea (how can sandwiches be stacked, why layer cake generally isn't a sandwich but lasagna is, etc) as well as any hungry person knows how to hold messy food using less messy food. So by natural progression we are pushed on to nobler pursuits: a möbius strip of bread can be made into an actual sandwich that still fits the geometric definition. The proof is simple, make a möbius strip with strip of paper folded length wise. A knife can cut the crease all the way around while staying inside.
The origins of theoretical gastronomy
https://www.attalus...info/athenaeus.html
[pertinax, Jul 12 2023]
Möbizza
[pocmloc, Jul 31 2023]
The Cube Rule of Food, the Grand Unified Theory of Food Identification
https://kottke.org/...food-identification
quick review of the theory [Loris, Jul 31 2023]
The Cube Rule of Food Identification
https://cuberule.com/
Original (very tall and scrolly) article. [Loris, Jul 31 2023]
Copenhagen interpretation
https://en.wikipedi...agen_interpretation
[hippo, Aug 01 2023]
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I'll have the Möbius Chicken* strips with a side of recursive fries. |
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*Special breed, native to east-central Canuckistan; you should see the eggs! |
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// Halving a möbius strip? // |
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Sure just cut it in half. No, the other half. No, the other other half... |
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This is important research. However, the author defines a sandwich as "food contained by two opposite and discontiguous pieces of bread" and this definition I believe contradicts their proposed innovation in sandwich design. If you halve a möbius strip by cutting along it, this does not result in two interlinked möbius strips but instead a single contiguous new strip which makes two loops and which thus does not fulfill the author's definition of a sandwich. |
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[a1] this could be tested by cutting a bagel in half, but rotating the angle of the cut while going round the bagel |
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Since the sandwiches are non-Orientable, I guess you cannot follow them up with a topologically difficult fortune cookie (which may or may not have a fortune written inside, if there actually is an inside...) |
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// this does not result in two interlinked möbius strips but instead a single contiguous new strip which makes two loops and which thus does not fulfill the author's definition of a sandwich. // |
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This is only true when cutting orthogonal to the reference plane. By slicing along the plane itself the strip can in fact be completely bissected without affecting the topology of either half. |
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This was in fact how Jesus fed the 5000. Some applied topology, limit theorems and a bit of calculus and wala, infinite fish sandwiches. |
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[mace] - hmm, we may need a diagram... what I'm saying is that if you do the cutting action you suggest: "The proof is simple, make a möbius strip with strip of paper folded length wise. A knife can cut the crease all the way around while staying inside." - then you'll end up with a single piece of paper which loops twice, rather than two separate single-loop pieces of paper. |
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I think what mace is saying is thicken the paper. Perhaps have 2 sheets that you twist together as one mobius strip. What happens then? |
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// Hard to do with paper strips. Try it :D // |
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I did it, no problem. The paper I used had markings on one side (put that inside to represent the condiment side). [hippo] is 100% correct. Note the new single piece is not a mobius strip. It has two sides, one condiment side, one was the original outside. |
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So either we need to redefine a sandwich, or put sandwich in quotes in the marketing, then explain to the customer that while this may _look_ like a sandwich at first glance, it really isn't a sandwich. I don't know the correct category of the top of my head. |
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Regarding keeping it together, I don't think tension will help because of the half-twist. I recommend those toothpicks with the decorative ends. Just stab them into the top of the sandwich and push them in until they come out the top of the sandwich... |
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Only hard if you're limiting yourself to 3 dimensions. |
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IMPORTANT BREAKTHROUGH: I'M AN IDIOT |
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[hippo] is correct, my instructions are bad. I just tried it out and actually taped it instead of trying to hold pieces together. It does in fact produce a single double-twisted torus no matter the cutting angle. The bread halves would need to be made individually and then simply used together. There is no way to cut an non-orientable surface in two without destroying that property |
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Non-orientable... that's racist. |
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//IMPORTANT BREAKTHROUGH: I'M AN IDIOT// |
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Congratulations; you're one of us now. |
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[mace] As [pertinax] said, welcome to the club! This is how the frontiers of science get pushed forward. Now all you have to do is to redefine a sandwich, going from your original "food contained by two opposite and discontiguous pieces of bread" to something more like "food contained between two yeasted bread surfaces, normally held in the hand with the bread serving as a barrier between the filling and the hand" - you have to include the word 'yeasted' to avoid the definition including non-canonical sandwich items such as a kebab in a pita bread. |
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When I was a child we would make a sandwich by taking a slice of bread, spreading with butter, adding filling, and finally folding the entire edifice in half (valley fold, obviously) |
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//This is important research. However, the author defines a sandwich as "food contained by two opposite and discontiguous pieces of bread" and this definition I believe contradicts their proposed innovation in sandwich design.// |
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After a little thought, it occurred to me that contrary to the above claim, it should be possible to make a möbius sandwich, and experimentation confirms this. |
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It turns out you need one möbius strip, interlinked with a second (non-möbus) loop twice as long - with two full twists.
The möbus strip is wrapped by the longer loop along the full length of its side, which creates a 'double decker' möbus sandwich. |
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Production of the two discontinuous, interlinked loops of bread is straightforward, assuming one already has a suitable
möbius loaf (or suitably thick möbius slice) and sufficient dexterity - one simply cuts it all the way around, one-third of the way through. |
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<person giving directions>: "Ah, I think you're looking for the Department of Möbius Club Sandwich Topology. This is the Department of Möbius Sandwich Topology" |
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The topological problem here is that "sandwich" is defined in a non-unitary way, in that the definition offered is //food contained by two opposite and discontiguous pieces of bread// |
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The implication of this definition is that the concept "sandwich" refers properly to the food, not to the bread which contains the food. |
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I have already demonstrated by my childhood reminiscence that the discontiguousity of the containing bread is incidental rather than necessary to the containment of the food, in that food contained by a contiguous envelope of bread can still fall into the category of "sandwich" if the containment is of an appropriate geometrical nature. |
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I also note that the bread is not considered a part of the food which is interesting from a nutrition science point of view, but we can disregard that here. |
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Anyway the point is that if the sandwich is defined by the food, then the precise form of the bread is somewhat incidental so long as the bread is topologically toroidal. For example take a bagel, pierce it with a knife, and then cut along the length of the torus whilst gradually rotating the knife through 180˚ around the toroidal axis. You will then have a single piece of bread, in the form of a double twisted loop folded onto itself, but the cut (assuming we can define the cut as an object in the world) is möbiform. |
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(I maintain that this definition of a cut is permitted, since when I was a child and a pristine cake was about to be sliced, I would eagerly request to eat "the first cut", and my father would proceed to insert the knife and make a single radial cut and then say "yes there is the first cut, just for you".) |
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Anyway, after making the möbiform cut into the toroidal bread, you can now carefully insert any möbiform sandwich filling, such as a möbius strip of ham or cheese, or a series of slices of tomato and cucumber (as long as you don't mind the orientation of the slices having a disjunct at at least one point of the filling). |
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Yes, I agree. The problem with the definition "food contained by two opposite and discontiguous pieces of bread" (which, I note in passing, does not allow a club sandwich to be called a sandwich) is not that the requirement for discontiguity disallows a Möbius-form bread substrate for your sandwich filling, but that the definition needs to be changed to something more generic such as "food contained between two bread surfaces" (or the wordier definition in my 14th July annotation) |
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Right but my text became too verbose. It is the filling that should be möbiform, not the bread. |
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// The problem with the definition "food contained by two opposite and discontiguous pieces of bread" (which, I note in passing, does not allow a club sandwich to be called a sandwich)// |
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What is it about a club sandwich which is disallowed?
I'm not sure if you're using this definition, but wikipedia says:
::A club sandwich, also called a clubhouse sandwich, is a sandwich consisting of bread (traditionally toasted), sliced cooked poultry, fried bacon, lettuce, tomato, and mayonnaise. It is often cut into quarters or halves and held together by cocktail sticks. Modern versions frequently have two layers which are separated by an additional slice of bread.:: |
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To my mind the cutting into four as part of production is irrelevant. (For what it's worth, I also disagree with the concept entertained above that 'sandwich' describes the filling only. The definition is /inclusive/ of the bread, which is also eaten in this case.) A sandwich cut into four can still be considered one sandwich total. I agree that this is a context-sensitive, slightly messy condition. But consider the alternative - if not, theoretically one could never eat a sandwich.
So I think your objection is the second qualifier; the third slice of bread (creating what I described above as a 'double decker' sandwich). I think this fits within the definition; one merely has to consider the internal slice as part of the filling to see this. |
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//What is it about a club sandwich which is disallowed?// - I was thinking of the kind of club sandwich with three slices of bread, whereas the definition quoted only permitted two slices of bread |
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I am also going to bang on at tedious length about the various types of sandwiches which consist of one single contiguous piece of bread which has planar extents encompassing both sides of the filling, e.g. by folding a single slice of bread around the filling, or by cutting 99% of the way through a baguette or bap and inserting the filling between the (still mutually attached) halves. |
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Also I never said that "sandwich" only described the filling; I was proposing that "möbius" could describe only the filling, allowing the bread to be non-möbioid. |
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Let us not get distracted by open sandwiches. |
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//I am also going to bang on at tedious length about the various types of sandwiches which consist of one single contiguous piece of bread which has planar extents encompassing both sides of the filling, e.g. by folding a single slice of bread around the filling, or by cutting 99% of the way through a baguette or bap and inserting the filling between the (still mutually attached) halves.// |
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The problem with these is that they fall into FCT group 2B (along with hot-dogs), in mace's schemata. |
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//Let us not get distracted by open sandwiches.// |
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Nah, they're obviously FCT group 1, along with toast. |
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Obviously, mace's system varies from the seminal cube rule of food identification (see: links), which would put hot-dogs in group 3. But there is of course scope for different scholars to purport alternative systems. |
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//Also I never said that "sandwich" only described the filling; I was proposing that "möbius" could describe only the filling, allowing the bread to be non-möbioid.// |
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I see.
I now understand your claim here, but I'm not sure whether I agree. I will need to think on this some more. |
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// //Let us not get distracted by open sandwiches.// - Nah, they're obviously FCT group 1, along with toast.//
I would like to propose that this is the 'Copenhagen interpretation' of sandwichology, as they're very keen on those smørrebrød open sandwiches there |
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// It turns out you need one möbius strip, interlinked with a second (non-möbus) loop twice as long - with two full twists. The möbus strip is wrapped by the longer loop along the full length of its side, which creates a 'double decker' möbus sandwich. // |
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It's alive! [Loris], I believe your research into proving the existence of a true möbius sandwich is biggest FCT breakthrough of the year! |
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To update our definition to include the bread itself (including additional pieces required for clubs), the possibility of them being interlinked, we could say that a sandwich is "at least two pieces of discontiguous bread aligned to contain food between them" |
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[pocmloc]'s toroidally cut toroid is either a 2B or 3 depending on if food is actually packed into the hinge, though this ambiguity might be enough to disprove the existence of 2B as a distinct group from 3. However, I would argue [hippo]'s Copenhagen interpretation of single-bread sandwiches is merely a descriptor of FCT group 1 (Toast) as an intermediary state for group 2 |
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OK here's another possibility. Bake a loaf such that it is in the form of a möbius strip. Or alternatively bake a toroidal loaf and cut a slice of bread from it by making two parallel möbioid slices around the length of the torus. This will generate a true möbioid slice of bread, with one crusty edge and one crummy face. Make this slice nice and thick (at least 1cm, prefereably more, like a proper "doorstep") |
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Now to make a sandwich out of this möbius strip of bread, you place the möbioid slice of bread onto a cutting table, and then you slice horizontally in a single plane which divides the slice in two. The slice is pefectly planar, and simply parts the twisted inversion part of the möbius strip like Gordion untying his shoelaces. |
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Then you can use any planar ingredient to fill the sliced zone (as long as you trim it neatly to align with the edges of the cut). |
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Reassemble the two halves and you have a sandwich, in the form of a möbius strip, made of two discontiguous pieces of bread, with food inbetween them. |
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[pocmloc] you've brought us full-möbioid and reinvented a topologically normal sandwich, fully orenientable with distinct top and bottom sides |
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