2658: Coffee Cup Holes

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Coffee Cup Holes
Theoretical physicist: At the Planck length, uncountably many.
Title text: Theoretical physicist: At the Planck length, uncountably many.


Ambox notice.png This explanation may be incomplete or incorrect: Created by a CAFFEINE MOLECULE WITH A HOLE DRILLED IN ITS SIDE. Do NOT delete this tag too soon.
If you can address this issue, please edit the page! Thanks.

This comic depicts people in different fields of study answering the question, "How many holes are there in a coffee cup?" and also compares this to what a normal person would say.

This question has different interpretations, entirely dependent upon the definition of a hole. The type of coffee cup shown in the comic is with a handle (like a mug), but Randall calls it a cup and there are also cups with handles on the Wikipedia page for coffee cups. Most people would recognize that there is a hole through the handle.

The comic explores the answer to the question through several peoples’ avenues of thought, and is funny because of the ambiguity:


The coffee mug and donut shown in this animation both have topological genus one.

Ponytail, a topologist, states the coffee cup belongs in the genus of one hole. From the topologist's point of view, the coffee cup definitely has one hole, which corresponds to the opening created by the cup handle. A cup without a handle would have zero holes, as it is equivalent to a dinner plate, just an indentation in the surface. See 2625: Field Topology for more information about topology.

The panel as a whole references an academic joke wherein topologists can't tell the difference between a coffee cup with a handle and a doughnut since they're homeomorphic to each other — meaning they have the same genus (i.e one hole).

Normal person[edit]

Hairy, representing a normal person, is not sure (the acronym "IDK" stands for "I don't know") and asks for clarification about whether the opening at the top counts as a hole. This shows flaws in the question, which suffers from the mathematically imprecise, ambiguous common usage of the word "hole." Topologists would refer to the opening as a concavity, not a hole, and while they consider such geometrical properties generally outside their field, most practical applications of topology do involve geometric components. By contrast, in everyday usage many concavities are called holes, such as a hole dug into dirt with a shovel. Hairy would say one for the handle, and two if the opening counts as a hole, which he is not certain the one asking the question thinks.

A genus two surface


Hairbun, a philosopher, answers the question with an elucidating counter-question, considering a hypothetical scenario. One might expect that drilling a new hole would also increase the number of holes by one. However, as illustrated, some people would consider that the new arrangement has three holes (in addition to the handle, there is a hole at the top where coffee can be poured in, and one at the bottom where it can run out), while others would consider it to have only two (the new hole forming a continuous hole with the original opening at the top, through which coffee can run). Some might in fact say that the coffee cup now has one hole because it is leaky, disregarding the handle topology at this point. In this way she requires her interlocutor to confront the ambiguities and consider what they mean by the word 'hole' in different contexts. An interesting point about Hairbun's response is that she doesn't actually answer the question, a trope often found in philosophical replies.

A point cloud of a genus one surface


Cueball, a chemist, looks at the coffee in the cup on a molecular level. He envisions a ball-and-stick model of the caffeine molecules in the coffee, and estimates a total number of holes of all the coffee molecules. He comes up with a truly massive number: 1,000,000,000,000,000,000,000 (1021 or 1 sextillion) “in the caffeine alone.” One molecule of caffeine has two rings of bonds with holes in them, multiplied by 500 quintillion molecules, or 0.00083 moles. As the molecular mass of caffeine is about 194 grams per mole, Randall must think that the mass of caffeine in a typical cup of coffee is 161 milligrams. The coffee could have other holes, depending on the type of coffee; for example, espresso contains significant amounts of niacin and riboflavin, which have one and three rings in their chemical structure, respectively.

This estimation depends on taking the ball-and-stick model of molecules somewhat literally. However, real molecular bonds are not solid sticks, but shared electron clouds between atoms. The "holes" in the middle of a molecule's rings are not completely empty but instead merely have lower electron probability density through the middle than other parts of the bonds. So the point-cloud duality of electron orbitals and bonds might not satisfy a topologist's, normal person's, or philosopher's criteria for a connected substrate in which holes may be formed.

String theory describes the worldlines of point-like particles as worldsheets of "closed strings," forming topological holes; shown here as a genus two surface.

Theoretical physicist[edit]

In the title text, a theoretical physicist looks even deeper, at the subatomic scale of Planck units. Since fundamental particle interaction is governed by fundamental forces and collision (per the Pauli exclusion principle) instead of tensile or ductile solid connectedness, the theoretical physicist posits that any definition providing for a single hole would also describe a number of holes akin to the factorial of the number of particles in the universe,[1] or at least within the cup's light cone, which is a number impractical to accurately count, but not uncountable in a mathematical sense.

Practical considerations[edit]

The main joke is that the number of holes depends on both the scale and perspective from which you are looking at the world. From a topological standpoint, when someone digs into the ground it should go all the way through (or easier, down and up again another place) before it is considered a hole, since a hole is something that some other thing should be able to pass through. But from a common usage perspective, if people dig in the ground the result is called a hole, because functionally it creates a discontinuity in to which, for example, things can be placed or fall. Similarly, the opening in a coffee cup without a handle or a bottle of beer is called a hole, even though they are topologically equivalent to a dinner plate, which normal people would never say had a hole.

A cavity in a surface could also be considered a physical barrier, preventing movement along the surface in certain scenarios (e.g. a sinkhole opening up in the middle of a road) even though it may be topologically 'flat' in the most general way, and so is very open to context, and such a hole might be considered more a 'thing' than the surface that has been removed to create it. And a concavity in a vessel that can hold liquid (or a drilled hole which removes that ability) is of a different nature from the holes in the molecules that are part of the liquid therein. And such holes very different from the string-theoretical holes at the Planck scale, which don't necessarily involve barriers, containment, or any other aspects of topological connectivity. This conceptual ambiguity of what a hole is or means is demonstrated by the fictional portable hole, which obeys and defies a normal person's preconceptions of a hole.

The topological discussion here regarding cups and doughnuts is related to the question of how many holes there are in a human, which is excellently answered in Vsauce's video How Many Holes Does a Human Have?. This also takes a good look at the topological difference between a paper cup and a mug with handle, and how one could be morphed into a plate and the other into a doughnut.


[The first panel has text only and is phrasing a question:]
How many holes are there in a coffee cup?
[Each of the next four panels has a caption at the top to indicate the kind of person answering the question. In the first of these Ponytail stands holding a coffee cup in its handle.]
Caption: Topologist
Ponytail: One.
[In the next panel Hairy stands to the right of Ponytail, holding the coffee cup in its handle at an angle so he can to look into it.]
Caption: Normal person
Hairy: IDK, does the opening count as a hole?
[In the next panel Hairbun is shown in closeup, holding her hand out palm up to indicate two drawings of coffee cups with handles to her left. The top drawing is larger and shows the cup with coffee inside, and a hole drilled at the bottom part of the side away from the handle. Coffee pours out of this hole. Beneath and further left is a smaller version of the same cup, but now without coffee. Instead two curved arrows goes from above to below through the hole of the handle and the hole now drilled in the bottom part of the cup. Each arrow is labeled with a question-mark.]
Caption: Philosopher
Hairbun: To answer that question, consider another: If we drill a hole in the side, how many holes are there now?
[Cueball, without any cup, stands with a drawing of a caffeine molecule above and to the right of him. It has two "rings" with 5 and 6 atoms. Those rings are connected along one side. There are 9 "edges" on this, three of those has one atom attached to it and 3 others have four atoms attached to them (one atom with three others attached). The two that are at the end of the edge that belongs to both rings have no atoms attached, and the final of the 9 also has no atom.]
Caption: Chemist
Cueball: 1021 in the caffeine alone

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I was confused for a moment. That's a coffee mug. And the correct answer is either one (the handle) or none (because below the macroscopic level (and above the theoretical sub-Planck scale of string-theory loops) it's increasingly not even mostly holes but very, very barely anything 'solid' jostling about in empty space giving no real impediment to any theoretical quantum-scale cheesewire without even being cut through). A coffee cup has no holes (regardless) if you don't count any form of sippy-lid it might have. 22:25, 12 August 2022 (UTC)

Actually, the mug has two at the macro level (the hole that makes up the handle and the hole on the top). There could conceivably be more shallow holes inside the mug where the handle connects to the cup. At a plank-length level, the atoms could be viewed as holes in the vacuum bending space time around it.
You're not a topologist, certainly. And a hydrogen-nucleus is approximately 10^20 times the planck-length. The whole atom on the order of 10,000 times larger, and the constiuent quarks 'only' 1,000th, or so, smaller, with the differences being the space betweenn that anything that cares isn't going to consider much of an obstruction. 23:43, 12 August 2022 (UTC)
There is no "hole" at the top - at best it count as an indention in the surface (talk) 23:38, 12 August 2022 (please sign your comments with ~~~~)
Hole has multiple meanings. A hole in the ground doesn't have to go all the way through the Earth. The point of panel three is that we don't know what definition the question is using, which makes it impossible to answer correctly.Zzyzx (talk) 00:47, 13 August 2022 (UTC)
Do coffee “cups” not have handles wherever you are? Google image search shows white ceramic cups with rounded bottoms, wider than they are high, with round handles that a finger or two can pass through, on saucers; and that is indeed what I think of when I hear “coffee cup”. Wikipedia shows similar examples in other colours and materials. In my understanding, it is entirely equivalent to a mug-with-a-handle topologically and has the same one hole. Oh, are you perhaps thinking of those cardboard cups you get from vending machines and cheap coffee shops? I wouldn’t call them “coffee cups” at all; just “paper cups”. Chortos-2 (talk) 13:01, 13 August 2022 (UTC)
For my part, "wider than tall and rounded (or even very tapered)" is a cup (it cups the liquid), hence "teacup", and they mostly do have handles, whilst the shape held in the comic is a mug for being more a height-dominant cylinder (or close to it). Topologically the same, but distinct in fully-fleshed form (at least for those of either not morphologically distorted towards the other, a tall cup or a wide mug, say).
A "paper cup that coffee comes in" (or a similar re-usable "cup-for-life") that does not have a handle is, however, always a cup even if it's taller than wide, for reasons clearly more descriptivist than prescriptivist in origin. There are no "paper mugs", that I'm aware of; I know you have plastic cup-holding things that give you a (re-usable) handle to hold the thing that the cup sits in so that you don't have to grip a thin, fragile and very heated disposable/vendable cup skin-on-'skin', but that's a holder for a cup and it's still a cup that it holds.
I have no compunction in calling the comic's container a mug, based entirely upon its appearance, though obviously applying my own cultural/learnt distinctions to this. YMMV. 19:08, 13 August 2022 (UTC)
In my experience, it's almost never possible to get even a single finger through the handle on one of those cups. So from a finger perspective they have no holes. 10:19, 15 August 2022 (UTC)
Where it does have a hole (rather than be a solid blade with a thick rim for ripping purposes) the intention is clearly to have the skin-on-skin between finger and thumb as a part of the grip-enhancement. As the hand is (ignoring blood vessels in its interior) not topologically a loop yet is touching then that qualifies the loop of plastic (however unnavigable by any whole digit) as a hole through which such contact can be made. Much more so than the fuss with what loops there are in an Alexander horned sphere, certainly.
Contact between components may also count, especially as the typical 'basket' form of such a cup-holder (definite holes) is now part of the cup-assemblage unit leaving no (or even more insignificant) gaps where those holes were in the holder-alone. In which case you would indeed consider the pinched digits to be looped (finger/thumb/inter-digit-'webbing' forming the hole) and then the handle that they loop through to form a must in turn be a loop to go though the interossic(?)-hole that has a hole. Which may then topologically create a two-domain composite topology (both parts of which are genus-1-ish toroidality) for which I can't currently imagine the terminology. But it'd be interesting to look at the Borromean Rings object and work out what professional topologists think about that (three loops, none of which are individually linked to any of the other two, but they are inseperable from both of the other two). ...sorry, just idly musing about that, not sure it's entirely relevent to the coffee-mugs/etc here. ;) 12:23, 15 August 2022 (UTC)
That may be the intention, but it's not the reality - to all intents and purposes the handle might as well be a solid blade in most cases. Also, plastic?? Philistine!! [Edit] Wait - I think you're talking about the things for use with coffee shop takeaways? I was responding to the earlier comment about round-bottomed china cups. 12:59, 15 August 2022 (UTC)
Randall uses Coffe Cup and those type of cups are shown on wikipedia for coffee cups, so we should use coffee cup in the explanation and I have corrected this and just mentions that it is a coffee cup of the Mug type. --Kynde (talk) 10:50, 15 August 2022 (UTC)

[2] 23:54, 12 August 2022 (UTC)

For something to be a hole, you need to consider what is capable of passing through the hole. For instance, a mesh screen might have no holes that my fingers can pass through, but it is full of holes for water or air to pass through. And while atoms might be mostly space, other atoms can't usually just pass through that space, although high-energy particles may. Also, the space can be considered filled with forces, which may act as barriers to certain things. 00:36, 13 August 2022 (UTC)

Sure, for one definition of “hole.” That’s the whole point of the comic: there are multiple definitions, and no single definition is correct. Szeth Pancakes (talk) 01:01, 13 August 2022 (UTC)

Is “cup” or “mug” better for the explanation? “Mug” is a better descriptor, but it’s described as “cup” in the comic, so that would be more faithful to what Randall intended. Szeth Pancakes (talk) 01:25, 13 August 2022 (UTC)

Coffee Cup in the explanation with mention of Mug. I have done that --Kynde (talk) 10:50, 15 August 2022 (UTC)

Linguist: Zero to Two... mostly. Given linguistic variation and local functional style the object being referred to may not have a closed handle, or any handle at all (Cup vs Mug), and the top may be considered a hole in the common usage. --- 01:33, 13 August 2022 (UTC)

You've left out the deep dark hole of despair at your existence that's reflected back at you if it's your first coffee of the day. 10:24, 15 August 2022 (UTC)

Part of the joke is that all five methods don't discern between a cup and a mug, the original cliché being that topologists are unusual because they don't. 03:06, 13 August 2022 (UTC)

All methods dicern and topologist especially notices the difference so this sentence makes no sense --Kynde (talk) 10:50, 15 August 2022 (UTC)

Someone should mention that part of the joke is that when the topologist says it has one hole, they're referring to the hole in the handle, while in the next panel the "normal person" assumes the one hole they mentioned is the opening and questions its validity. 03:25, 13 August 2022 (UTC)

Has been done --Kynde (talk) 10:50, 15 August 2022 (UTC)

All frames except the first and last depict a mug; a topologist most definitely discerns between a a cup and a mug because they give different answers, the "normal" person is only questioning a specific feature, and the philosopher is clearly considering a mug. If it's part of the joke the only contrast is the question. Seems way too subtle for Mr Munroes normal style. probably just what he is used to calling it. 07:04, 13 August 2022 (UTC)

Yes it is a coffee cup of the Mug type. A shame he drew it like this because the mug/cup discussion has nothing with the comic to do at all. --Kynde (talk) 10:50, 15 August 2022 (UTC)

We have a lot of visual aids for topology in this comic, and none for the article about 2625: Field Topology. That seems backwards to me. 22:47, 13 August 2022 (UTC)

That's a good point. Please find photos of the various sports fields and edit them to overlay brightly colored and contrastive lines showing where their holes are, link to them on the admin noticeboard, and I'm sure someone will upload and add them. I think they turned off uploads by IPs and new users to discourage troll vandals. 01:07, 14 August 2022 (UTC)

In the physicist paragraph, I put an Actual Citation Needed tag after "factorial of the number of particles in the universe" because, while I see what is being got at, with string theory of force mediation e.g. photons (and gravitons? or Higgs bosons?) it would be really nice to have a reference for that topic. 01:37, 14 August 2022 (UTC)

Gotchu fam [3] 18:19, 14 August 2022 (UTC)

In LQG, at each instant of time, geometry is concentrated on one dimensional structures, called graphs, which can be arbitrarily complicated. But I don't think this implies uncountable holes?

Look, I know you're all having a super-important topology discussion or whatever you call it, but did you know today is Star Trek day on 2636: What If? 2 Countdown? 18:27, 14 August 2022 (UTC)

The philosopher may be referencing the following thought experiment: If you add a hole to a balloon, the result is equivalent to a flat disk that has 0 holes. Therefore, a balloon has -1 holes. (See this Stand Up Maths video for instance.) 18:51, 14 August 2022 (UTC)

Dear topologists, which interior is the inside of a balloon? Relative interior or one of its see-alsos, or something else? 21:46, 14 August 2022 (UTC)
IDK - but I do not think so. It is just Randall's way of showing us that on our scale a coffee cup with a handle has exactly one hole. I'm sure he is on the topologist side, but think it is a stupid question to ask regular people. --Kynde (talk) 10:50, 15 August 2022 (UTC)
IDKE (E=either.) 10:33, 19 August 2022 (UTC)
What you're looking for is the interior in the sense of the Jordan curve theorem --Light rays (talk) 23:03, 29 August 2022 (UTC)

I think the philosopher explanation is a bit misleading? "Now" refers to the hypothetical scenario in which there is an extra hole. Natural language philosophy is often (always?) ambiguous, but philosophers generally accept that some interpretations are misreadings and apply the principle of charity. They are not known for "gotcha" trick questions. Rather, Hairbun's question is an example of an "intuition pump", a hypothetical scenario intended to test the reader's intuitions with regard to the concept. So in this case, we might expect the reader to answer that there are now n+1 holes, where n = the number of holes before we made a new hole. You could poll people with this question to get data on the popular understanding of "hole" and "cup" as used in everyday language. Wordnerd (talk) 01:56, 15 August 2022 (UTC)

Must have been changed since this comment was written... --Kynde (talk) 10:50, 15 August 2022 (UTC)

Coincidentally, a recent survey asked people "how many holes are there in a straw". )Most people were evenly split between 1 and 2.) https://news.yahoo.com/voices-many-holes-straw-answer-095803773.html Shamino (talk) 15:20, 15 August 2022 (UTC)

Presumably that was a straw poll? 15:37, 15 August 2022 (UTC)
How dare you 03:42, 16 August 2022 (UTC)

All this reminds me of the joke of sending the new guy to the stores to fetch a bag of holes. 08:52, 16 August 2022 (UTC)

I have broken plenty of cheap earthenware cups in my lifetime, and it seems to me the material invariably contains visible voids! Do those count as "holes"? Also: dear chemist, the contents of a cup are not part of the cup! Koskinoman (talk) 18:47, 21 August 2022 (UTC)

“Most people would recognize that there is a hole through the handle.” No? If I had to imitate a normal person, I would never call the handle a hole. Reason: it is a handle. A handle is not a hole. You do not put fingers “through” it, you grab it. Therefore it is a handle, not a hole. — So much for my imagination of normal people. ;) -- 20:23, 29 August 2022 (UTC)