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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.

Explanation

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.

This comic depicts people in different fields of study answering the question, "How many holes are there in a coffee cup?" This question can have multiple interpretations, in particular concerning the definition of a hole.

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. A common joke is that topologists can't tell the difference between a coffee cup and a donut since they're homeomorphic to each other — they have the same genus. From the topologist's point of view, the coffee cup definitely has one hole. See 2625: Field Topology for more information about topology.

Hairy, 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.

A genus two surface

Hairbun, a philosopher, answers the question with an elucidating counter-question, considering a hypothetical scenario. Drilling a new hole should increase the number of holes by one, and after the hole has been drilled, a common teacup or mug has two holes according to topologists. Since drilling a hole increases the number of holes by one,[citation needed] the philosopher's question requires the original questioner to reveal the answer to their own question. (Also, she asks how many holes there are now rather than after we do that, an ambiguity.)

A point cloud of a genus one surface

Cueball, a chemist, looks at the coffee in the cup on a molecular level, which means it has very many holes: 1,000,000,000,000,000,000,000 (1021 or 1 sextillion) “in the caffeine alone.” An implication is that there are more holes in the cup itself, with which normal people and topologists would probably disagree. Also, the coffee itself could have other holes, depending on the type of coffee. For example, espresso contains significant amounts of niacin and riboflavin, each of which has at least one hole in its chemical structure. However, this ignores the fact that bonds are not discrete sticks as portrayed in many molecular models. The "holes" in the middle of a caffeine molecule are not completely empty but instead merely have lower electron densities/probabilities. 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.

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, or at least within the cup's light cone, which is a number impractical to accurately count, but not uncountable in a mathematical sense.

Part of the joke could be that all five methods of inquiry don't discern between a cup (as described) and a mug (as depicted), the cliché being that topologists are unusual because they don't. Or, as many people use the terms interchangeably, Randall may too.

Transcript

Ambox notice.png This transcript is incomplete. Please help editing it! Thanks.
[The first panel has text only. The "Q:" below is a large letter Q representing a question, not a character name.]
Q:
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.]
Caption: Topologist
[Ponytail stands holding a coffee mug.]
Ponytail: One.
Caption: Normal person
[Hairy stands to the right of Ponytail, holding a coffee mug at an angle to look into it.]
Hairy: IDK, does the opening count as a hole?
Caption: Philosopher
[Hairbun is shown in closeup, with two drawings of coffee mugs to her left.]
Hairbun: To answer that question, consider another: If we drill a hole in the side, how many holes are there now?
Caption: Chemist
[Cueball stands with a drawing of a caffeine molecule above him and to the right.]
Cueball: 1021 in the caffeine alone


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