804: Pumpkin Carving
Title text: The Banach-Tarski theorem was actually first developed by King Solomon, but his gruesome attempts to apply it set back set theory for centuries.
This comic is a reference to the custom of carving pumpkins to set out on porches and front steps for the holiday of Halloween, which occurs on October 31. The pumpkin has the inside emptied out and a face or design carved in the side. Then a light is placed inside (usually a candle). These are called "Jack-O'-Lanterns". The Jack-O'-Lantern in the 3rd frame is the typical and standard design for a carved pumpkin.
Beret Guy, naturally, stays oddly on-topic by carving a pumpkin in his pumpkin.
In the 2nd frame, Black Hat is putting nitroglycerin (a non-igniting explosive) into his carved pumpkin, which would cause it to explode when agitated. Teenagers are a rather impulsive and rebellious lot; as Halloween is a night with lots of meticulously erected decorations and more lax parental supervision, troublemaker teens see it as an enticing time to engage in rampant vandalism, including but not limited to pumpkin-smashing. Hence, the off-panel character presumes that Black Hat is setting up a trap to get back at these ne'er-do-wells.
Black Hat, rather unconvincingly, insists that his pumpkin is suffering from chest pains, and that he is merely treating them with nitroglycerin. While it is true that this chemical is used to treat angina (chest pain due to blocked arteries in the heart), nitroglycerin used for this purpose is dispensed in small spray bottles and controlled by prescription. Also, pumpkins do not contain nervous or circulatory systems of mammalian complexity; even if they did, the process of pumpkin carving involves hollowing them out, making it a moot point.
To top it off, Black Hat plans to put up a sign warning passers-by to not smash the pumpkin. This would only serve to tempt impulsive teenagers to disturb it, which is very likely what the sadistic and chaos-loving Black Hat is hoping for.
In the 3rd frame, Megan is our typical emotional xkcd comic character. She is projecting herself onto the jack-o'-lantern as she tries to distract herself with holiday traditions that won't work to distract her.
In the 4th frame, Cueball is referencing the Banach-Tarski paradox, a theorem which states that it is possible to carve a three-dimensional ball, in this case a pumpkin, into a finite number of "pieces," and then reassemble the "pieces" into two different balls identical to the original. This paradox has been proven for just about anything theoretically, but requires infinitely complicated pieces, which are impossible for anything made of physical atoms rather than mathematical points. The person off-screen in that frame references the Axiom of Choice, which says that given a set of buckets or bins, each containing one or more objects, it is possible to select exactly one object from each bucket. The Banach-Tarski rests on several axioms which are fairly well respected, but also requires the Axiom of Choice to work correctly. So a person who does not believe in the Axiom of Choice would not have been able to do what Cueball managed to do.
The title text says that King Solomon developed the Banach-Tarski theorem first. This is a reference to the story of two women being brought before him. Both were arguing that a particular child was their own. Solomon said that the solution was to cut the baby in half and give each woman one of the halves. One of the two women said that the other should have the baby whole. Solomon then knew she was the true mother, and gave her the child. The joke is that Solomon may have actually intended to cut the child, but, believing that two whole children could be made from the one, intended to give a baby to each woman, and the Banach-Tarski paradox states that, were the baby infinitely divisible, it should have been possible.
- [Beret guy stands next to a pumpkin with a picture of a pumpkin carved into it.]
- Interlocutor: So what did you—
- Beret Guy: I carved a pumpkin!
- Interlocutor: ...
- [Black Hat stands next to a pumpkin and a box labeled "Nitro-glycerin. Do not shake."]
- Interlocutor: Taking on teen vandals, I see.
- Black Hat: Heavens, No. My pumpkin simply has chest pains. In fact, I'll leave a note warning them not to smash it.
- [Megan stands next to a jack-o' lantern.]
- Megan: My pumpkin's name is Harold. He just realized that all the time he used to spend daydreaming, he now spends worrying. He'll try to distract himself later with holiday traditions, but it won't work.
- [Cueball stands next to two pumpkins and a knife.]
- Cueball: I carved and carved, and the next thing I knew I had two pumpkins.
- Interlocutor: I told you not to take the axiom of choice.
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