Sunday, August 10, 2025

My Tom L post inspired a mathematical definition of Rabbithole

NICK: I read and enjoyed your blog post on Tom L (see here). I then spent 40 minutes down a rabbithole listening to his music on YouTube.

BILL: You call that a rabbit hole?!  A while back I spent 3 hours reading questions and answers on quora ranging from is Michelle Obama actually a man? to Is Donald Trump the Anti-Christ? (The answer to both is no.) THATS a rabbithole. Listening to Tom L is not.

NICK: SO... what is and isn't a rabbit hole? Also, is it rabbithole or rabbit hole?

BILL: Spellcheck is happy with rabbithole, hence so am I. As to your question,  we need a function \(f\) and a threshold \(T\) such that

if you spend \(x\) minutes on A, and 

you get y enjoyment out of it, where \(0\le y\le 10\), and

 \(f(x,y)\ge T\)  

then A is a  rabbithole.

(Note- I was kidding. There can't possibly be a function f that works.)

ONE DAY LATER

NICK: I asked Gemini  what \(f\) and \(T\) are  and it told me:

 \(f(x,y) = \frac{x}{y+1} \) and  

\(T=20\).

  It also gave me some examples:


1) The Tom L. Example: 40 mins, 8/10 enjoyment. Score 4.44. NOT a Rabbit Hole!

2) Doomscrolling: 90 mins, 1/10 enjoyment. Score  45. Rabbit Hole!

3) Binge-Watching Mediocre TV: 3 hours (180 mins), 4/10 enjoyment. Score = 36. Rabbit Hole!

4) Incredible Documentary: 3 hours (180 mins), 9/10 enjoyment.  18. Worthwhile but its close.





Gemini also output a heatmap, see here.

BILL: Uh- I was only kidding.

NICK: Well, the jokes on you. 

5 comments:

  1. I don't think this definition captures the essence. I think rabbit hole requires more than just time vs. enjoyment.

    Merriam-Webster says, "a complexly bizarre or difficult state or situation conceived of as a hole into which one falls or descends; especially: one in which the pursuit of something (such as an answer or solution) leads to other questions, problems, or pursuits".

    That seems correct to me.

    I didn't know it before, but the etymology is from Alice!

    The dictionaries spell it "rabbit hole", which is how I would spell it.

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  2. I have heard the phrase ``I fell down a rabbit hole' to mean exactly what I described- spending to much time on the web- often going from link to link and getting far away from your original interest- e.g., you look up general relativity and end up reading the Wikipedia entry on HOT TUB TIME MACHINE.

    Words and phrases change their meaning, or have two common uses, over time. EITHER NOT that many people use it as I describe and you are correct, OR the meaning I have is spreading and at some point the dictionaries will change.

    Henceforth I will use `Rabbit hole' not `Rabbithole'

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    Replies
    1. The "getting far away" part isn't in f. So, I think I agree with your definition, but not with Gemini.

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  3. I think f(x,y) (rabbitholeness? depth?) cannot be inversely proportional to enjoyment.

    I prefer to think of rabbitholeness as a property of a path through a graph.

    When I share this link of Tom Lehrer discussing Kurt Weill,

    https://www.youtube.com/playlist?list=PLNXjZzwMqcB8TyvQV_SnwaXHGhQ--uiRb

    I add an edge connecting the vertex that is this blog to the vertex that is that playlist of recordings of Tom Lehrer discussing Kurt Weill. We can add a weight to this edge that describes the probability that a blog reader navigates their attention from this blog to the Tom Lehrer playlist.

    Falling down a rabbit hole, then, is like taking a biased random walk through a graph, and the rabbitholeness of the path taken is one over the product of probability weights assigned to each edge of the path.

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  4. AH- i think of a rabbit hole as a long period of time where you say WHY DID I READ ALL OF THAT CRAP! You seem to be definiing it as JUST spending a long time on the web-perhaps its more of a rabbit hole if you visit more sites. So if you go from Quantum To Relativiity To time-space-contiuum TO Hot tub time machine TO a list of movies that had terrible sequels TO a list of bad movies made into great TV shows to Buffy the Vampie Slayer to Bram Stroker to Physics in Bram Strokers Time to Relatiivey THAT IS A RABBIT HOLE even if you enjoyed it. Am I correct in saying what I think you mean? I hope so since if not then I wasted a lot of time (a rabbit hole?) coming up with all of those connections.

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