# The Best Resources To Build Any Intuition

Many textbooks, tutorials or ... tapes leave out the ways people actually think about a subject, abandoning you to fumble your way to your own intuition. They don't even attempt to help you build intuitions. (Looking at you, Bourbaki.) This sucks. I love it when explanations try to tap into what I can touch, feel and see and every other bit of my [native architecture](https://www.overcomingbias.com/p/use-the-nativehtml). Yes, even with great resources, you still have to put in work to understand why an idea is intuitive. But you'll be left with much richer than when you started. 

I've occasionally found Luke's *The Best Textbooks on Every Subject* thread useful[^cxgi0ta9f8n], and [this tweet](https://x.com/norvid_studies/status/1953882767462220241) reminded me that I *don't* have an analogue for intuitive explanations. Time to fix that. 

Rules
=====

Share links to resources below! Share them frivolously!  I will add the shared resources to the post. Here are the loose rules:

1.  Recall a resource you've seen that conveys an intuition (or even a group of intuitions). A rule of thumb for sharing: did you actually get the intuition from this resource? Even in part? If so, share it.
2.  The resource can be anything, for any intuition in any subject. You can write it down, yourself, if you can't find a link.
3.  If it can be linked, post the title and (optionally) a link to the resource.
4.  Give a reason: indicate what intuition this resource helps build, and why it's useful. Provide a bit of info on why this resource is useful, e.g. you found the explanations intuitive.[^xksbn58gh6b] 
5.  Describe the intuition/idea.

To make the comments easy to navigate, please format your comment as follows:[^xksbn58gh6b]

> **Link**: [Proofs and Refutations](https://dl1.cuni.cz/pluginfile.php/730446/mod_resource/content/2/Imre%20Lakatos%3B%20Proofs%20and%20Refutations.pdf)
> 
> **Idea**: V+F=E+2
> 
> **Creator**: Imre Laktos   
> 
> **Reason**: A (fantastic) essay on mathematical philosophy, how mathematics is actually done, the difficulty of coming up with good definitions, Euler's characteristic, and how we came to prove V+F=E+2 for convex polyhedra. After reading through it, I came up with two additional proofs on the spot, using the core proof idea. My gold-standard for this thread. 

List of Resources
=================

*(last updated 24-08-2025)*

### [Bayes Rule: Guide](https://www.lesswrong.com/w/bayes-rule?lens=bayes-rule-guide)

**Idea:** Bayes Rule

**Creator:** Eliezer Yudkowsky, Eric Rogstad, Alexei

**Reason:** Clearly written, rigorous, works through a number of examples, gives a number of ways to think about the topic, provides mental images for picturing Bayesian updates, tests your understanding, and lets you customize the depth and breadth of the explanation. 

### [Diarc's belt trick](https://www.youtube.com/watch?v=ACZC_XEyg9U)

**Idea**: Spin 1/2 particles rotate twice to return to their start. 

**Creator**: NoahExplainsPhysics (Youtube Channel)

**Reason:** Clarifies Dirac's belt-trick, a famous intuition that Dirac provided for why spin 1/2 particles need to be rotated 720 degrees to return to their starting point. After this, I felt like I could *see* the mental picture Dirac was attempting to communicate with his belt. 

### [Thinking and Explaining](https://mathoverflow.net/questions/38639/thinking-and-explaining)

**Idea**: Various bits of mathematics

**Creator**: Bill Thurston (thread creator), et. al (responses)

**Reason**: This is a thread with some of the world's greatest mathematicians explaining the bits of their mental models that they leave out of their usual explanations. Many of these I've never heard of anywhere else. Just look at this example (chosen for humour) from Terrence Tao: 

>   
> In one extreme case, I ended up rolling around on the floor with my eyes closed in order to understand the effect of a gauge transformation that was based on this type of interaction between different frequencies.

If that's not enough for you, Bill Thurston, who defined an era in geometry and topology, wrote a lot of interesting comments in the thread. Self-recommending.

### [Proofs and Refutations](https://dl1.cuni.cz/pluginfile.php/730446/mod_resource/content/2/Imre%20Lakatos%3B%20Proofs%20and%20Refutations.pdf)

**Idea**: V+F=E+2

**Creator**: Imre Laktos 

**Reason**: A (fantastic) essay on mathematical philosophy, how mathematics is actually done, the difficulty of coming up with good definitions, Euler's characteristic, and how we came to prove V+F=E+2 for convex polyhedra. After reading through it, I came up with two additional proofs on the spot, using the core proof idea. My gold-standard for this thread.  

### Undergrad++ math; [The Napkin Project ](https://web.evanchen.cc/napkin.html)

**Idea**: high-quality explanations motivating undergrad++ math to bright high schoolers

**Creator**: Evan Chen (US IMO coach)

**Reason**: "I'll quote Evan himself:

> The philosophy is stated in the preamble:
> 
> > I’ll be eating a quick lunch with some friends of mine who are still in high school. They’ll ask me what I’ve been up to the last few weeks, and I’ll tell them that I’ve been learning category theory. They’ll ask me what category theory is about. I tell them it’s about abstracting things by looking at just the structure-preserving morphisms between them, rather than the objects themselves. I’ll try to give them the standard example Gp, but then I’ll realize that they don’t know what a homomorphism is. So then I’ll start trying to explain what a homomorphism is, but then I’ll remember that they haven’t learned what a group is. So then I’ll start trying to explain what a group is, but by the time I finish writing the group axioms on my napkin, they’ve already forgotten why I was talking about groups in the first place. And then it’s 1PM, people need to go places, and I can’t help but think:
> > 
> > *Man, if I had forty hours instead of forty minutes, I bet I could actually have explained this all.*
> > 
> > This book is my attempt at those forty hours.
> 
> This project has evolved to more than just forty hours.

Caveat that I didn't actually do a math degree so I'd be curious to see takes from math folks saying disliking it, but I've enjoyed dipping in and out of its 1,048 pages over the years." - [@Mo Putera](https://www.lesswrong.com/users/mo-putera?mention=user) 

### [FFT (Fast Fourier Transform)](https://www.youtube.com/watch?v=h7apO7q16V0)

**Idea**: Motivates FFT as accelerating polynomial multiplication in a special representation. 

**Creator**: [Reducible](https://www.youtube.com/@Reducible) (YouTube Channel)

**Reason:** "The video that made FFT finally click for me." - [@jaan](https://www.lesswrong.com/users/jaan?mention=user).

### \[Geometry of\] The Determinant

Also [Linear Algebra Done Right](https://linear.axler.net/) by Sheldon Axler

**Idea:** The determinant of a matrix tells you the (signed) volume of a unit cube after applying the matrix transformation

**Creator:** Grant Sanderson (3Blue1Brown), Sheldon Axler

**Reason:** This geometric interpretation makes properties that seem arbitrary in formula-based definitions suddenly obvious. For example,

*   Why det(AB) = det(A) det(B)
*   Why det(A^T) = det(A)
*   Why det(A^-1) = 1/det(A)
*   Why det(kA) = k^n det(A)
*   Why det(A) = 0 means a matrix is not invertible
*   Why rank < n means det = 0
*   Why swapping matrix rows multiplies the determinant by -1

### [Thinking Physics ](https://archive.org/details/ThinkingPhysicsPracticalLessonsInCriticalThinking_201808)

Idea: core intuitions in mechanics, optics, electromagnetism, fluids e.g. conservation laws, frames of reference, work-energy relationships etc.

Creator: Lewis Carrol Epstein

Reason: focuses on physical reasoning and intuition rather than computation. Isolating a skill is the best way to improve it. 

[^cxgi0ta9f8n]: Likewise for The Best Software For Every Need, and The Best Tacit Knowledge Videos on Every Subject, both of which I plagarized horribly.  

[^xksbn58gh6b]: Feel free to leave out the "why" section.