Visual group theory

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1. impendiaJun 2026
   Math professor here. If you want to learn math, then for the most part I recommend choosing time-tested avenues, using popular materials.

   There are two reasons for this:

   (1) Popular materials are usually popular for a reason: they reflect an approximate consensus, across a significant fraction of the mathematical community, that their approaches are more-or-less the best.

   (2) If you learn the same way everyone else does, you'll have an easier time talking to others and finding materials on the internet.

   I know some very innovative books which I highly recommend, for example Visual Group Theory by Nathan Carter:

   https://bookstore.ams.org/clrm-32/

   But the innovation is pedagogical, in what Carter chooses to emphasize and how he presents everything. At the book's core, Carter agrees with everyone else about what the foundations of group theory are and should be.

   Even Sheldon Axler's Linear Algebra Done Right (another excellent book), with its hilariously provocative title, only differs in its choice of emphasis and order of presentation. His choices are quite compatible with everyone else's.

   https://linear.axler.net/LADR4e.pdf

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2. godelskiAug 2025
   I highly encourage you to talk to a mathematician about this stuff. In high dimensions everything is all about caveats and edge cases. Your {1,2,3}D intuition leads you astray rather than helps you closer to the answer. In fact, math is specifically an overly pedantic because nuances are so critical.

   I'll use an example that we can reason from low dimensions but that many people make the wrong connections. What is the probability that in N dimensions two i.i.d randomly drawn vectors are orthogonal? We can easily intuit in low dimensions that this is a pretty unlikely event. But in high dimensions exactly the opposite is true. An easy mistake to make is to not think about this question from a continuous perspective, by reframing the question to ask what the probability that the angle between to i.i.d drawn vectors are within some epsilon bound. If we don't, then the answer is trivial in all cases: 0. A good student would be able to reason through this from looking at the 2D case and 3D case but this is only a sample of 2 and it would be naive to generalize such a notion without doing the proof.

   So let's look at something a little harder. A classic problem is looking at the volume of a n-ball. Our 2D and 3D understanding make this problem look easy. The volume of the 3D ball is much larger than the 2D ball. We can even verify this my analytically solving the 4D case. Same with the 5D case! But that's when things go awry. For the unit n-ball, 5D is where the maximum volume is. By 10 dimensions you have less volume than our 2D case and by 20 dimensions your volume is quickly converging to 0.

   Now let's look at something wild. Consider a hypersphere with radius r inscribed within a hypercube with sides of length 2r, what proportion of the hypersphere resides within the hypercube for a given dimension D? This sounds like an absolutely dumb question if we use or 2D and 3D logic. What amount of the sphere is inside the cube? 100%! It's right there in the problem description! Right? But this is entirely wrong. As dimensionality increases, very quickly, ~0% of the hypersphere resides within the hypercube and this is all due to the fact that hypercubes and hyperspheres have very different representations in high dimensions. Yes, there is still visual understanding we can draw to help reason, but again, this requires us to look at the problem from a different vantage point (the key thing I stressed in my original post btw). We have to look at the dual of the problem and instead ask "what is the ratio of the volume of a hypersphere to the volume of the hypercube it is inscribed within?" This will help but there's also a reason I put this question after the previous one because we know that there is not a clean linear relationship here and that at least with the sphere the volume increases and then decreases. Our visual intuition only helps if we can recognize that the relationship has everything to do with the corners of the cube.

   It is easy to look at these 3 cases and see how visual intuition from low dimensions can help, but be careful about post hoc trivialization. It's easy when you know these things and after they have been told to you but they are not so clear when you're being presented with the problem. All 3 of these problems are deeply related to the nearest neighbor problem and the concentration of measure. Where we run into the problem that it becomes nearly impossible to distinguish the nearest point to the furthest point. Leveraging our lower dimensional visualization can provide help but it is important to remember that they are a small part of the much larger story here. Which of course should make sense because as the dimensionality grows our 2D/3D slices of those are a smaller and smaller portion.

   This is why I referenced the blind man and the elephant. Not because you can't use visualization to aid you, but because to see the elephant there you have to sample that elephant at many different places. And just like the blind man, you don't know how big that elephant is and where things are changing.

   But this stuff isn't going to help with even more abstract concepts. Like how there is no division algebra in 3D and how the largest is represented via the Octonion. How by moving to quaternions we lose commutativity and how moving to octonions we lose associativity. These are very fundamental features of math that we are losing while moving to higher dimensions.

   So I'm not sure why you're pointing to Farin and Hansford's book. Their highly concentrated on low dimensions and only briefly discuss generalization at the very end. They miss many important concepts. Not because the book is poorly written but because they are far out of scope from what is being taught. My caution about high dimensional spaces does not mean I don't absolutely love Needham's books on Visual Differential Geometry and Forums or his book Visual Complex Analysis or even Carter's inspired Visual Group Theory. These are masterpieces and books I highly recommend. But it is about what is being communicated. We work frequently in lower dimensions and there is a wealth of information there. But this does not mean higher dimensions are not absolutely littered with pitfalls and paradoxes. The true study of high dimensional spaces is relatively new and drove people like Cantor and Boltzmann insane. Every small mistake that is easy to make in low dimensions becomes exponentially more important in high dimensions. Yes, there are dimensional reduction techniques and these help with varying degrees (utility being critically dependent upon the latent dimensionality of the data) but every single one of these comes with major concessions. There is often even very bad science performed due to this lack of understanding. Techniques like t-SNE and UMAP are routinely unknowingly abused to draw inaccurate conclusions about these spaces, as it is possible to perform nearly any clustering you wish (see Lior Pachter's Picasso). Even PCA is frequently abused in this area due to there being subtle assumptions being invalid on different problems.

   My point is: you absolutely should warn people that there are many pitfalls when dealing with high dimensions. Anyone brushing these differences off as inconsequential nuances is just ignorant of this rich space. I don't blame anyone for being unaware of these things because it requires some very complex mathematics to truly understand, and we're talking about math that the average scientist is never going to be introduced to. You likely won't even see this in a given science PhD. Not because it doesn't matter, but because it is just very difficult math. And we learn math through a game of telephone that is often hyper focused (not a great way to learn it tbh)

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3. godelskiMay 2025
   It think this is silly on multiple accounts. I'll claim that there's not real thing such as a "language brain" or "math brain." I'll also claim that most people don't know what math is, and that their evidence supports a "math brain".

   Math isn't about calculations/computations, it is about patterns. You get to algebra and think "what are these letters doing in my math" but once you get further you think "what are these numbers doing in my math?"

   A great tragedy we have in math education is that we focus so much on calculation. There's tons of useful subjects that are only taught once people get to an undergraduate math degree or grad school despite being understandable by children. The basics of things like group theory, combinatorics, graphs, set theory, category theory, etc. All of these also have herculean levels of depth, but there's plenty of things that formalize our way of thinking yet are easily understandable by children. If you want to see an example, I recommend Visual Group Theory[0]. Math is all about abstraction and for some reason we reserve that till "late in the game". But I can certainly say that getting this stuff accelerates learning and has a profound effect on the way I think. Though an important component of that is ensuring that you really take to heart the abstraction, not getting in your own way by thinking these tools only apply in very specific applications. A lot of people struggle with word problems, but even though they might involve silly situations like having a cousin named Throckmorton or him wanting to buy 500 watermelons, they really are part of that connection from math to reality.

   This is why "advanced" math accelerating my learning, because that "level" of math is about teaching you abstractions. Ways to think. These are tremendously helpful even if you do not end up writing down equations. Because, math isn't really about writing down equations. But we do it because it sure helps, especially when shit gets complicated.

   [0] https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53N...

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4. impendiaFeb 2025
   (Math professor here)

   What do you mean by "foundational"?

   I love Calculus Made Easy, but in my opinion the reason it's so good is that it doesn't try to be foundational. It relies on quick and dirty tricks -- especially, treating dy/dx as a fraction when it can't be formally defined as such, but where doing so makes the subject much more intuitive and easy. It's not really a good foundation for further study, you need something that doesn't cut corners for that, but it's fantastic for what it is!

   My favorite foundational book is Linear Algebra by Jim Hefferon:

   https://hefferon.net/linearalgebra/

   It is free, and I've seen the author on HN occasionally. It's foundational in the sense that it builds up the subject brick by brick, assuming the minimum realistic prerequisites needed, and it does an outstanding job of this.

   But it's a slow read, essentially by design. If you want something quicker, you might try Interactive Linear Algebra by Margalit and Rabinoff:

   https://textbooks.math.gatech.edu/ila/

   Again free, with lots of widgets to play with.

   Some other textbooks I really like:

   Epp's Discrete Mathematics is fantastic. The newest edition is astronomically expensive, so go looking for used copies of previous editions.

   Carter's Visual Group Theory is tremendous fun. Once again not foundational per se, but if you want to learn what group theory is all about without slogging through lots of formalities, that's a great choice. I'd say that roughly it's in the spirit of Calculus Made Easy.

   Also, if you enjoy poker, check out Dan Harrington's books on the subject. In the course of explaining hold'em strategy, they will teach you lots about probability, counting, expected value, and game theory in an applied setting. They're fantastically written.

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5. impendiaJan 2024
   For group theory, I love Nathan Carter's Visual Group Theory -- very example and picture based. Rather than treating groups abstractly from the beginning, it treats them as "groups of actions" which in my opinion is the best way to view the subject.

   For a more traditional, formal approach, Saracino's "Abstract Algebra: A First Course" is my favorite among this genre.

   I have also heard good things about Pinter's "A Book of Abstract Algebra", and it is available quite cheaply. I have not read it though.

   Good luck!

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6. hintymadJan 2023
   For what level and in which area? Books like _Methods of mathematical physics_ can be both too hard and irrelevant to your needs. For starters,

   To become a better problem solver with high-school level maths:

     - Polya's How to Solve It.
     - Books of your choice about math contests.
     - Concrete Maths. I understand that this book is taught in college, but it requires very little advanced maths, and its techniques are hugely useful for high school students too.
   
   

   To hone my intuitions. I learned it the hard way that college maths were different from high school math: in high school, my teachers painstakingly drilled intuitions into us with very targeted explanations and tons of well designed exercises. In college, we won't get such luxury. So, it's really up to us to understand mathematical concepts intuitively before diving into technical details. For that matter, the following books helped me a lot:

     - The visual series. Visual Complex Analysis and Visual Group Theory, for instance
     - Pinter's A book of Abstract Algebra
     - Strichartz's The Way of Analysis
     - Linear Algebra Through Geometry by Wermer. The book offers a comprehensive geometric interpretation to linear algebra concepts. It's especially helpful for me to understand quadratic forms.
   
   

   To understand Analysis better. This area is vast, so I'll skip recommendations of excellent text books:

      - _Counterexamples in Analysis_. Those counterexamples in Analysis play a huge role in helping me truly appreciate the intricacies of Analysis. Similarly, books like _Counterexamples in Probability and Real Analysis_ are of great help too.
      - The Way of Analysis by Robert S. Strichartz. This books is AMAZING for laymen like me. You'd want someone to *explain* how concepts emerge, and how intuitions evolve.
   

   To become good at maths by doing maths, so the following books used to help me a lot:

     - Problems and Proofs in Real Analysis
     - Putnam and Beyond. I still suck at maths, but those well designed problems in Putnam really taught me how to seek insights in higher maths.
     - Piotr's Problems in Mathematical Analysis. But really, any problem books that challenge you will do. I'd recommend you find problem sets from the website of university courses. They cover essential techniques, and will not be as overwhelming as the books.

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