Immersive Linear Algebra (2015) | Hacker News Reader

Immersive Linear Algebra (2015)

immersivemath.com

876 pointsoumua_don1771 comments

Very cool!

Does anyone have a list of other similar texts?

There's:

- Geometry: Joyce's Java version of Euclid's _Elements_: https://mathcs.clarku.edu/~djoyce/java/elements/elements.htm...

- Physics: https://www.motionmountain.net/

- Chemistry: The Elements by Theodore Gray https://apps.apple.com/us/app/the-elements-by-theodore-gray/...

A nifty thing my kids enjoyed was the website version of the book, _Bembo's Zoo_ (which sadly is no longer on-line: https://soundeffects.fandom.com/wiki/Bembo%27s_Zoo_(Websites... )

https://schoolyourself.org/ with different ideas about

https://www.youtube.com/watch?v=4cQIbZa7P80 Computer-Assisted Instruction

http://dmentrard.free.fr/GEOGEBRA/Maths/export4.25/golf.html some people manage to create some applets in geogebra

https://xenaproject.wordpress.com/2022/07/29/teaching-formal... a way to teach theorem proving

maybe I'll remember something else and add it here

https://explorabl.es/

https://github.com/blob42/awesome-explorables

I started on two resources on different topics. One day, I want you invest a good amount of time into building them out.

https://cstheory.avikdas.com/

https://nesdev.avikdas.com/

One on complex analysis - https://complex-analysis.com/

This was the core premise on which Innings2 books was made. In this day and age textbooks can be made interactive. So, the CBSE math syllabus (main syllabus in India) was made interactive for 6th to 10 grades.

https://books.innings2.com/demo

its not animated but its really nicely illustrated in html friendly layout

https://www.c82.net/euclid/book1/

and this is animated, and interactive, for polyhedra

https://polyhedra.tessera.li/dodecahedron/operations

You might like these-

- Matrix Arcade (https://yizhe-ang.github.io/matrix-explorable//)

- Computational Discovery on Jupyter (https://computational-discovery-on-jupyter.github.io/Computa...)

- Nature Inspired Programming Recipes (https://github.com/clever-algorithms/CleverAlgorithms)

- A Little Calculus (https://papl.cs.brown.edu/2018/func-as-data.html#%28part._.A...)

- The Matrix in Computer Science (https://cs.brown.edu/courses/cs053/current/lectures.htm)

- Discrete Mathematics and Functional Programming (https://cs.wheaton.edu/~tvandrun/dmfp/)

- A Programmer's Introduction to Mathematics (https://pimbook.org/)

- Complexity Explorer (https://www.complexityexplorer.org/)

- Learn Physics with Functional Programming (https://nostarch.com/learn-physics-functional-programming)

- Seeing Theory (https://seeing-theory.brown.edu/)

- Coding The Matrix (https://codingthematrix.com/)

I created a list of links to various freely available online learning resources about computer programming, math, and science:

https://github.com/bobeff/programming-math-science

Most of them are collected from HN posts.

I created a list of Interactive Machine Learning, Deep Learning, and Statistics websites (https://p.migdal.pl/interactive-machine-learning-list/), which includes Immersive Linear Algebra, among other things.

For a broader range of subjects, there is a list of Explorables (https://explorabl.es/).

Here's a modern version for Euclid's Elements- https://www.c82.net/euclid/

Thank God people are moving towards this modality. I have a fervent hatred for textbooks and publishers. The former are antiquated, static, badly formatted, and often ride with distracting garbage in the margin, or worse inline. It makes actually reading them far more difficult than needs be, with unremarkable asides that may span pages and that are easy to get pulled into. While I understand that they have a purpose, they aren't for everyone so having a platform with the dynamism of a webpage is something that I hope will inevitably lend itself to future development along this course. Not to mention being able to have interactive questions that give quick feedback rather than requiring turning through pages to find out if you're on the right track... And this interactive stuff is just an excellent means to drive meaning from terse and difficult to explain systems.

Cheers to the folks that put this together, a thousand thanks for the travails you've been through to blaze these trails!

See Seskir et al., "Quantum games and interactive tools for quantum technologies outreach and education", Optical Engineering 61(8), 081809(2022), https://doi.org/10.1117/1.OE.61.8.081809 (open access, but if you prefer ArXiv, https://arxiv.org/abs/2202.07756).

While it is focused on quantum mechanics & quantum computing, sections 3.2 Science-based Games, and 3.3 Explorable Explanations, discuss this medium (full disclaimer - I am one of the lead authors).

My undergrad math professor created one of the first fully online linear algebra texts: http://linear.ups.edu/html/fcla.html It's integrated with Sage, a Python library for studying (among other things) number theory. Another prof at the same university also wrote his own linear book, using a lot more illustrations, but as a traditional textbook.

I see this book as a solid evolution in both directions. Nicely done!

My gut reaction to this sort of thing is that textbooks work great and have a timeless quality. But with a little more thought, they certainly aren't optimal. Being able to see a video, hear a sound, or play with a figure can be powerful learning. I think my hesitation comes from the fact that this stuff just doesn't seem to have longevity (think educational java applets) and tends to be proprietary (shitty Pearson learning hw). How can we get newer media as robust as text and pngs?

It does not have to do with the book or any other medium. It comes from the learner. You are right about the problems mentioned.

1. Stuff does not have longevity. This is because most of the new tools are outside curriculum. This does not encourage students to learn something extra.

2. Proprietary. Books providing add ons have a different goal. They want to sell their books. So the addons or just that. Add ons. They are not complete.

These were some of the thoughts which encouraged us to build [1]. Keep it curriculum focused so kids don't learn anything extra. No books to sell :)

[1] https://books.innings2.com/

If only there was some kind of magic device that has basic input facilities coupled with some kind of simple graphics output.

I jest, but I think a very basic computational platform - think 6502-style - with simple graphics could be standardized upon and anointed as universal educational computation base.

If you don’t touch the instruction set this can go a long way. The outputs of course need to be universal, but we have some experience with outputting stuff on machines. I think we are able to come to some sort of useable situation.

Think PDF or whatever with built-in NES emulator.

Is that what you have in mind?

Yes. And the latest iteration of JavaScript/canvas has had longevity, but it doesn’t have the robustness of a JPEG. What you describe would be interesting.

WASM is a totally different path that’s also viable (I think). You can build whatever you want, as long as it compiles down to WASM you should be fine.

I’d like the standardization to be slightly lower in the abstraction tower. If people want JS or whatever, they can build it on top of WASM. Using JS as a stable basis is.. insane (but practical).

Wasm + canvas might be the worst longevity and robustness.

It's not really a textbook but at this stage, 3Blue1Brown courses almost needs an obligatory mention for how fantastic they are.

https://www.3blue1brown.com/

Previous posts on HN:

1) 2015 (78 comments):

https://news.ycombinator.com/item?id=10183725

2) 2019 (140 comments):

https://news.ycombinator.com/item?id=19264048

Thanks! Macroexpanded:

Immersive Linear Algebra - https://news.ycombinator.com/item?id=26294588 - Feb 2021 (3 comments)

Immersive Linear Algebra (2016) - https://news.ycombinator.com/item?id=19264048 - Feb 2019 (140 comments)

Immersive Linear Algebra – textbook with fully interactive figures (2015) - https://news.ycombinator.com/item?id=13904881 - March 2017 (74 comments)

Immersive Linear Algebra – A free interactive online book - https://news.ycombinator.com/item?id=10183725 - Sept 2015 (78 comments)

I hope that this sets a standard for future textbooks/publications. I haven’t been able to grasp several concepts in math unless I was able to properly visualize it, which most modern textbooks do with a terribly compressed and unsaturated JPG.

This looks like a fantastic resource. After a quick scan, I couldn't find any information on how this book was programmed / created. Does anyone know if a particular framework was used, or if this was all coded by hand?

Looks like the authors rolled out their own https://immersivemath.com/ila/javascript/samillustration3d.j...

I studied physics-based rendering from a book by one of the two authors (T. A-M), and it was written excellently. I'll have a look at this for sure, as I need a refresher every now and again.

A naive question: I see that they are using mathjax for the displaying mathematics content. But what are they using to create the interactive diagrams?

looks like the future of education. especially with VR

I remember Jaron Lanier describing this vision of learning through VR, where a student in class would "become the triangle" in learning trig, etc.

Feel like a treatment of SVD is a must for an entry level Linear Algebra course

The navigation links at the bottom of the home page are too small on my phone.

Maybe a CSS issue ...

Would love to see this done with geometric algebra AKA Clifford algebra.

The web site says math, but it seems to be only about linear algebra.

Which "subject" would you normally ascribe linear algebra to?

It looks like this is a first module within the Immersive math system. No doubt there will be more.

Why not look at what is presented rather than what is not?

Are there such immersive math courses geared more towards middle / high schoolers?

Cool to see stuff like this which makes math fun for everyone. I love stuff like this because it brings together two things I love, math and programming.

I mean textbooks are cool. But with the tools available to us we should be able to make almost any textbook interactive. It will need effort, pedagogy, programming skills and design. But it's certainly worth it.

Making an effort with this. Started with 6th to 10th grade math. Let's see how this goes.

Hell yeah. Linear algebra was a hole in my math education and has turned out to be far more relevant to my work than the calculus I spent so much time on. I’ve been teaching myself piecemeal on an as-needed basis but this looks like a great opportunity to finally get a cohesive overview.

I like this. I feel like I’m too old, at 45, to relearn and this may help!

The biggest barrier to learning is having a real motivating need to learn the thing. Young people aren't as dissolutioned about the uselessness of what they are learning relative to all the demands of adult life.

I've learned amazing things when I had a real reason pop pup, after a decade of idly wanting to learn it.

Love the bootstrap 3. Clean, it works, and embraces the principle of least surprise.

ah sick people from my uni

it's pretty messy tbh but it's a good idea

What’s messy about it?

It’s possible I’m alone in this, but the textbook / vaguely “academic” style of site always comes across as messy to me. That said, I think we’re spoiled by vaporware that invests huge money into style and flashy visuals — as anyone who’s been in school during the Blackboard/Piazza era knows, education software does not have to be pretty to be popular.

maybe the layout just messed up on the resolution I looked at it with, there were boxes all over the place, the spinny thing on the main page scaled out of its bounding box. also I think it would be helpful to have some consistent design cues for which elements are interactive or what you can do with them.

if nothing else I feel each interactive canvas should have some kind of description of what you can do with it so you don't have to trial & error

I found if I resize my browser window the spinny thing on the landing page starts spinning really fast. I like it :)

oh interesting it actually works that way whether you zoom in or out

I tried look at this page on an iPhone, and the figures kept resizing and the page was reflowing whenever I tried interacting with the diagrams, sometimes looking messy. That makes me thing it’s possible the parent was at an odd resolution that does look bad.

Looks incredible! All this needs is a little bit of conversational AI magic in the background to filter and modulate the content according to plain-English student questions and its go time.

Note that this was finished in 2019, so now would be the perfect time for someone to polish this up and expand it to the rest of math! Assuming this is threeJS, you could get an open-source file format going for simulations, and even host crowdsourced applications of it to existing popular math textbooks by Figure/page #. I mean, linear algebra is cool, but the market for good free geometry education is limitless

Does anyone know if the big names in math education offer simulations yet, or is it all animations/images/videos still?

EDIT: definitely ThreeJS — love the vector chapter. What this needs is true spatial computing support - not pages with nested simulations, but site-wide (SPA-wide) simulated objects. What if every student in geometry class could have their own simulation on their Chromebook as they read/follow along? I can’t wait.

It really pains me to see someone suggesting adding AI to a book like this. Current AIs are infamously bad at math. The last thing we need is ChatGPT misplacing a minus sign and confusing readers or setting back their understanding by weeks.

I remember a friend reviewing some math before starting grad school was stymied by a typo in her textbook for an inordinate amount of time. It’s really vital that instructional materials avoid errors as much as humanly possible. AI right now ain’t it.

True tho detecting untrue maths is a key skill. Unlike software there is no compiler tests whatever step to filter out your own minds flagrant errors.

In a graduate course, we used a horrible Russian translation of

Jacques Neveu, Mathematical Foundations of the Calculus of Probability https://www.amazon.com/dp/B0006BNQSQ

with a huge amount of misprints in formulas. I spent lots of time hunting for those misprints, and I think it really helped me understand and remember the material.

In my last homework the professor omitted a required assumption and I nevertheless proved the false assertion. Extremely embarrassing. This happened earlier in the semester and I correctly failed to finish the homework problem. I am getting tired I guess.

Yeah, that was my secret superpower. She was fresh out of engineering school and I’d been a nerd school drop out who hadn’t taken a math class in four years (and never studied the material she was reviewing), but I was able to look at the text with a critical eye at least in part because I was spending a lot of time on math typesetting and had learned that the math rarely never gets properly proofread.

Please read the comment carefully before parroting a canned retort. It doesn't say what you think it says.

(Not the person you replied to, but) I just re-read it, and the "canned retort" still looks completely accurate and relevant. Can you elaborate on why you think that AI's (known, admitted, and inherent) propensity for hallucination _wouldn't_ be disastrous in the context of pedagogy?

If the original comment had _just_ proposed to direct students to locations _within_ the original content ("filter"), it would have been less-impactful - being directed to the wrong part of a (non-hallucinated) textbook would still be confusing, but in the "this doesn't look right...?" sense, rather than the "this looks plausible (but is actually incorrect)" sense. But given that the comment referred to "Conversational AI", and to "modulat[ing]" the content (i.e. _giving_ answers, not just providing pointers to the original content), hallucination is still a problem.

Hey it’s the original commenter himself! I appreciate you taking my comment seriously enough to analyze, but I think I missed the mark; I totally agree that LLMs shouldn’t be giving the answers to literal arithmetic problems, or be anywhere near designing the materials (digital textbooks) themselves.

I was indeed referring mostly to something like filtering, but I think there’s plenty of room for an LLM to help out there. With something as relatively complex as simulation parameters, theres lots of room for them to support the users choices by making changes to machine-readable formats.

Thus the LLM would be “tweaking” or “framing” or “instantiating” the content without getting near the fundamental signal, which here is the specific pedagogical intent of that diagram in the context of the current lesson. I used “modulate” to try to express this idea somewhat clumsily, would love suggestions on a better one though from lurkers!

IMO simulations are hard to justify as embedded content of a pedagogical site because they’re so engaging, which makes them dangerous in a situation where close attention to the teacher/problem set/text is the much more important goal in the background. They’d have to be low cognitive load to use individually during class time, ideally so low they’re practically ambient, and I think LLMs are the only practical path in that direction.

TL;DR I didn’t mean writing LaTEX pedagogical content, I meant writing JSON objects that do stuff like highlighting, variants, scaling, inputting specific equations to a general sim, etc.

Oh, fascinating! OK, yeah, I fully misunderstood your intent, then - I thought you were suggesting the LLMs should be summarizing the content in response to queries from students ("How do I find the determinant of a matrix?" // "Well, first you..."), which I think we both agree that they're not ready for (and, while hallucination remains a problem, never will be).

So if I'm understanding it right, your proposal is for the LLM instead to be a "control layer" over the simulation object, so that a student could say something like "what happens if I increase the scale factor by 2?" and the LLM interprets that natural-language request and outputs the simulation-control-variables that correspond with the student's request (and then either feeds them into the simulation directly, or outputs them for the student to read, understand, and input)? Makes sense to me!

GP's comment has been edited since my post. The original said something like "regenerate diagrams according to student questions". It's obviously a bad idea if you're trying to learn vectors and the entire diagram is flipped over the X axis, for example.

Nonetheless, today's AIs still regularly contradict themselves from one sentence to the next. Even if they're only generating text and "modulating" (which I take to mean rephrasing/summarizing), mistakes can and will happen. I stand by my comment even as it applies to the edited GP.

Filtering Modulating means selecting relevant excerpts. Think "AI for Search", not conversational chat generation. This is what LLM has been exceedingly good at, in the Alpha (DeepMind) series of projects.

You clearly have no idea how effective an interactive conversation with a text can be. An AI doesn't have to be "good at math" to be useful. People (and programs) who are "good at math" are a dime a dozen. To be useful to a student, a language model just has to be good at answering questions about math.

That part works, right now. Try it. Go to ChatGPT4 and pretend you're a student who is having trouble grasping, say, what a determinant is. See how the conversation unfolds, then come back and tell us all how "infamously bad" the experience was. Better still, ask it about something you've had trouble understanding yourself.

The key is to use GPT4, not the free version.

Yes, that makes a big difference.

Many people on HN formed their opinions on the basis of GPT3.x-generation models, though. They asked it a question, they got the nonsensical or hallucinated answer they expected, they drew the conclusion they wanted to draw all along, and by golly, that settles it, once and for all.

The AI isn't doing math, the AI is curating the textbook material. In the same way that you have a host of different faculties that enable you to excel in all that you excel at, there is more to math (and math pedagogy) than arithmetical consistency

Just taking something like the threejs GLTFExporter and combining it with modelviewer.dev on the fly could enable a 'view in AR' button compatible with both SceneViewer and Quick Look (i.e. most mobile devices available today).

+1 for Spatial Computing here -- I see immersive here and just think 2D animations of 3D concepts, good start though it may be, is leaving possibilities on the table. 3D consumed inside a fully 6DOF 3D animated space is a better environment to transfer meaning. These collections of links could be piped into WebXR with just a little tweaking and really be immersive.

One critique that glares in my face is "you changed the textbooks via hackers and now my children's development is your fault". These accusations fly when money (e.g. big textbook starts a media campaign) is on the line. As such, these interactive apps need to be able to handle that level of criticism if in a public school (before college).

These applications probably need to be client side only enabled and a hash that agrees to the hash of the "approved curriculum" for most parents to think that this might be okay, or a read-only USB rental from the school library that gets verified when it is checked back in.

I will not let the government install unknown tech on my computer, but I am willing to download a read only file. I do not want to have my children learning "new math" and be able to point at the technology as the problem "because hackers" or some other BS that has sounds true but has a high amount of effort to disprove coming while chatting with parents at a bus stop or PTA meeting.

https://en.wikipedia.org/wiki/Brandolini%27s_law

It’s a web page.

I’ve never seen this particular accusation anywhere, do you have examples? You’re worried that someone will hack a math textbook to teach incorrect math that’s not obviously wrong or obviously hacked, but misleading and damaging to math education somehow? Or what kind of threat exactly are you imagining? Who’s got a secret plan to corrupt math education, and how does that work when math involves exercises and proofs? Brandolini’s Law certainly finds application in politics, but it’s news to me if anyone has ever taken the time to intentionally develop fake but believable academic curricula. Why would someone go to that level of effort, what would be the political agenda, and what makes you think it would get very far without getting caught and taken down?

I don't necessarily think that math is where this critique is the strongest. As you say, this happens most in political arguments, and this critique is strongest when talking about history books. I listened to a rant about this once and I think it is adjacent enough to be relevant as a critique here:

While I attended a church in a different state, the Sunday school teacher decided to get on a talking point aka rant about how history books are going to be changed via an opaque technology process to save kids backs from heavy textbooks and this was the BIG BAD. The entire topic kinda blindsided me at the time as I was leaning toward khan academy and other progressive teaching techniques at the time. (Still do, but i have had a number of bad experiences with subscription models for knowledge that is a prerequisite for certain jobs).

I bet that math textbooks are the more exact science, so it will probably pass over well, but some people (conservatives who don't trust change) will probably mount resistance if not addressed.

I just wanted to say that there are two sides (monetary incentives) to the question "should we adopt", and it directly shapes the narrative (and may cause a narrative to change into a political one).

Regarding "never heard of deliberately fake curriculum", i agree that i havent seen a deliberately fake math curriculum, but i have seen a number of "montressouri schools" that have no credentials and have not updated based on science for years. Most "charter schools" have a similar standard.

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