3 ms·

Visual explanations of mathematics (2020)

193 points by thunderbong 2y ago

mncharity 2y ago
I've seen physical-sciences research talks by people from Caltech, where the format is putting up a single equation, and then walking through it piece by piece. That's the entire talk, all the slides, and how the talk is organized. I've never seen that from elsewhere, and so think of it as a "caltech style". Does this have a real name? Can anyone flesh this out?
- tsumnia 2y ago
  I can't speak to Caltech, but it sounds like what you're describing is a perfect example of a "worked example". The idea is that you are walking through an algorithm, piece by piece even to the most basic levels. Its the style I use for my AI classes, though I've never been to Caltech; barely even to the west coast. I think its just something educators realized should get highlighted because all those symbols get complicated quickly like what the heck do η and τ mean for Ant Colony Optimization[1]? [1] They mean desirability and # of ants that picked a path
WillAdams 2y ago
There is also Joyce's Java Version of Euclid's Elements: https://mathcs.clarku.edu/~djoyce/java/elements/elements.html https://mathcs.clarku.edu/~djoyce/java/elements/elements.htm... and I'm still impressed by the custom Unity tools which Freya Holmér uses for her videos: https://www.youtube.com/watch?v=jvPPXbo87ds https://www.youtube.com/watch?v=jvPPXbo87ds Wish Geogebra was both more capable and widely used: https://www.geogebra.org/ https://www.geogebra.org/ That said, these days if I need to plot out something I just use OpenSCAD: https://openscad.org/ https://openscad.org/ (or, the Python-enabled version: https://pythonscad.org/ https://pythonscad.org/ )
- windows_hater_7 2y ago
  Grant Sanderson's YouTube channel 3blue1brown and his Manim package have made it significantly easier to visualize math computationally: https://www.youtube.com/c/3blue1brown https://www.youtube.com/c/3blue1brown https://3b1b.github.io/manim/index.html https://3b1b.github.io/manim/index.html There's also a well supported community version as well: https://www.manim.community/ https://www.manim.community/
- AdieuToLogic 2y ago
  > and I'm still impressed by the custom Unity tools which Freya Holmér uses for her videos Speaking of Unity... What blew my mind making a 3D interactive system using it was Quaternions[0]. It makes sense to me now, but that's after many discussions with a coworker having a PhD in mathematics. 0 - https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Quaternion
deleted 2y ago
[deleted]
mncharity 2y ago
> We have lots of tools to write better mathematics: LaTeX, markdown, Jupyter Notebooks, and so on. But it feels like nothing has really converged yet. Technology that seamlessly mixes symbolic equations, illustrative-and-explicative annotation, and runnable code is, I am sure, not far off. [2020; emphasis added] Back in the 1990's, we had lots of tools to write better mathematics: LaTeX, HTML, Mathematica notebooks, and so on. But it felt like nothing had really converged yet. Technology that seamlessly mixes symbolic equations, illustrative-and-explicative annotation, and runnable code is, I then expected, not to be anticipated before my distant retirement and death. Given the preceding decades of cripplingly underfunded and glacial programming language evolution - that seemed a similar kind of society-hobbling infrastructure challenge that we just don't handle well. Now here we are, 3 decades later, and languages are in a far better place... though with pace still throttled by poor funding. Perhaps someday math tech will be too. Large-scale hypertext, and GUIs, also managed the "and now, what you've lonnnng been waiting for, tada!" transition.
- fuck_google 2y ago
  [dead]
- caspper69 2y ago
  Do you believe that existing open source symbolic kernels are the limitation here, given that we can now easily (for some value of n) render the notation and the notes/annotations? I mean, there are a ton of math libraries available for most programming languages, but symbolic solving is pretty important here, because presumably people would want to (for example) show the integral or summation symbols and use algebraic notation rather than, say, Mathematica functions.
  - mncharity 2y ago
    > Do you believe [...] are the limitation here [a late reply, but...] I strikes me as having a flavor I associate with PIMs, rich text formats (eg markdown etc, and their assorted extensions), literate programming, and book writing systems. A high-dimensional sparse space of possibilities and taste and needs and nopes, which scatters effort and users, avoids critical mass, and thus discourages use and convergence. LaTeX and github-flavored markdown and pandoc and Jupyter are all achievements of convergence within multi-decade slogs. Ongoing slogs (how do you edit that LaTeX, make a book using pandoc, wrap a development process around Jupyter, etc). So yes, symbolic math is another slog. With 1970's macsyma stalling on licensing, Mathematica suing to prevent open-source implementations, and more. But I'm not sure that's a critical bottleneck here. I'm guess I'm suggesting that these wins of convergence often have long slogs not just behind, but ahead too, making it easy to overestimate the pace of future progress when looking at a collection of recent wins. Sometimes not - rust as C successor. But until time, or some famous app, creates user expectations for notes/annotations, their future looks a slog?
Jun8 2y ago
I don’t think visual explanation should be about writing colored versions of formulas or adding notes, it’s providing a visual that immediately unlocks the core idea. Many examples are given here: https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-that-are-easy-to-explain https://math.stackexchange.com/questions/733754/visually-stu...
- programjames 2y ago
  Or, the famous AoPS Proofs Without Words: https://artofproblemsolving.com/wiki/index.php/Proofs_without_words https://artofproblemsolving.com/wiki/index.php/Proofs_withou...
- kwinkunks 2y ago
  I agree that you can't beat a good visual. Sometimes, or for some people, the symbolic representations unlock other insights though, and I think adding them helps develop new scientists too. For me, code also helps a lot. I guess the main point is: add modalities.
mncharity 2y ago
Might a multimodal LLM be taught to generate illuminated equations and descriptions?
jackallis 2y ago
link to the book in the website takes you here https://lacathospital.com/ https://lacathospital.com/ ?
- JadeNB 2y ago
  I thought "huh, some hospital link," but it appears to be a scam site, so be wary of clicking. (Good advice anyway, but I naively thought I could trust a link on HN.) Which link did you follow, by the way? I clicked a few, and the link to https://www.edwardtufte.com/tufte/books_visex https://www.edwardtufte.com/tufte/books_visex is broken, but none took me to your link.
corysama 2y ago
Want to see a thousand more? Copy-paste the example image (not the URL) https://images.squarespace-cdn.com/content/v1/549dcda5e4b0a47d0ae1db1e/1582600213850-LS91KKPFTN5BRVNKG94J/annotated_Fourier_Transform.png https://images.squarespace-cdn.com/content/v1/549dcda5e4b0a4... into https://yandex.com/images/ https://yandex.com/images/ Or, the URL into https://www.bing.com/images https://www.bing.com/images -> "[o] Search using an Image" Unfortunately, Google Images is too smart. Even, search by image or image URL yields results about the Fourier transform :P
m_kos 2y ago
I am teaching intro stats and would love to have a textbook that introduces equations in this manner. I have been trying to incorporate this style into my slides but it is not as easy as it looks. (Same issue with syntax highlighting in PowerPoint/Google Slides.)
- abhgh 2y ago
  I have had some success with PowerPoint animations. Put up the equation, and for each mouse click surround the symbol(s) you want to explain in a red box or something and have a infobox show up (which can also contain images, such as plots - very helpful). As you transition to the next symbol(s), fade away the previous infobox and surrounding box.
- kwinkunks 2y ago
  You might be interested in the work of Andrew Head, in particular his team's Formula Formatting Language: https://andrewhead.info/assets/pdf/ffl.pdf https://andrewhead.info/assets/pdf/ffl.pdf - I have not tried it myself though.
vivzkestrel 2y ago
If you are the owner of the website, could you please add a page where I can browse all the articles easily without scrolling 10 kms on every single page. It is a very interesting blog with a tonne of seismic machine learning articles that I would like to read but I have to keep hitting next and next browsing through every page. Can I just see an index or all the titles somewhere?
- kwinkunks 2y ago
  That's me, sorry it's frustrating. I don't maintain the site anymore, but I just added some new views to this page, in case they help at all > https://agilescientific.com/index https://agilescientific.com/index
  - vivzkestrel 2y ago
    i am a full stack dev, i can help create an index for that page if you want
voidhorse 2y ago
I use this "annotated equation" technique in my own notes when I am working through mathematics. It's honestly kind of crazy to me that it isn't already standard practice in mathematics texts. I can see how mathematicians might feel that it is inelegant, but it certainly helps with establishing an understanding.
fjarlq 2y ago
Tom M. Apostol's Project Mathematics videos circa 1988-2000 are wonderful: https://en.wikipedia.org/wiki/Project_Mathematics https://en.wikipedia.org/wiki/Project_Mathematics! https://www.youtube.com/watch?v=vpxWyJg4_1A&list=PL8_xPU5epJdchhIkbjCPJM7m2anGUq9JT https://www.youtube.com/watch?v=vpxWyJg4_1A&list=PL8_xPU5epJ...
- 3abiton 2y ago
  I remember seeing these videos on youtube long time ago, but couldn't remember the sources. Thanks for sharing!

193 points by thunderbong 2y ago

I've seen physical-sciences research talks by people from Caltech, where the format is putting up a single equation, and then walking through it piece by piece. That's the entire talk, all the slides, and how the talk is organized. I've never seen that from elsewhere, and so think of it as a "caltech style". Does this have a real name? Can anyone flesh this out?

tsumnia 2y ago
I can't speak to Caltech, but it sounds like what you're describing is a perfect example of a "worked example". The idea is that you are walking through an algorithm, piece by piece even to the most basic levels. Its the style I use for my AI classes, though I've never been to Caltech; barely even to the west coast. I think its just something educators realized should get highlighted because all those symbols get complicated quickly like what the heck do η and τ mean for Ant Colony Optimization[1]? [1] They mean desirability and # of ants that picked a path

WillAdams 2y ago

There is also Joyce's Java Version of Euclid's Elements: https://mathcs.clarku.edu/~djoyce/java/elements/elements.html https://mathcs.clarku.edu/~djoyce/java/elements/elements.htm... and I'm still impressed by the custom Unity tools which Freya Holmér uses for her videos: https://www.youtube.com/watch?v=jvPPXbo87ds https://www.youtube.com/watch?v=jvPPXbo87ds Wish Geogebra was both more capable and widely used: https://www.geogebra.org/ https://www.geogebra.org/ That said, these days if I need to plot out something I just use OpenSCAD: https://openscad.org/ https://openscad.org/ (or, the Python-enabled version: https://pythonscad.org/ https://pythonscad.org/ )

windows_hater_7 2y ago
Grant Sanderson's YouTube channel 3blue1brown and his Manim package have made it significantly easier to visualize math computationally: https://www.youtube.com/c/3blue1brown https://www.youtube.com/c/3blue1brown https://3b1b.github.io/manim/index.html https://3b1b.github.io/manim/index.html There's also a well supported community version as well: https://www.manim.community/ https://www.manim.community/
AdieuToLogic 2y ago
> and I'm still impressed by the custom Unity tools which Freya Holmér uses for her videos Speaking of Unity... What blew my mind making a 3D interactive system using it was Quaternions[0]. It makes sense to me now, but that's after many discussions with a coworker having a PhD in mathematics. 0 - https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Quaternion

deleted 2y ago

[deleted]

mncharity 2y ago

> We have lots of tools to write better mathematics: LaTeX, markdown, Jupyter Notebooks, and so on. But it feels like nothing has really converged yet. Technology that seamlessly mixes symbolic equations, illustrative-and-explicative annotation, and runnable code is, I am sure, not far off. [2020; emphasis added] Back in the 1990's, we had lots of tools to write better mathematics: LaTeX, HTML, Mathematica notebooks, and so on. But it felt like nothing had really converged yet. Technology that seamlessly mixes symbolic equations, illustrative-and-explicative annotation, and runnable code is, I then expected, not to be anticipated before my distant retirement and death. Given the preceding decades of cripplingly underfunded and glacial programming language evolution - that seemed a similar kind of society-hobbling infrastructure challenge that we just don't handle well. Now here we are, 3 decades later, and languages are in a far better place... though with pace still throttled by poor funding. Perhaps someday math tech will be too. Large-scale hypertext, and GUIs, also managed the "and now, what you've lonnnng been waiting for, tada!" transition.

fuck_google 2y ago
[dead]
caspper69 2y ago
Do you believe that existing open source symbolic kernels are the limitation here, given that we can now easily (for some value of n) render the notation and the notes/annotations? I mean, there are a ton of math libraries available for most programming languages, but symbolic solving is pretty important here, because presumably people would want to (for example) show the integral or summation symbols and use algebraic notation rather than, say, Mathematica functions.
- mncharity 2y ago
  > Do you believe [...] are the limitation here [a late reply, but...] I strikes me as having a flavor I associate with PIMs, rich text formats (eg markdown etc, and their assorted extensions), literate programming, and book writing systems. A high-dimensional sparse space of possibilities and taste and needs and nopes, which scatters effort and users, avoids critical mass, and thus discourages use and convergence. LaTeX and github-flavored markdown and pandoc and Jupyter are all achievements of convergence within multi-decade slogs. Ongoing slogs (how do you edit that LaTeX, make a book using pandoc, wrap a development process around Jupyter, etc). So yes, symbolic math is another slog. With 1970's macsyma stalling on licensing, Mathematica suing to prevent open-source implementations, and more. But I'm not sure that's a critical bottleneck here. I'm guess I'm suggesting that these wins of convergence often have long slogs not just behind, but ahead too, making it easy to overestimate the pace of future progress when looking at a collection of recent wins. Sometimes not - rust as C successor. But until time, or some famous app, creates user expectations for notes/annotations, their future looks a slog?

Jun8 2y ago

I don’t think visual explanation should be about writing colored versions of formulas or adding notes, it’s providing a visual that immediately unlocks the core idea. Many examples are given here: https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-that-are-easy-to-explain https://math.stackexchange.com/questions/733754/visually-stu...

programjames 2y ago
Or, the famous AoPS Proofs Without Words: https://artofproblemsolving.com/wiki/index.php/Proofs_without_words https://artofproblemsolving.com/wiki/index.php/Proofs_withou...
kwinkunks 2y ago
I agree that you can't beat a good visual. Sometimes, or for some people, the symbolic representations unlock other insights though, and I think adding them helps develop new scientists too. For me, code also helps a lot. I guess the main point is: add modalities.

mncharity 2y ago

Might a multimodal LLM be taught to generate illuminated equations and descriptions?

jackallis 2y ago

link to the book in the website takes you here https://lacathospital.com/ https://lacathospital.com/ ?

JadeNB 2y ago
I thought "huh, some hospital link," but it appears to be a scam site, so be wary of clicking. (Good advice anyway, but I naively thought I could trust a link on HN.) Which link did you follow, by the way? I clicked a few, and the link to https://www.edwardtufte.com/tufte/books_visex https://www.edwardtufte.com/tufte/books_visex is broken, but none took me to your link.

corysama 2y ago

Want to see a thousand more? Copy-paste the example image (not the URL) https://images.squarespace-cdn.com/content/v1/549dcda5e4b0a47d0ae1db1e/1582600213850-LS91KKPFTN5BRVNKG94J/annotated_Fourier_Transform.png https://images.squarespace-cdn.com/content/v1/549dcda5e4b0a4... into https://yandex.com/images/ https://yandex.com/images/ Or, the URL into https://www.bing.com/images https://www.bing.com/images -> "[o] Search using an Image" Unfortunately, Google Images is too smart. Even, search by image or image URL yields results about the Fourier transform :P

m_kos 2y ago

I am teaching intro stats and would love to have a textbook that introduces equations in this manner. I have been trying to incorporate this style into my slides but it is not as easy as it looks. (Same issue with syntax highlighting in PowerPoint/Google Slides.)

abhgh 2y ago
I have had some success with PowerPoint animations. Put up the equation, and for each mouse click surround the symbol(s) you want to explain in a red box or something and have a infobox show up (which can also contain images, such as plots - very helpful). As you transition to the next symbol(s), fade away the previous infobox and surrounding box.
kwinkunks 2y ago
You might be interested in the work of Andrew Head, in particular his team's Formula Formatting Language: https://andrewhead.info/assets/pdf/ffl.pdf https://andrewhead.info/assets/pdf/ffl.pdf - I have not tried it myself though.

vivzkestrel 2y ago

If you are the owner of the website, could you please add a page where I can browse all the articles easily without scrolling 10 kms on every single page. It is a very interesting blog with a tonne of seismic machine learning articles that I would like to read but I have to keep hitting next and next browsing through every page. Can I just see an index or all the titles somewhere?

kwinkunks 2y ago
That's me, sorry it's frustrating. I don't maintain the site anymore, but I just added some new views to this page, in case they help at all > https://agilescientific.com/index https://agilescientific.com/index
- vivzkestrel 2y ago
  i am a full stack dev, i can help create an index for that page if you want

voidhorse 2y ago

I use this "annotated equation" technique in my own notes when I am working through mathematics. It's honestly kind of crazy to me that it isn't already standard practice in mathematics texts. I can see how mathematicians might feel that it is inelegant, but it certainly helps with establishing an understanding.

fjarlq 2y ago

Tom M. Apostol's Project Mathematics videos circa 1988-2000 are wonderful: https://en.wikipedia.org/wiki/Project_Mathematics https://en.wikipedia.org/wiki/Project_Mathematics! https://www.youtube.com/watch?v=vpxWyJg4_1A&list=PL8_xPU5epJdchhIkbjCPJM7m2anGUq9JT https://www.youtube.com/watch?v=vpxWyJg4_1A&list=PL8_xPU5epJ...

3abiton 2y ago
I remember seeing these videos on youtube long time ago, but couldn't remember the sources. Thanks for sharing!