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I am looking for 3d-math animation software that is specifically meant for

  1. Visualization: That is geometry-based software that can aid in making visualizations as in https://www.youtube.com/watch?v=x7d13SgqUXg (on the sphere-eversion process).

  2. Presenting concepts: That is software that can help in making videos as in https://www.youtube.com/watch?v=rB83DpBJQsE (on divergence curl), or https://www.youtube.com/watch?v=r18Gi8lSkfM (on Fourier Transform).

So my question is which mathematical software is freely available/cheap, that can help in achieving the above task?

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The math videos creator 3blue1brown has a webpage and in the FAQ section he goes into detail about how we went about creating his videos. Here's his answer for the question "What do you use to animate your videos?"

I create most of the animations programmatically, using a python library named "manim" that I've been building up. I’ve also used Grapher for a number of 3d animations.

It’s open-source on github, and a small community has emerged of people who use it, but before you dive in I feel like a warning is in order. This is something I put together with my own use case in mind, never really meant to be a professional product that’s consistently maintained for others. It's not that I want to discourage others from doing similar things, quite the contrary I love when that happens, but often my workflow and development with manim can make it more difficult for an outsider to learn than other better-documented animation libraries, like Matplotlib and Mathematica, or other tools built for math visualization like Geogebra, Desmos, Grapher and more.

I also get worried when I hear people ask things like “how do I sync up narration into manim”. This is just a tool for spitting out the individual mathematical animations to be edited together later. For goodness sake, use traditional video editing software for as much as you can!

There are aspects of producing videos with a self-made tool like this that I find quite pleasing, but which are pleasing precisely because it's my own. It enforces a uniqueness of style, for example, which is by its very nature a benefit that can't be shared. There's also a certain freedom in being able to tear up the guts of the tool every now and then when I feel a change is in order, since backward compatibility needs are very limited when you only care about videos moving forward. Not exactly the best practice from a collaborative standpoint.

The channels I know who took inspiration from 3b1b and did it best, like Primer or Jazon Jiao, found or created the tooling that worked best for their particular content.

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I have not personally made videos, although I have used Maple (not free) to create many visualizations and Inkscape (free) for some diagrams. I usually prefer to write mathematics by hand, but from time to time I have used PowerPoint for slides when many visuals are required that would be difficult or time-consuming to draw by hand.

I am impressed by the materials Robert Ghrist has produced. If you are interested in knowing more about how he creates them, check out his blog post but be aware that you could be in for a steep learning curve. To summarize a little of what he writes, he advises investing the money and time if you want to create pro-quality graphics. Also, learn basic graphics, design principles and compositing. You might also need to learn coding and/or game engines, especially if you want to simulate systems. You could get started using Processing, and there are always languages such as Python and Julia. A list of 3D animation software can be found here, some of which is free.

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You could try VPython (https://vpython.org/). It's free and does 3D animation. It was built for teaching physics, and can visualize vector fields, graphs, various primitive objects, etc. as well as animating them, and letting you interact with them.

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There are many good options for the "presentation" one. If you are doing multivariate calc stuff, then CalcPlot3D is a great option. (I also recommend many others such as Sage Math, which created this amazing graphic using Tachyon for image production, but CP3D is the easiest to start with.)

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