In light of my 16th wikiversary today, I'll post some "creators's commentary" on a few of the stuff I created for Wikipedia. I hope you guys find this interesting/helpful!

One of the reasons I use my own drawing library is to make arrows, surfaces and shapes feel "physical". This is so I can play with our visual/spatial intuition.

Here's a good example: the "thick" surface in this animation allows you to instantly "get" the complex 3D shape.

Another great example of this technique to convey physical structure to 3D surfaces is in this animation depicting a "Fourier transform surface", something I never saw fully visualized before. Without this approach, using thin surfaces, this would likely look like garbage.

I always try to find a middle ground between flatness/3D. Naive 3D rendering, with shadows & etc, produce unnecessary visual noise. You can convey depth by simply treating flat shapes as actual tridimensional objects, like the arrows here. The grid adds extra information.

This cross product GIF also illustrates how the flat-rendered vectors are always drawn as physical, 3D objects. If I didn't treat the arrowheads as cones, you wouldn't get the same depth effect.

Notice that I also don't typically treat the arrow body as cylinders unless needed.

Another way I mix flatness with depth is to smoothly alternate between orthographic and perspective projections. You'd probably never notice it, but I do it all the time.

The camera motion in this line integral animation switches from orthographic to perspetive, then back.

The torus is my favorite 3D solid, which explains why I have so many torus stuff. This one showing Villarceau circles, also using physical intuition with the "slicing", was inspired by the "Villarceau Circles" rendering in the POV-Ray Hall of Fame, by Tor Olav Kristensen.

I've been trying to visualize General Relativity for a while. Since my more formal attempts at it didn't seem very enlightening so far, I resort to conceptual illustrations. They may be a bit misleading, but it's easier to add formality to intuition than the reverse, IMO.

The highest point of ballistic trajectories at constant speed form what I call the "ballistic ellipse". I discovered this when I was in high school, but apparently this was only published about in 2004! https://arxiv.org/abs/physics/0402020

I love traditional animation, and timing there is key. If you want to convince viewers that something is "physical", you have to make sure to never use plain linear interpolation. Nothing ever instantly starts/stops moving! Easing functions can help you achieve that effect.

Speaking of timing, pauses are essential when explaining things through animations. A lot of people overlook this and make things too fast paced. Every step should have a clear purpose and its own moment to be observed and understood.

Fun fact: in my mind, angles are yellow.

I've been told that some of my animations have become so iconic that almost everyone knows them. Apparently, this animation showing the homotopy between a torus and a coffee cup (an old topology example and joke) is shown in many intro topology classes worldwide now.

I always try to give a shot at visualizing things people haven't visualized before. Line integrals over scalar fields have been visualized before, but I have never seen something similar for vector fields before I made this animation.

I also have a fondness for obscure/underappreciated concepts, like the Villarceau Circles or Pappus' Centroid Theorem (this one), which is much cooler than the sidenote given in calculus textbooks. I wrote a post about it:

https://1ucasvb.tumblr.com/post/90160448163/mathematics-is-full-of-wonderful-but-relatively

One of my crowning achievements is completely ubiquitous now, and can be found all over the internet.

Back in the early days Wikimedia held a contest to pick a new redirect arrow graphic. I won by proposing something simple.

This particular arrow has made its way everywhere!

Anyway, just a few comments behind some of my work. I hope that was amusing enough!

I haven't done much in a while, but don't worry, I haven't given up. There's still a lot of subjects I want to illustrate and visualize, and I have great plans for the future!

Sadly, Twitter absolutely butchered the quality of these animations! Make sure to check my Wikipedia gallery & Tumblr blog if you want to see them properly!

https://en.wikipedia.org/wiki/User:LucasVB/Gallery
https://1ucasvb.tumblr.com/archive 

Also, I'm always open to requests. If you have a cool idea of something that could use a visualization, let me know!

Some bonus stuff!

I like to explore obvious "what ifs". For example, what if you plotted a f(x) function as r(θ)? Here's how to connect the two ideas.

This helps us understand the difference between Cartesian and polar plots. Notice the use of surprise 3D!

Sometimes I just want to do a conceptual animation to convey a few ideas so that they are memorable.

In this conceptual animation, I illustrate how wavelength is related to color of light, and how different colors propagate at different speeds inside a medium, spreading out.

Completing the square is one of those "magical" techniques we learn in high school that are rarely explained to us.

But math is supposed to make sense. As long as you put the effort to understand things you can make sense of them, and once you do you'll never forget it.

I love rigid body physics, which explains the motion for rotating bodies. A key concept is the notion of moment of inertia: how much objects resist changing their rotation. This animation illustrates that, with different objects rolling downhill. Try this experiment at home!

Sine & cosine are always poorly explained, but they are super simple: imagine a point at an angle θ around the unit circle. Then sin θ is the y coordinate of the point, and cos θ is the x coordinate.

Memorization is nothing compared to understanding. Anyone can "get" math!

Fun fact: the circle MUST be on the lower right corner if this animation is to be accurate. Any other position reverses one or both of the functions, as in cos(-θ) or sin(-θ), and you get the wrong graphs!

That's something similar animations fail to take into account!