Interesting Math Videos Mega-Thread

[Null Simplex]

Usually when people think of noneuclidean geometry, they think of hyperbolic space or sometimes spherical.  But for 3 dimensional spaces, their are 8 different types of geometries which are possible (if I understand the theorem correctly).  This first video is a tour through these 8 geometries using balls to demonstrate how space and light curves in these different geometries. 

In at least one of these geometries, shapes which in our flat, Euclidean space are deemed paradoxical or merely optical illusions created by playing with perspective on a 2D image are in fact viable objects in these realities.  Here is one such shape, the Penrose Triangle. 

Clearly this shape cannot work in our world.  However, in Nil Geometry, the shape is perfectly valid due to the twisty nature of the space.

Here is another such example of an impossible shape which is possible in Nil Geometry, the Penrose stairs.  These stairs have you either constantly ascending or descending depending on which direction you circle them, and were featured prominently in the film Inception.  This video shows what it would be like to walk up a set of Penrose stairs in Nil geometry where it is possible to constantly ascend or descend.  When your grandparents talk about walking up hill both ways to and from school, they actually lived in a Nil reality.

 

Edited November 16, 2023 by Null Simplex

[CARDOZZO]

Cool thread 

[Yimpa]

 

[Null Simplex]

Everyone knows about circular portals.  Here's a generalization of that idea using knot portals.  In the first video a trefoil knot, the second simplest knot after a circle, is used to connect six different universes depending on which angle you view the knot from.

Here's a lecture on knot portals from the mathematician who created the concept.

 

[Sincerity]

Very interesting videos guys.  I loved the Y'all Are Nerds one, and You @Null Simplex posted some amazing content.

[Null Simplex]

Here are some videos on game theory, one of the most practical branches of mathematics.  I believe that all children should learn some form of the Prisoner's Dilemma by the time they are in high school, if not sooner.  What's important about the Prisoner's Dilemma is that it shows why, mathematically, selfishness emerges even when selflessness yields a more optimal situation for everyone, which feels like a relevant concept for this forum.

What I found most interesting about this video is that even in this crude, overly simplified mathematical model about a game involving two entities which choose to either compete or cooperate, the most successful entities had 4 qualities:  They were nice, forgiving, retaliatory, and predictable.  This mimics the qualities of someone in the real world who would be good to cooperate with without being a pushover.  Specifically, tit-for-tat, the most natural and obvious strategy, was the most effective overall.  I absolutely love when simple mathematical models are able to capture intuitive aspects of physical reality. 

Counter intuitively, despite the fact that Tit-for-Tat can never outscore it's opponent in an individual match, it outscores everyone when all of the scores in the tournament are summed together.  On the opposite end of the spectrum, the most selfish strategy can never lose an individual round, however it scores very few points throughout the entirety of the tournament, mimicking how selfishness may be a good short term strategy but a poor long term strategy.

Here are some game theory videos by Primer showing how selfishness and selflessness could have evolved through natural selection.

I was already planning on sharing Primer's videos on game theory, but as soon as I saw Veritasium posted a video on the topic, I realized now would be the perfect time to share the game theory videos.

Edited December 27, 2023 by Null Simplex

[Null Simplex]

By far the simplest explanation I’ve encountered for the intuition and fundamentals for what a neural network really is.  If, like me, you’ve found yourself wondering how a neural network works but would get lost when the person explaining represents the neural network as a graph with nodes and edges connecting the nodes, then this is the video for you!  It succinctly illustrates how an AI can take massive amounts of data and output something useful.

PS:  Thanks for the videos and Youtuber @LastThursday.  Very interesting. 

[MuadDib]