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Some experiments in youtube mathematics videos

January 3, 2021 math and physics play No comments , , , , , , ,

A couple years ago I was curious how easy it would be to use a graphics tablet as a virtual chalkboard, and produced a handful of very rough YouTube videos to get a feel for the basics of streaming and video editing (much of which I’ve now forgotten how to do). These were the videos in chronological order:

  • Introduction to Geometric (Clifford) Algebra.Introduction to Geometric (Clifford) algebra. Interpretation of products of unit vectors, rules for reducing products of unit vectors, and the axioms that justify those rules.
  • Geometric Algebra: dot, wedge, cross and vector products.Geometric (Clifford) Algebra introduction, showing the relation between the vector product dot and wedge products, and the cross product.
  • Solution of two line intersection using geometric algebra.
  • Linear system solution using the wedge product.. This video provides a standalone introduction to the wedge product, the geometry of the wedge product and some properties, and linear system solution as a sample application. In this video the wedge product is introduced independently of any geometric (Clifford) algebra, as an antisymmetric and associative operator. You’ll see that we get Cramer’s rule for free from this solution technique.
  • Exponential form of vector products in geometric algebra.In this video, I discussed the exponential form of the product of two vectors.

    I showed an example of how two unit vectors, each rotations of zcap orthonormal \(\mathbb{R}^3\) planes, produce a “complex” exponential in the plane that spans these two vectors.

  • Velocity and acceleration in cylindrical coordinates using geometric algebra.I derived the cylindrical coordinate representations of the velocity and acceleration vectors, showing the radial and azimuthal components of each vector.

    I also showed how these are related to the dot and wedge product with the radial unit vector.

  • Duality transformations in geometric algebra.Duality transformations (pseudoscalar multiplication) will be demonstrated in \(\mathbb{R}^2\) and \(\mathbb{R}^3\).

    A polar parameterized vector in \(\mathbb{R}^2\), written in complex exponential form, is multiplied by a unit pseudoscalar for the x-y plane. We see that the result is a vector normal to that vector, with the direction of the normal dependent on the order of multiplication, and the orientation of the pseudoscalar used.

    In \(\mathbb{R}^3\) we see that a vector multiplied by a pseudoscalar yields the bivector that represents the plane that is normal to that vector. The sign of that bivector (or its cyclic orientation) depends on the orientation of the pseudoscalar. The order of multiplication was not mentioned in this case since the \(\mathbb{R}^3\) pseudoscalar commutes with any grade object (assumed, not proved). An example of a vector with two components in a plane, multiplied by a pseudoscalar was also given, which allowed for a visualization of the bivector that is normal to the original vector.

  • Math bait and switch: Fractional integer exponents.When I was a kid, my dad asked me to explain fractional exponents, and perhaps any non-positive integer exponents, to him. He objected to the idea of multiplying something by itself \(1/2\) times.

    I failed to answer the question to his satisfaction. My own son is now reviewing the rules of exponentiation, and it occurred to me (30 years later) why my explanation to Dad failed.

    Essentially, there’s a small bait and switch required, and my dad didn’t fall for it.

    The meaning that my dad gave to exponentiation was that \( x^n\) equals \(x\) times itself \(n\) times.

    Using this rule, it is easy to demonstrate that \(x^a x^b = x^{a + b}\), and this can be used to justify expressions like \(x^{1/2}\). However, doing this really means that we’ve switched the definition of exponential, defining an exponential as any number that satisfies the relationship:

    \(x^a x^b = x^{a+b}\),

    where \(x^1 = x\). This slight of hand is required to give meaning to \(x^{1/2}\) or other exponentials where the exponential argument is any non-positive integer.

Of these videos I just relistened to the wedge product episode, as I had a new lone comment on it, and I couldn’t even remember what I had said. It wasn’t completely horrible, despite the low tech. I was, however, very surprised how soft and gentle my voice was. When I am talking math in person, I get very animated, but attempting to manage the tech was distracting and all the excitement that I’d normally have was obliterated.

I’d love to attempt a manim based presentation of some of this material, but suspect if I do something completely scripted like that, I may not be a very good narrator.

Falling victim to youtube clickbait: Sociologist claims “Math is racist”

January 24, 2020 Incoherent ramblings No comments , , , ,

Retrospective note:

It appears that the youtube thought police have struck, and the video linked to below is no longer available: “This video is no longer available because the YouTube account associated with this video has been terminated.”  I’ll leave the link in place below in case of the unlikely event that the original posting account is ever reinstated.

The video I watched may have been a clip from “Watter’s world”, interviewing Anne Delesio-Parson (linking to rumble instead of youtube to avoid further censorship.)  I had some doubts that it was the same interview, since I remembered the interviewer being more verbally adapt, and this interviewer Watter is anything but verbally adapt at the beginning of this video.  However, I think that Watter is playing dumb (rather condescendingly) at the beginning of the interview for theatrical effect, so this may be the same video after all.  It’s definitely as annoying as the original, and makes many of the same points.

Original blog post:

I made the mistake of listening to the following stupid interview while eating lunch today:

This was a stupid interview, and was probably just designed to piss people off:

  1. The premise itself is asinine.  There have probably been racist applications of all fields of study, but that does not imply any intrinsic racism.  Individuals can be racist, but it takes extraordinary circumstances to make a subject racist.
  2. The interview format was ridiculous.  If one makes the unlikely assumption that there is some sort of nuanced view to the thesis, how can somebody be expected to explain it in 4 minutes in an aggressive and confrontational interview?

Sadly, it sounded like the interviewee actually did want to make the claim that “math is racist”.  However, she was actively trying to bend language to her will, redefining racism in the process, which is both lazy and pathetic.  It seems to me that it is profoundly immoral to attempt to use words that have historical baggage, words that invoke an emotional reaction because of that history, and then do a bait-and-switch redefinition of the word under the covers.  It’s like playing the magician’s game, distracting somebody with the left hand, while the tricky right hand palms the coin.

What would racist fields of study actually be?  How about the research programs of the Nazi doctors, or US military radiation and disease experimentation on blacks in the ’50s [1].  Those I’d call racist research programs.  To use abuses of math to call the subject itself racist weakens the term to the point that it is meaningless.

The 4 minute constraint on this interview was also pointless.  I don’t have any confidence that the interviewee would have been able to provide a coherent argument, but this sound bite format made that a certainty.  Calling that an interview is as ridiculous as the thesis.  Kudos to the interviewer for quickly calling her on her BS as it was spouted, but he should be ashamed of trying to fit that “discussion” into a couple of minutes.

<h1>References</h1>

[1] William Blum. <em>Rogue state: A guide to the world’s only superpower</em>. Zed Books, 2006.

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