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As a sort of continuation of last week's short, we apply last week's identity to derive a new identity for the inner product of a vector with the outer product of two other vectors. This little formula is surprisingly useful, and while I don't cover it here, there are some interesting geometric facts you can get from it. Next week we'll look at an application of this identity!
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As a sort of continuation of last week's short, we apply last week's identities to derive some new ones, this time about the symmetric and antisymmetric parts of the geometric product. Next week we'll look at an application of this!
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While the equations u · v = v · u and u ∧ v = -v ∧ u aren't true in general, there are some special cases where something similar is true. What about the inner or outer product of a vector with a multivector?
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This short is about the grade involution, a neat little operation used in geometric algebra. It can make many expressions and equations much simpler. While I've mentioned it in a video before, it was deep in a particularly dry video, so I thought I would make a short that talks just about it.
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Here's a surprising fact that I discovered a little while ago: in an infinite-dimensional space, the geometric product is not continuous! While infinite-dimensional geometric algebra is perfectly fine and usable, this suggests that infinite-dimensional geometric calculus is practically useless. Interestingly, I came across this fact back when I was trying to prove that the geometric product IS continuous. I kept running into roadblocks, and eventually started wondering if it actually isn't continuous. After running a bit of code, I found that the numerical evidence suggested that it's not, and soon afterwards I found a comment online about something similar (https://math.stackexchange.com/questions/816092/infinite-dimensional-clifford-algebras#comment6841432_818115), and I adapted that argument into the one shown in the short.
You can find more counterexamples in geometric algebra in this document: https://drive.google.com/file/d/1BMnv9aZlDcsh4AnivVt7ZIGqmdTcp8tn/view
Discord: https://discord.gg/3Zj59zA2Rg
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Last week, I talked about projections, and how geometric provides a very general formula for them. However, that's only one side of the story! There are also rejections, which are the counterpart to projections. Geometric algebra provides a simple and general formula for rejections as well. Honestly, I made last week's short as a lead-up to this one.
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While the projection formula for vectors is well-known, in geometric algebra, this formula works for any geometric object!
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How can you add or multiply no numbers? As nonsensical as this may sound, it actually is well-defined and makes sense! It even extends to other monoids as well.
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In making my series on geometric algebra, I ran into a problem that I solved using...geometric algebra! Here's how I did it.
Sorry that this is a day late. To be honest, I completely forgot to upload/schedule this one. It was already made like a week ago :|
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Many people (including myself sadly) have described the inner and outer products as the commutative and anticommutative parts of the geometric product, but this is not true in general. In this short, I provide several counterexamples to this idea, showing that "commutative part" and "anticommutative part" are just not useful ideas in general.
Discord: https://discord.gg/3Zj59zA2Rg
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While most people consider two isomorphic objects to be the same thing, there are situations where an isomorphism is not "strong" enough and it doesn't preserve important structure. I recently heard someone call this an "isoblurism", and I liked the term enough that I made this short presenting the idea. I didn't present a formal definition here, but I think a good definition is two objects that are isomorphic in one category (after applying a forgetful functor) but not in another.
The interesting thing is, there are cases where isoblurisms are useful, as long as you don't use them to think of the structures as identical. For example, there is an isoblurism between geometric algebra and exterior algebra. This allows you to think of the elements of both algebras as being the same, but the products are still different.
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In this video, I show that contrary to popular belief, you can add scalars and vectors! While this particular idea isn't terribly useful, with a bit of generalization it can lead the way to many useful things, such as multivectors.
Sorry for how long this video took to make! While the final product isn't that long, I rewrote parts of it so many times. I even removed a significant section at the last moment because it was too confusing. I hope that in the end, it was worth it and that this video is informative.
This video is a part of "From Zero to Geo", a series where we formulate geometric algebra, an incredibly powerful branch of mathematics, from the ground up. Full playlist here: https://www.youtube.com/playlist?list=PLVuwZXwFua-0Ks3rRS4tIkswgUmDLqqRy
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Sections:
00:00 Introduction
00:39 Questions
01:14 Analogy with basis vectors
02:16 Exercise
02:48 Paravectors as a linear space
03:12 Zeros
03:58 General direct sums
05:10 Properties of direct sums
06:11 Conclusion
06:40 Rigorous Construction
One of the most important aspects of geometric algebra is the grading. However, for many people, this is the first time they encounter the concept of grading, so they don't know the relevant definitions. In this short, I show these definitions, and how they imply that while geometric algebra is both an algebra and a graded space, it is not a graded algebra.
Also, I know that under the Z/2Z grading, GA is a graded algebra. However, we don't use that grading too much, and the N-grading is much more useful.
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Patreon Supporters:
Christoph Kovacs
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Richard Penner
Rosario
trb
In this short, I show a pattern that many people have noticed connecting geometric algebra with Pascal's triangle. As a bonus, this also provides a proof that the algebraic dimension of an n-D geometric algebra is 2^n.
Discord: https://discord.gg/3Zj59zA2Rg
Patreon: https://www.patreon.com/sudgylacmoe
Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
trb
In this short, I describe how to find the inverse of some of the simpler types of objects in geometric algebra. While not all multivectors are invertible, a good number of the ones we care about are, and this short shows how to calculate a lot of those.
For more information on the complicated inverse algorithm, read https://arxiv.org/abs/2005.04015.
Discord: https://discord.gg/3Zj59zA2Rg
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Christoph Kovacs
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Jason Killian
p11
Richard Penner
Rosario
trb
In this short, I describe how the cross product can be represented in geometric algebra, and how it's a little more confusing than it initially seems.
Discord: https://discord.gg/3Zj59zA2Rg
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Christoph Kovacs
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Richard Penner
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trb
In this short, I give a geometric explanation for a formula relating the inner product to the length of vectors. This formula can be useful theoretically, but it reveals some interesting geometric ideas as well.
Discord: https://discord.gg/3Zj59zA2Rg
Patreon: https://www.patreon.com/sudgylacmoe
Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
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trb
In this short, I provide another version of the random number paradox from last week that removes one of the most common objections. Please see the comments for a few more notes on this paradox.
Discord: https://discord.gg/3Zj59zA2Rg
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In this short, I talk about an interesting number paradox about random numbers that I've heard before. Stay tuned next week for part 2, which will make things even more confusing!
Discord: https://discord.gg/3Zj59zA2Rg
Patreon: https://www.patreon.com/sudgylacmoe
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Christoph Kovacs
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In this short, I talk about the universal properties of various number systems. The most interesting one to me is the complex numbers, which is not that well-known.
Discord: https://discord.gg/3Zj59zA2Rg
Patreon: https://www.patreon.com/sudgylacmoe
Patreon Supporters:
Christoph Kovacs
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p11
Richard Penner
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trb
While the scalar product in geometric algebra is commutative, that doesn't necessarily imply that you can arbitrarily reorder products when taking the scalar part.
Discord: https://discord.gg/3Zj59zA2Rg
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A common term I hear beginners in geometric algebra say is "odd subalgebra", but this is actually incorrect! Odd multivectors are not closed under multiplication, so they don't form a subalgebra.
Discord: https://discord.gg/3Zj59zA2Rg
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trb
When is the outer product of a multivector with itself zero? In this short I show that there are many examples where this is not true, and that (as far as I know) there is no easy characterization of the multivectors for which this is true.
You can look at this document to find more counterexamples in geometric algebra: https://drive.google.com/file/d/1BMnv9aZlDcsh4AnivVt7ZIGqmdTcp8tn/view
Discord: https://discord.gg/3Zj59zA2Rg
Patreon: https://www.patreon.com/sudgylacmoe
Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
trb
In my most recent video, I showed an animation involving changing between different perspectives of 3D and 4D cubes. I thought that the way I made this animation was interesting, so I thought I would share it here.
Here's the video this animation is from: https://www.youtube.com/watch?v=_WWoG7sSBTo
Discord: https://discord.gg/3Zj59zA2Rg
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What actually is a bivector? While they are commonly described as oriented plane segments, many people get confused once they start seeing more details about bivectors. Hopefully this short clears some things up.
Discord: https://discord.gg/3Zj59zA2Rg
Patreon: https://www.patreon.com/sudgylacmoe
Patreon Supporters:
Christoph Kovacs
David Johnston
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trb
Created 3 years, 6 months ago.
30 videos
Category None
This channel is, for the moment, going to be somewhat mirroring my youtube channel (I go by sudgylacmoe there). It's mainly just for whatever I want, but I'm planning on having some more math educational videos soon, so that might be a lot of what this channel is about.