A Swift Introduction to Geometric Algebra @sudgylacmoe

sudgylacmoe This video is an introduction to geometric algebra, a severely underrated mathematical language that can be used to describe almost all of physics. This video was made as a presentation for my lab that I work in. While I had the people there foremost in my mind when making this, I realized that this might be useful to the general public, so I also tried to make this useful to others as well.

This video was made using manim (github.com/ManimCommunity/manim), a math animation library.

Several things in this video were incorrectly simplified. Please watch the video before reading the rest of the description.

Before I get into the things that were incorrectly simplified, I need to state a few definitions. The grade of a k-vector is defined to be k. A blade is defined to be the outer product of vectors. A k-blade is a blade of grade k. Some of the terminology between linear algebra and geometric algebra can be confusing, such as the use of the terms vector space and dimension. Because multivectors form their own vector space, but we don't consider all of them to be vectors, some people use the term linear space instead of vector space. While we think of dimension in terms of spatial degrees of freedom, the mathematical definition means that the 3D geometric algebra shown here is actually eight-dimensional. This is part of the reason for the distinction between the terms grade and dimension.

The following are the things that I purposefully got wrong to make things simpler:

The biggest thing I glossed over is the distinction between blades and vectors. A lot of the ways that I described k-vectors intuitively actually only applied to k-blades, not k-vectors. For example, in four or more dimensions, all 2-blades can be represented as an oriented area, but not all 2-vectors can. I didn't mention this fact for two reasons: first, for a first look, the distinction is not that important. Second, the distinction doesn't even matter until you reach four or more dimensions. In three dimensions, all k-vectors are k-blades.

My use of the term basis was not quite right, and I should have used the term "orthonormal basis". I wanted this video to be understood by people who had learned about vectors in a physics class where they don't go into too much detail about the exact definitions a linear algebra class would give you, so I didn't want to use the term orthonormal. I tried to mitigate this somewhat by always using terms like "the basis" and assuming that it was known that I meant the standard orthonormal basis. Hopefully this doesn't cause any confusion.

I mentioned that the inner product can get complicated when generalized to generic blades. What I didn't mention is that there is a large amount of disagreement on what this generalization is. I've seen at least five different extensions of the inner product to higher-grade blades. I didn't mention this because it was not pertinent to the discussion. However, even though I didn't talk much about the inner and outer products in this video, they are essential to the usage of geometric algebra in many applications.

This wasn't necessarily wrong, but I assumed that space is Euclidean throughout the video. In some applications, especially in relativity, space is not Euclidean, and a few things aren't the same.

In general, the pseudoscalar for a geometric algebra is the highest-grade element, but this does not always square to -1. In the three main applications of geometric algebra (2D Euclidean space, 3D Euclidean space, and 4D Minkowski space), this does happen to be the case, but in general it is not true.

When I mentioned dividing vectors, I didn't mention the fact that when dividing, you have to "divide on the left" or "divide on the right" because multiplication is not commutative. I actually prefer just to divide by multiplying by the inverse, because then you don't have to worry about how you're dividing. I didn't mention this because I never mentioned inverses again after I introduced them.

Another issue with inverses is that while vectors have inverses, not all multivectors have inverses. Also, in other spaces, such as the Minkowski space used in relativity, even some vectors don't have inverses. Again, this was because I didn't really use inverses much after this.

This one is small, but we assumed several properties of the geometric product that we didn't prove or state, such as distributivity and associativity. To be precise, the set of multivectors along with addition and the geometric product forms an associative algebra. I defined the geometric product in a bit of a roundabout way and I couldn't find a good place to insert these facts. The kind of people this video is aiming at are those who don't know a lot of abstract algebra and so I assumed that they would assume associativity and distributivity.

Rotors actually have many different definitions, so don't always think of them as a complex exponential.

updated 4 years ago

A Swift Introduction to Geometric Algebra

sudgylacmoe 2020-08-18 | This video is an introduction to geometric algebra, a severely underrated mathematical language that can be used to describe almost all of physics. This video was made as a presentation for my lab that I work in. While I had the people there foremost in my mind when making this, I realized that this might be useful to the general public, so I also tried to make this useful to others as well.

This video was made using manim (github.com/ManimCommunity/manim), a math animation library.

Several things in this video were incorrectly simplified. Please watch the video before reading the rest of the description.

Before I get into the things that were incorrectly simplified, I need to state a few definitions. The grade of a k-vector is defined to be k. A blade is defined to be the outer product of vectors. A k-blade is a blade of grade k. Some of the terminology between linear algebra and geometric algebra can be confusing, such as the use of the terms vector space and dimension. Because multivectors form their own vector space, but we don't consider all of them to be vectors, some people use the term linear space instead of vector space. While we think of dimension in terms of spatial degrees of freedom, the mathematical definition means that the 3D geometric algebra shown here is actually eight-dimensional. This is part of the reason for the distinction between the terms grade and dimension.

The following are the things that I purposefully got wrong to make things simpler:

The biggest thing I glossed over is the distinction between blades and vectors. A lot of the ways that I described k-vectors intuitively actually only applied to k-blades, not k-vectors. For example, in four or more dimensions, all 2-blades can be represented as an oriented area, but not all 2-vectors can. I didn't mention this fact for two reasons: first, for a first look, the distinction is not that important. Second, the distinction doesn't even matter until you reach four or more dimensions. In three dimensions, all k-vectors are k-blades.

My use of the term basis was not quite right, and I should have used the term "orthonormal basis". I wanted this video to be understood by people who had learned about vectors in a physics class where they don't go into too much detail about the exact definitions a linear algebra class would give you, so I didn't want to use the term orthonormal. I tried to mitigate this somewhat by always using terms like "the basis" and assuming that it was known that I meant the standard orthonormal basis. Hopefully this doesn't cause any confusion.

I mentioned that the inner product can get complicated when generalized to generic blades. What I didn't mention is that there is a large amount of disagreement on what this generalization is. I've seen at least five different extensions of the inner product to higher-grade blades. I didn't mention this because it was not pertinent to the discussion. However, even though I didn't talk much about the inner and outer products in this video, they are essential to the usage of geometric algebra in many applications.

This wasn't necessarily wrong, but I assumed that space is Euclidean throughout the video. In some applications, especially in relativity, space is not Euclidean, and a few things aren't the same.

In general, the pseudoscalar for a geometric algebra is the highest-grade element, but this does not always square to -1. In the three main applications of geometric algebra (2D Euclidean space, 3D Euclidean space, and 4D Minkowski space), this does happen to be the case, but in general it is not true.

When I mentioned dividing vectors, I didn't mention the fact that when dividing, you have to "divide on the left" or "divide on the right" because multiplication is not commutative. I actually prefer just to divide by multiplying by the inverse, because then you don't have to worry about how you're dividing. I didn't mention this because I never mentioned inverses again after I introduced them.

Another issue with inverses is that while vectors have inverses, not all multivectors have inverses. Also, in other spaces, such as the Minkowski space used in relativity, even some vectors don't have inverses. Again, this was because I didn't really use inverses much after this.

This one is small, but we assumed several properties of the geometric product that we didn't prove or state, such as distributivity and associativity. To be precise, the set of multivectors along with addition and the geometric product forms an associative algebra. I defined the geometric product in a bit of a roundabout way and I couldn't find a good place to insert these facts. The kind of people this video is aiming at are those who don't know a lot of abstract algebra and so I assumed that they would assume associativity and distributivity.

Rotors actually have many different definitions, so don't always think of them as a complex exponential.

Inverse of a 3D PGA Bivector

sudgylacmoe 2024-10-13 | How do you invert a 3D PGA bivector, as required by some of my last shorts? It's actually more involved than you might think.

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Decomposing a 3D PGA Rotor

sudgylacmoe 2024-10-06 | A couple weeks ago I talked about how to calculate the invariant decomposition of a 3D PGA bivector into simple commuting parts. But what about decomposing a 3D PGA rotor into simple commuting factors? Once again, these ideas came from Roelf and De Keninck's paper "Graded Symmetry Groups: Plane and Simple".

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An Alternative Way to Find Rotors

sudgylacmoe 2024-09-29 | As a sort of generalization of Euler's formula, we can think of simple rotors as being the sum of a generalized cosine and a generalized sine, which are the scalar and bivector parts respectively. This also leads to the generalized tangent, which leads to a new way to find rotors.

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Calculating the Invariant Decomposition in 3D PGA

sudgylacmoe 2024-09-22 | Last week I talked about how bivectors in 3D PGA can be written as the sum of a line in space and the line at infinity around that line. But how do you actually calculate these two lines given an arbitrary bivector? A general algorithm for doing this was found recently in Roelf and De Keninck's paper "Graded Symmetry Groups: Plane and Simple", and while the general algorithm is too complicated for a short, I present the special case of the algorithm for 3D PGA.

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Decomposition of a 3D PGA Bivector

sudgylacmoe 2024-09-15 | I've talked before about how non-simple bivectors can be written as the sum of simple commuting bivectors, making calculating exponentials much easier. But what does this mean geometrically? In this short, I show what this looks like in 3D PGA, which is one of the simplest cases where this is important.

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The Logarithm of a Rotor

sudgylacmoe 2024-09-08 | Last week, I talked about how to interpolate rotors. But to do that, you need to know how to exponentiate bivectors and find the logarithm of rotors! I covered the exponentiation part a while ago: youtube.com/shorts/YnzQBC5zx9U. In this short, I show how to calculate logarithms of rotors.

The idea in this short is based on a paper by Roelfs and De Keninck, and a preprint for the paper, including algorithms for finding these simple commuting decompositions, can be found here: arxiv.org/abs/2107.03771

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Interpolating Rotors

sudgylacmoe 2024-09-01 | How do you interpolate rotors? The most straightforward idea doesn't work. This short is the first in a series about some of the math behind how my new animation library, ganim, works.

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Solving a Particular Vector Equation

sudgylacmoe 2024-08-25 | There's a problem in David Hestenes' book New Foundations for Classical Mechanics that many people have a hard time with. In this short, I show the solution.

You might notice that something looks just a bit different in this short. That's because it was made with my new animation software, ganim! It's just barely at the point that if I pick the right topics, I can barely make it work, after running into several bugs and missing features. Since it's still very early on in development I'm not going to be helping anybody use it yet (and the code for this video is terrible since I want to add several features that would make it much better), but if you want to look at the library you can find it here: github.com/sudgy/ganim

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Are Geometric and Exterior Algebra Isomorphic?

sudgylacmoe 2024-08-18 | Geometric Algebra and Exterior Algebra both use multivectors, but given that they are different non-isomorphic algebras, how exactly can we formalize this idea?

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Zorns Lemma Demystified

sudgylacmoe 2024-08-16 | Zorn's lemma states that in a partially ordered set, if every chain has an upper bound, then the set contains a maximal element. But what in the world does that mean? Many people have been confused by Zorn's lemma, and the fact that it's mainly mentioned just in the context of the axiom of choice only makes the problem worse. In this video, I dispel the confusion around Zorn's lemma by showing what it's actually saying and when it's used. This video is my #SoMEπ submission, and just in time too.

Links:
The simple proof of Zorn's lemma: https://digitalcommons.kennesaw.edu/cgi/viewcontent.cgi?article=2161&context=facpubs
My own write-up of it, since I ended up changing more than I initially thought: drive.google.com/file/d/1aRHrLP1WtYkmLqhPMokDmWVdECHsUoHD/view
My formalization of it: github.com/sudgy/math-from-nothing/blob/master/src/Set/zorn.v
My construction of the real numbers: drive.google.com/file/d/1fb1pNJVMuROmpLq-kDwXXeY2QxzCV0hy/view
My formalization of it: github.com/sudgy/math-from-nothing/tree/master/src/Number/Real/Zorn

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Sections:
00:00 Introduction
01:24 Basis Introduction
01:48 Basis Argument
03:14 Well Order Introduction
06:23 Well Order Argument
07:47 Real Number Introduction
09:27 Real Number Argument
10:48 Stating Zorn's Lemma
14:13 Using Zorn's Lemma
16:37 Basis Proof
18:26 Well Order Proof
21:30 Real Number Proof
25:30 Proof of Zorn's Lemma
32:44 Conclusion

An Interesting Construction of Polynomials (Part 3)

sudgylacmoe 2024-08-11 | Last week, I showed how polynomials of an infinite number of arguments can be seen as the free vector space on the positive natural numbers, with multiplication being multiplication of natural numbers extended by linearity. But what about zero? What happens if we include it?

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An Interesting Construction of Polynomials (Part 2)

sudgylacmoe 2024-08-04 | Last week, I showed how polynomials of a single argument can be seen as the free vector space on the natural numbers, with multiplication being addition of natural numbers extended by linearity. But what happens if we define multiplication to be multiplication of natural numbers extended by linearity?

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An Interesting Construction of Polynomials (Part 1)

sudgylacmoe 2024-07-28 | In this short, I show an interesting construction of polynomials as the free module on the natural numbers. This is the construction that I ended up using in my Coq project developing math from the ground up. While I used the real numbers as my scalars in this video, this construction works for any commutative ring.

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The Reverse in the Sandwich Product

sudgylacmoe 2024-07-21 | Most people, when talking about the sandwich product in geometric algebra, write it like RAR†, where the reverse is on the right. However, in my videos, I've written it like R†AR, where the reverse is on the left. In this short, I explain why I use this convention.

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The Limits of the Cartan-Dieudonné Theorem (Part 2)

sudgylacmoe 2024-07-14 | The Cartan-Dieudonné theorem is a fundamental result in geometry connecting orthogonal transformations to reflections. How far can we push it? In this short, I discuss another counterexample that appears when you try to apply the theorem to degenerate spaces, even if you try to modify the theorem to remove the previous counterexample. If you want to see the proof that you can't write this orthogonal transformation as the product of at most three vectors, it's near the bottom of my document on counterexamples in geometric algebra: drive.google.com/file/d/1BMnv9aZlDcsh4AnivVt7ZIGqmdTcp8tn/view

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The Limits of the Cartan-Dieudonné Theorem (Part 1)

sudgylacmoe 2024-07-07 | The Cartan-Dieudonné theorem is a fundamental result in geometry connecting orthogonal transformations to reflections. How far can we push it? In this short, I discuss what one of the limits of the theorem is, and one of the counterexamples that appears when we push past that limit.

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Null Vectors vs. Degenerate Vectors

sudgylacmoe 2024-06-30 | Symmetric bilinear forms are generalizations of inner products that allow for vectors to square to any scalar value, not just positive numbers. This leads to some new kinds of vectors: null vectors and degenerate vectors. While many people think they are the same, it is possible for a null vector to not be degenerate.

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Hyperplane Reflections in Geometric Algebra

sudgylacmoe 2024-06-23 | The Cartan-Dieudonné theorem shows how important hyperplane reflections are in geometry. With a minor adjustment, we can convert the line reflection formula from geometric algebra into a hyperplane reflection formula, allowing us to utilize geometric algebra when applying the Cartan-Dieudonné theorem.

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The Cartan-Dieudonné Theorem

sudgylacmoe 2024-06-16 | The Cartan-Dieudonné theorem is an important result in geometry relating orthogonal transformations and reflections. In this short, I provide a quick sketch of a proof of the theorem.

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Line Reflections in Geometric Algebra

sudgylacmoe 2024-06-09 | From the 2D rotation formula in VGA, we can derive the fundamental GA reflection formula. It reflects one vector across another, creating a line reflection. Because this formula only involves two vectors, it applies in any number of dimensions, even though the derivation was limited to two dimensions!

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What Is a Reflection?

sudgylacmoe 2024-06-02 | The idea of reflections is well-known. But what actually is a reflection? This short gives several examples of different kinds of reflections, and the common point between them all.

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2D Rotations in Geometric Algebra

sudgylacmoe 2024-05-26 | In two-dimensional vanilla geometric algebra, multiplying a vector by the product of two other vectors rotates the vector by the angle between them. But why is that? That's the question I answer in this short.

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Why Multiplying by i Causes Rotation

sudgylacmoe 2024-05-19 | The geometric picture of multiplying by the imaginary number i is one of rotating by a right angle. But why is that? In this short, I show one possible answer to that question using geometric algebra, where i is considered to be the unit pseudoscalar of 2D VGA.

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Products of Vectors in Two Dimensions

sudgylacmoe 2024-05-12 | As a sort of continuation of last week's short, we apply last week's identity to show that in two dimensions, the product of three vectors is the same forwards and backwards. This has applications to the way that geometric algebra describes rotations.

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The Antisymmetric Part of the Product of Three Vectors

sudgylacmoe 2024-05-05 | As a sort of continuation of last week's short, we apply last week's identity to derive a new identity for the "symmetric" and "antisymmetric" part of the geometric product of three vectors. While the symmetric part is a little too complicated to be useful, we will be seeing an application of the antisymmetric part next week.

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The Product of Three Vectors

sudgylacmoe 2024-04-28 | As a sort of continuation of last week's short, we apply last week's identity to derive a new identity for the geometric product of three vectors. While this identity isn't as useful as some of the other recent ones, we'll see an application of it next week.

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The Inner Product of a Vector and a Simple Bivector

sudgylacmoe 2024-04-21 | 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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The Symmetric/Antisymmetric Parts of the Geometric Product

sudgylacmoe 2024-04-14 | 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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When the Inner/Outer Products are Commutative

sudgylacmoe 2024-04-07 | 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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The Grade Involution

sudgylacmoe 2024-03-31 | 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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The Geometric Product Is Not Continuous (In Infinite Dimensions)

sudgylacmoe 2024-03-24 | 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 (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: drive.google.com/file/d/1BMnv9aZlDcsh4AnivVt7ZIGqmdTcp8tn/view

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Rejections

sudgylacmoe 2024-03-17 | 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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Projections in Geometric algebra

sudgylacmoe 2024-03-10 | While the projection formula for vectors is well-known, in geometric algebra, this formula works for any geometric object!

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Empty Sums and Products

sudgylacmoe 2024-03-03 | 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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Putting Labels on the Outside of a Triangle

sudgylacmoe 2024-02-26 | 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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The Commutative and Anticommutative Parts of the Geometric Product

sudgylacmoe 2024-02-18 | 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: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
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You can add scalars and vectors! From Zero to Geo 1.11

sudgylacmoe 2024-02-12 | 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: youtube.com/playlist?list=PLVuwZXwFua-0Ks3rRS4tIkswgUmDLqqRy

Discord: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
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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

When Isomorphism Is Not Equality

sudgylacmoe 2024-02-11 | 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.

Discord: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
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Geometric Algebra and Grading

sudgylacmoe 2024-02-04 | 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.

Discord: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
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Pascals Triangle in Geometric Algebra

sudgylacmoe 2024-01-28 | 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: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
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Simple Inverses in Geometric Algebra

sudgylacmoe 2024-01-21 | 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 arxiv.org/abs/2005.04015.

Discord: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
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Cross Product Correspondences in Geometric Algebra

sudgylacmoe 2024-01-14 | 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: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
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The Inner Product and Length

sudgylacmoe 2024-01-07 | 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: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
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A Random Number Paradox (Part 2)

sudgylacmoe 2023-12-31 | 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: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
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A Random Number Paradox (Part 1)

sudgylacmoe 2023-12-24 | 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: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
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Universal Properties of Number Systems

sudgylacmoe 2023-12-17 | 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: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
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Is the Scalar Product Commutative?

sudgylacmoe 2023-12-10 | 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: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
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There Is No Odd Subalgebra

sudgylacmoe 2023-12-03 | 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: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
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When is the outer product of a multivector with itself zero?

sudgylacmoe 2023-11-26 | 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: drive.google.com/file/d/1BMnv9aZlDcsh4AnivVt7ZIGqmdTcp8tn/view

Discord: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
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How to Change Your Perspective

sudgylacmoe 2023-11-19 | 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: youtube.com/watch?v=_WWoG7sSBTo

Discord: discord.gg/3Zj59zA2Rg

Patreon: patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
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