The quaternions | Geometric algebra episode 10

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The quaternions | Geometric algebra episode 10

All Angles 14.05.2026 1 687 просмотров 129 лайков

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#geometricalgebra #quaternions #complexnumbers Earlier, we discovered the complex numbers as the even-graded elements in 2D geometric algebra. In a very similar way, the quaternions turn out to be the even-graded elements in 3D. We explore some of their properties, and we ask ourselves why the quaternions use a sandwich product. The answer to this intriguing question will be given in upcoming videos. You can support us on Patreon, where you can already watch all the Geometric Algebra videos and get access to exclusive content. Unfortunately, we won't be publishing any new videos after Geometric Algebra, at least not in the near future. Your support is still more than welcome of course: https://www.patreon.com/user?u=86649007 I have collected many interesting resources for you, where you can learn much more about quaternions in general, and specifically for how they emerge out of geometric algebra: [3B1B 1] https://www.youtube.com/watch?v=d4EgbgTm0Bg This one is not about the algebra of the quaternions, but about their geometry. In particular, it tries to give you a sense for the 4-dimensional space that they live in. [3B1B 2] https://www.youtube.com/watch?v=zjMuIxRvygQ A short follow-up about quaternions, and in particular how they help us describe rotations in 3D. [PENG 1] https://www.youtube.com/watch?v=jTgdKoQv738 Good introduction to the sandwich product and why it performs rotations. [MOM 1] https://www.youtube.com/watch?v=jlskQDR8-bY Quick introduction to the algebra of the quaternions, without any geometric insight. [MOM 2] https://www.youtube.com/watch?v=nJnWM7R6Y5A Very good overview of how the even-graded sub-algebra of 3D GA gives you the quaternions. The only unfortunate thing, is that the video uses e_31 instead of e_13 as a basis bivector, so that it requires an extra minus sign on one of the quaternion components. [WIKI 1] https://en.wikipedia.org/wiki/Quaternion#Quaternions_as_the_even_part_of_Cl3,0(R) The wikipedia page lists a large number of advantages of geometric algebra over quaternions. 0:00 Recap of complex numbers in geometric algebra 1:47 Every plane has a copy of the complex numbers 4:32 Discovering the quaternions 7:36 A few properties of quaternions 10:25 Advantages of geometric algebra over quaternions This video is published under a CC Attribution license ( https://creativecommons.org/licenses/by/4.0/ )

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Recap of complex numbers in geometric algebra

Today we will run into a well-known number system right in the middle of 3D geometric algebra. It's going to be very similar to how we discovered the complex numbers hiding in 2D geometric algebra. So let's quickly review how that happened. In 2D we have two basis vectors that square to one but their product the unit by vector squares to -1. Due to this property, we call this object I. The minus sign comes from the anti-ymmetry of the wedge product. The geometric product of two vectors always has the same form. It's a real number plus a real multiple of the unit by vector. But now that we know that the unit by vector is really the imaginary unit, what we have here is just a complex number. The real part has grade zero and the bvector part has grade two because it's the wedge product of two vectors. These even graded elements are the complex numbers. They are closed under geometric multiplication and they behave just like the complex numbers would. When you limit yourself to this even graded subalgebra the geometric product is complex multiplication. We can also write the real part as the dotproduct with a cosine and the B vector part as an area with a sign. This is the polar form of the complex number which can also be written exponentially thanks to Oiler. This was all a very quick recap. If you need more details, please see the earlier videos.

Every plane has a copy of the complex numbers

We are now going to see what happens in three dimensions. By definition, we have three basis vectors this time and they still square to one as always. We also have three basis by vectors. E1 E2, E2 E3, and E1 E3. They span the three main planes formed by the axes. And if you multiply these with themselves, a few simple steps of algebra will show you that they all still square to -1. This means that each of them serves as the imaginary unit for an entire complex number system that lives inside its own plane. In fact, every 2D plane inside every higher dimensional space contains its own copy of the complex numbers. Take this slanted plane for example. You can always find a unit by vector that lies fully inside the plane. It squares to minus1 and it rotates vectors by 90° inside the plane. Now just add real numbers to these B vectors and you have yourself a complex number system. Each unit by vector, no matter how it's oriented, rotates by 90°. But they all do so in their own individual planes. So we can now describe rotations in any direction we want. So take any two vectors in our plane. Their geometric product gives you a complex number with a real part and a bveector part which is basically its imaginary part. But it's better to write this in polar form where the modulus is the product of the vector lengths. And the angle is just the angle between the two vectors. This polar form is much more instructive because it talks about geometric properties such as lengths and angles rather than coordinates. Just as in two dimensions, the geometric product is still the same as the complex product. You can easily see this yourself. Just write two even graded multiff vectors, distribute their product and watch the formula for the complex product magically appear. So the amazing conclusion is that there's an infinite number of complex planes hiding inside of every higherdimensional geometric algebra. The imaginary unit is always a unit by vector and it tells you which plane we are looking at. Okay, this is already really cool, but we can go further. We can use all three

Discovering the quaternions

of the unit by vectors at once. So, we create multiv vectors with a scalar part and a bvector part that this time has three components. This is a linear combination of the unit of grade zero, that's the real number one, and the three unit by vectors of grade two. In other words, we are taking the even graded sub algebra again containing all elements with a mixture of grades 0 and two. When you multiply two of them together and you apply the usual algebraic rules, you will see that their product also contains only even grades. By now, I can leave this as an exercise for you because it's just the same algebraic manipulations we've done so many times before. So do try it yourself. You should be able to show that the even graded elements are closed under the geometric product. They really do form a sub algebra. What are the properties of this new subalgebra? It has three basis by vectors and one basis scalar. So it's fourdimensional. The three basis by vectors all square to -1. And if you multiply them pair-wise and then apply nothing but the same old rules for the geometric product, you get these relationships. These are incredibly famous. They are the defining multiplication rules for the quatronians. As a number system, the quatronians were discovered by the Irish mathematician Hamilton. He had spent years looking for an extension of the complex numbers to higher dimensions. He failed to find a three-dimensional extension, but then he suddenly realized that he could easily construct a four-dimensional extension instead. We now understand why this is the case. When we take the evenraded sub algebra, the numbers in the Pascal triangle force a total of four basis blades on us. That's the only way to get a system that is closed under multiplication. And so once again, we rediscover a well-known number system hiding inside of geometric algebra. We didn't have to invent the quaternians from scratch like Hamilton did. They very naturally present themselves inside a 3D Clifford algebra. And the properties of their product aren't exotic or weird. They just follow from everything we already know about the geometric product. There's no need to remember anything specific. You can rederive everything from the three rules we laid out all the way at the start of the series. The geometric product is quaton multiplication.

A few properties of quaternions

multiplication. Quoternians are very useful. Computer graphics programmers and game developers use them all the time because quaternons are very good at performing 3D rotations. Algebraically they rotate vectors by using a sandwich product. If you want to rotate a quatnon X, you have to sandwich it between a second quaternon Q and its inverse. This is one of the more mysterious behaviors of quaterians. Software developers are used to working with matrices which transform vectors by multiplying on their left. So, it isn't immediately obvious or intuitive why we now suddenly have to multiply on both sides and why the inverse is involved. This video linked in the description does a decent job of explaining where that sandwich product comes from. But, of course, you can already guess that we have a much better explanation in store for you. We will talk more about rotations in the upcoming videos on rotors and spinners. And this will give us not one but two independent justifications for the sandwich product. Here's another important property of quatronians. Their multiplication is not commutative. The complex numbers rotate inside a single plane. As long as you stay inside the same plane, you can always just add up the rotation angles. And since angle addition is commutative, rotations are commutative as well. But in three dimensions, you gain the ability to rotate in different planes. The main downside is that rotations in different planes don't commute with each other. Take a cube such as a die and do some experiments for yourself. If you flip up and then right, you get a different result than if you flip right and then up. This is why unlike complex multiplication, quatnon multiplication is not commutative. I have a hunch that non-commutativity is tightly related to the sandwich product. You may remember from our videos about group theory that conjugates are defined using a sandwich expression. And it just so happens that conjugates are only useful in non-commutative groups. So there is definitely a connection there, although I can't quite put my finger on it. Let me know in the comments if you have a good explanation. I'd be really interested.

Advantages of geometric algebra over quaternions

Geometric algebra has many advantages over quatnians. The Wikipedia article lists many of them. The main advantage, as far as I'm concerned, is that geometric algebra helps us understand what quirons really are. Traditionally, quatons are often described as having a scalar part and a vector part. The idea is that the vector part gives you the axis of rotation and the scalar part gives you the angle. But we know better than that. Rotations don't happen around an axis. They happen in a plane. And that's exactly what we get. A quaton is a scalar part and a bvector part. The bvector tells us in which plane the rotation takes place. The axis and the plane are orthogonal. They are dual to each other. So once again, hodgej duality has caused confusion between vectors and their dual bctors in 3D. And once again, geometric algebra clears up the confusion by making everything more consistent and precise. Another big advantage is that geometric algebra works the same way in any number of dimensions. Quatians can only describe 3D rotations, so they are much more limited. This was quite a short video. If you want to learn more about quaternians, I've collected a few interesting resources in the description below. In the next few videos, we will tackle rotations much more generally. We will construct them from simpler pieces like projections and reflections. We will see that they very naturally use the sandwich product, the ultimate power tool of geometric algebra. Subscribe if you don't want to miss that or watch the videos already on Patreon. Thank you for your support. See you next time and always keep learning.

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