Geometric Algebra, First Course, Episode 00: High Level Overview.

Transcript

hello and welcome to the stemce studio channel in this video i'm announcing a new series of videos that i'm calling geometric algebra in stem c studio geometric algebra if you've never heard of it is just probably one of the most fascinating mathematical tools that i have come across and by all account people like david hestonis and anthony lazenby are saying this is basically the 21st century mathematics for physics now if you've never heard of geometric algebra you can be forgiven because it's only been around for a couple of hundred years yeah you heard that right it has been around for a couple of hundred years and why it is not part of our current curriculum is probably a subject for historians um and politicians but we're going to put that all aside and we're going to in this video just i want to just give you a feel for what geometric algebra is all about and then tell you what this course is going to be all about geometric algebra is the idea that all physical quantities have a geometric object that can be associated with them so for example mass charge force angular momentum linear momentum all of these have an associated mathematical object and these mathematical objects can be combined with operators that is we can have a times b or a wedge b the operator goes in the middle we can combine all of these kind of quantities with operators to construct geometrical statements furthermore geometric algebra opens the possibility of these geometric objects which kind of comprise geometric numbers it opens a possibility for those to have a geometric and intuitive geometric representation so with this diagram here here let me give you um an idea of what i mean by that so this is three dimensional space and as you know things like velocity and momentum are represented by vectors mathematically and in a visual representation or geometric representation we represent those by arrows okay so the word vector comes from the french vector and it means to carry or to transport you may have heard it in connection with things like malaria and mosquito being the vector well that's kind of what we mean by a vector it's kind of like something that moves something from one place to another but geometric algebra opens the possibility that vectors aren't really the only objects that are in our interest and i've tried to illustrate that in this um animation here um by these paddles okay so the paddle is a new kind of object one at least that we don't kind of conventionally encounter uh certainly in education but it is a very appropriate object for representing things like torques angular momentums angular velocity and as we will see it actually is the correct representation for the electromagnetic field or at least it's a much more appropriate way to model the electromagnetic field than the way that we do it now using the electric field being a vector and the magnetic field being a vector now these two quantities that i've just shown you this paddle rotating and the arrow may not seem to have very much in common but in fact they actually do have some three common properties and i'm going to tell you what they are because then we can relate it to two other geometric objects in this diagram so the properties that they have in common are magnitude which is some measure of their size so for example the vector is the length of the arrow for this rotating object this paddle the the corresponding magnitude is really some kind of an area measure so there's no specific area that we should assign to this it's not a square it's not a circle but it is just an area so that's magnitude another quantity is called the aspect and this is to do with the sort of degree of freedom dimensions that this thing operates and i for want of a better word so here this vector here lies along a line so that's a one dimensional object right you can parameterize it with a single parameter tells you where you are on the line and in this quantity this paddle we actually have a plane okay so this is a this is a plane so this is really like a two-dimensional object now i should just add here that the conventional way to think about angular momentum and angular velocities is to use something that you might call the right-hand rule or depending on the situation you might be using a left-hand rule so you can imagine grabbing this white arrow here say with your right hand and your fingers would would give the direction of rotation of this well that works in 3d which is what we're in here but it doesn't work so well in 2d because in 2d in in the plane itself there is no third dimension that you can work with okay so we really find that this idea of a rotating kind of plane or spinning plane is much more generalizable to other dimensions so for example as we move to higher dimensions maybe four dimensions the direction here would be ambiguous and you wouldn't be able to just put your hand or your fingers on this direction of rotation and point to like a unique perpendicular so we've dealt with magnitude which is the length here the area here we dealt with aspect which is this is one dimensional arrow this is a two dimensional plane and then there's one other property which these have in common which is something we call orientation so for example here we have a line with the vector and this arrow is pointing to the right well it could point to the left so the orientation is basically like a plus or minus similarly for this rotating object it's maybe you know plus in this direction and minus in the other direction okay now what about things like mass and charge and things like that these are things we call scalar quantities and scalar quantities you know we're all we're always told you know scalar quantities don't have direction right they have a magnitude okay but they don't have a direction now the word direction you know it's an unfortunate sort of choice of word in the english language and it doesn't really fit very well but it actually turns out that um scalars actually have a lot more in quant in common with the vector and this spinning paddle than you might imagine so what we represent a scalar by is essentially a point because that kind of doesn't have a direction right but it also still has a magnitude right because i mean it's just a number okay so so the mass you know might have a certain certain numerical value that's the magnitude does it have an aspect recall the aspect is kind of like the dimensionality and the line here is one dimensional a plane here that's rotating that's two dimensional so a point which is like you know scale is kind of like a point this is actually zero dimensional okay so actually scalars in some ways you know that they're quite similar in in the way that they share these two properties they also share an orientation they can be positive they can be negative okay so actually a scalar and a vector and this rotating paddle here so the arrow and the rotating paddle they actually can all be described by the same set of properties now it doesn't stop there in three dimensions there's another quantity that um we can have and i've tried to represent it by this wire cube which is kind of like a volume element okay so let's see if we can kind of guess you know what the uh what the three properties are of this volume element well the magnitude i've given it away it's the volume itself okay what's the aspect like the sort of number of dimensions in here well it's kind of three-dimensional it's filling the whole of space okay so that's the sort of space that it occupies and what's its orient and does it have two orientations like everything else well i'll leave you to actually imagine this you know maybe you maybe kind of create little models at home but imagine you have a piece of wire that goes from this corner of the cube up to this corner the opposite corner of the cube how many different shapes of wire do you need to get from that corner to this corner to the opposite corner and you can kind of like go many different ways right but how many shapes of y do you have i'll uh you know let you guess what the answer might be okay so that's the sort of intuitive idea of what geometric algebra is it's a way of assigning mathematical objects to to the physical quantities connecting them all with operators and there are many interesting operators we have obviously addition and subtraction but we get into multiplication we can multiply these things together we can pull in a sense to factorize out pull out say we could pull out an arrow out of one of these believe it or not and we'd be left with another arrow that's not surprising right this is a one-dimensional object this is a two-dimensional object and if we pull an arrow out of this thing then maybe we're left with another arrow it's kind of like a subtraction similarly we can take two arrows here and we can perform an operation on them which we call the exterior product and that actually constructs for example one of these kind like paddles the official name for this paddle in the mathematical sense is that it's a bi vector because it's made of two vectors this big cube object is a tri vector because we think of it as being made up of three vectors not three vectors added together like this vector plus this vector plus this vector no they're not added together they're actually multiplied together and that's really the in a sense where the word geometric algebra gets its sort of geometric from it's called the geometric product which is the pro when you when you multiply vectors together and the algebra kind of comes from the idea that we have elements that we can multiply together using operators so the goal in the course will be really as follows we want to create an introduction to geometric algebra which is computational so you're able to use visualizations to help uh build your intuition but you will build your own geometric numbers as they call them and you'll give them also operators so that they can be kind of combined into these physical laws and then we'll demonstrate how we can use those in various applications so there's obviously a lot of theory that can go along with this but i will make the course self-contained so it won't be necessary to um quote um you know textbooks and things like that um and most of it the proof will be kind of in the pudding of you know what what kind of light works uh we'll start pretty simple we'll start in 2d and we'll do the geometric algebra of what we call the plane and then from there we can go in many different directions um we can add a dimension and go to three spatial dimensions we could add um we instead instead of just staying with spatial dimensions we could add time dimensions and that could take us into something called special relativity and so we could do that by going from two spatial dimensions to one spatial plus one one time or we could go from two spatial dimensions to two plus one time and in fact that's probably the best place to go because it's a good compromise we can visualize it because we have a total of three dimensions but also we can get to see most of the interesting effects so for example in two plus one space time those two space dimensions one time dimension we'll learn that you can only have a magnetic field with one component but you can have two electric field components so that should kind of automatically dispel the idea that in electromagnetism uh the way we're currently describing our elec electric and magnetic fields using vectors is not correct um how can a magnetic field in two dimensions only have one component it can be a vector so that's the program we um we're basically going to compute our way through geometric algebra adding complexity as we go and in the process we will learn how to do these various things in stem c studio i hope you found this interesting uh if you like this video and you're looking forward to the uh to the upcoming series please hit the subscribe button so you'll be notified and if you like this video itself um please hit the like button and tell your friends about it so these videos will be coming along very shortly probably at a fairly rapid pace so i look forward to seeing you there take care