A Swift intro to Geometric Algebra (2020) [video](youtube.com)
youtube.com
A Swift intro to Geometric Algebra (2020) [video]
https://www.youtube.com/watch?v=60z_hpEAtD8
20 comments
I am a bit confused hearing a vector is one dimensional. The way we represent a vector (x,y) or (r,θ) sounds like two dimensions to me.
It's a more abstract kind of dimension. The vector is only one dimensional because it has a factor of one unit vector (only one associated "direction". A Bivector has two factors of unit vectors to describe a 2 dimensional region.
That's independent of the dimension of the space it's embedded in. So you can have a one dimensional vector in 2space, 3space, nspace etc. But it only ever describes one direction, an oriented line.
That's independent of the dimension of the space it's embedded in. So you can have a one dimensional vector in 2space, 3space, nspace etc. But it only ever describes one direction, an oriented line.
(x, y) are the components of a vector expressed in a 2D space, where space as the medium the vector is in. The dimension of the vector and the dimension of the space it's in are different. You can have a 0-dimension point on the 2D space, a 1-dimension line on the 2D space, and a 2-dimension plane on the 2D space.
In a 3D space, a vector would be (x,y,z), in 4D (x,y,z,w), etc.
"One dimensional" simply means a vector is a straight line, as opposed to a plane or a solid.
"One dimensional" simply means a vector is a straight line, as opposed to a plane or a solid.
geometrically, a vector is direction that points from one point to another -- it is one-dimensional, like a line between the two points.
The geometric algebra over linear algebra people vs the tau over pi crowd. The two at 80km/h, who is faster?
No there is no similarity in those things.
The geometric algebra unifies many concepts from physics and mathematics that without it appear as a jumble of random unrelated things.
Using geometric algebra makes it much easier to understand and remember all those things and to predict relationships between them that are not obvious.
At least for me, learning about geometric algebra was the greatest leap in understanding the mathematical structure of physics. Only learning differential and integral calculus had a similar impact.
Without geometric algebra, you have to remember a lot of haphazard facts about scalars a.k.a. "real" numbers, "complex" numbers, "imaginary" numbers, quaternions, vectors a.k.a. polar vectors, pseudovectors a.k.a. axial vectors, pseudoscalars, tensors, pseudotensors, a very large number of arbitrary multiplication rules that give various kinds of products between all those entities and many other things.
With geometric algebra, it is possible to derive from a small set of easy to understand axioms all those kinds of mathematical objects and all the interesting kinds of operations that use them, without any additional arbitrary rules or definitions.
With geometric algebra, it becomes easy to understand not only why some mathematical objects are similar, but also why some that are superficially similar are nonetheless quite distinct, e.g. which is the difference between 2-dimensional vectors and "complex" numbers.
The geometric algebra unifies many concepts from physics and mathematics that without it appear as a jumble of random unrelated things.
Using geometric algebra makes it much easier to understand and remember all those things and to predict relationships between them that are not obvious.
At least for me, learning about geometric algebra was the greatest leap in understanding the mathematical structure of physics. Only learning differential and integral calculus had a similar impact.
Without geometric algebra, you have to remember a lot of haphazard facts about scalars a.k.a. "real" numbers, "complex" numbers, "imaginary" numbers, quaternions, vectors a.k.a. polar vectors, pseudovectors a.k.a. axial vectors, pseudoscalars, tensors, pseudotensors, a very large number of arbitrary multiplication rules that give various kinds of products between all those entities and many other things.
With geometric algebra, it is possible to derive from a small set of easy to understand axioms all those kinds of mathematical objects and all the interesting kinds of operations that use them, without any additional arbitrary rules or definitions.
With geometric algebra, it becomes easy to understand not only why some mathematical objects are similar, but also why some that are superficially similar are nonetheless quite distinct, e.g. which is the difference between 2-dimensional vectors and "complex" numbers.
Cool, thanks for the nice explanation. What is the prerequisite background for studying Geometric Algebra? Is there a resource that teaches basic Physics based on Geometric Algebra?
For studying geometric algebras, it should help a lot if one is already familiar with abstract algebraic structures, like groups, rings, fields, linear spaces a.k.a. vector spaces.
I have not found yet any book that I consider really satisfactory, mainly because all of them are more or less incomplete, which is understandable, because a complete presentation would require a huge amount of work for rewriting the manuals for all the branches of traditional physics to use models based on geometric algebras.
An older decent introduction is "Geometric Algebra for Physicists" by Chris Doran and Anthony Lasenby.
There are also several older books, which need more mathematical experience, by David Hestenes, who was responsible for the revival of the theory of geometric algebras, which had previously remained a niche domain of mathematics for about a century after the too early death of their discoverer, William Kingdon Clifford.
There are also many more recent books, which can be seen e.g. through a search on Amazon or other such sites, but I have not searched such books during the last years, so I do not know which of them are good.
The older books are good enough to provide an understanding of geometric algebras, but for practical applications one usually must go beyond them.
Even without applying in practice the theory of geometric algebras, it is still useful to understand it, because this removes most of the mystery from mathematical physics and it allows a more efficient organization in your head of the knowledge about it.
I have not found yet any book that I consider really satisfactory, mainly because all of them are more or less incomplete, which is understandable, because a complete presentation would require a huge amount of work for rewriting the manuals for all the branches of traditional physics to use models based on geometric algebras.
An older decent introduction is "Geometric Algebra for Physicists" by Chris Doran and Anthony Lasenby.
There are also several older books, which need more mathematical experience, by David Hestenes, who was responsible for the revival of the theory of geometric algebras, which had previously remained a niche domain of mathematics for about a century after the too early death of their discoverer, William Kingdon Clifford.
There are also many more recent books, which can be seen e.g. through a search on Amazon or other such sites, but I have not searched such books during the last years, so I do not know which of them are good.
The older books are good enough to provide an understanding of geometric algebras, but for practical applications one usually must go beyond them.
Even without applying in practice the theory of geometric algebras, it is still useful to understand it, because this removes most of the mystery from mathematical physics and it allows a more efficient organization in your head of the knowledge about it.
Thanks for the detailed answer. I have previously studied a bit from "Linear and Geometric Algebra" by Alan Macdonald (nice book) but I don't have any Physics background. I'm interested in learning more, however!
I still maintain that Alan MacDonald is the best expositor of the subject I've seen. His books are almost entirely self-contained. Here's a sampling: http://www.faculty.luther.edu/~macdonal/GA&GC.pdf
Adding another point; I read the first of his 2 books on GA and it was great! Easy recommendation.
Indeed, I have a copy of "Linear and Geometric Algebra" and its exposition is very friendly for self-learners!
Peano had a similar reaction. https://gaupdate.wordpress.com/2013/07/01/125-years-of-peano...
Great video!