4 ms·

A Swift intro to Geometric Algebra (2020) [video]

52 points by agnosticmantis 3y ago

lzralbu 3y ago
The geometric algebra over linear algebra people vs the tau over pi crowd. The two at 80km/h, who is faster?
- cyberax 3y ago
  Not really?
  - avmich 3y ago
    It's "four Maxwell equations" vs. "one Maxwell equation". Technically they allow to do the same, the argument about convenience and maybe underlying connections which aren't that visible in linear algebra but more pronounced in geometric algebra case.
    - deleted 3y ago
      [deleted]
- adrian_b 3y ago
  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.
  - neodypsis 3y ago
    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?
    - adrian_b 3y ago
      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.
      - neodypsis 3y ago
        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!
    - zrkrlc 3y ago
      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 http://www.faculty.luther.edu/~macdonal/GA&GC.pdf
tanvach 3y ago
Great video!
albert_e 3y ago
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.
- mecsred 3y ago
  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.
- jb1991 3y ago
  geometrically, a vector is direction that points from one point to another -- it is one-dimensional, like a line between the two points.
- ttctciyf 3y ago
  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.
- ww520 3y ago
  (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.
astrange 3y ago
The capitalization on here made it look like it was in Swift.
- dmvdoug 3y ago
  Exactly why I clicked on it. “Have you ever been confused…” the video begins. Why yes, yes, I have!

lzralbu 3y ago

The geometric algebra over linear algebra people vs the tau over pi crowd. The two at 80km/h, who is faster?

cyberax 3y ago
Not really?
- avmich 3y ago
  It's "four Maxwell equations" vs. "one Maxwell equation". Technically they allow to do the same, the argument about convenience and maybe underlying connections which aren't that visible in linear algebra but more pronounced in geometric algebra case.
  - deleted 3y ago
    [deleted]
adrian_b 3y ago
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.
- neodypsis 3y ago
  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?
  - adrian_b 3y ago
    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.
    - neodypsis 3y ago
      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!
  - zrkrlc 3y ago
    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 http://www.faculty.luther.edu/~macdonal/GA&GC.pdf

tanvach 3y ago

Great video!

albert_e 3y ago

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.

mecsred 3y ago
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.
jb1991 3y ago
geometrically, a vector is direction that points from one point to another -- it is one-dimensional, like a line between the two points.
ttctciyf 3y ago
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.
ww520 3y ago
(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.

astrange 3y ago

The capitalization on here made it look like it was in Swift.

dmvdoug 3y ago
Exactly why I clicked on it. “Have you ever been confused…” the video begins. Why yes, yes, I have!