I ran across Geometric Algebra a couple months ago, and am convinced it’s one of the most important things I’ve learned about in years.
Some quick summary / commentary to motivate reading the link:
- By the 19th century, the complex numbers had become a generally acceptable concept, for example with lots of attempts to prove the fundamental theorem of algebra. The idea that the complex numbers could be thought of as 1–1 mapped to a two-dimensional plane was widely known. Complex numbers used as an algebraic proxy for points in the plane were useful for solving geometry problems that Cartesian analytic geometry couldn’t manage.
- Many thinkers were trying to extend these ideas to three-dimensional space. Hermann Grassmann, who invented GA in the 1840s, was one of many, but in many ways he got things “right”. Another thinker who worked extensively on this problem was Hamilton, who devoted decades to the quaternions (a sort of generalization of the complex numbers with three imaginary units i, j, k instead of one) which can be quite useful for characterizing rotations and are still used in niche physics applications even though they’ve been otherwise marginalized by matrix approaches.
- It turns out that there is a quite natural relationship between imaginary numbers, the vector part of a quaternion, and geometric algebra. As Hestenes describes in the linked essay, the imaginary unit, a “bivector”, can be thought of either as an operator which performs a 90° rotation on vectors in the plane, or as a “directed area” with unit magnitude. A quaternion’s vector part is just a combination of three unit bivector components of geometric algebra in 3-dimensional space. This gives complex numbers and quaternions a nice intuitive geometrical interpretation, and makes them a lot less foreign seeming to students.
- The traditional/mainstream conception of vector algebra (which ended up winning the 19th century fight among tools for the algebra of space) uses a dot product and a cross product, which are defined to be, respectively, a scalar and a vector. Cross products constantly come up in the physics of rotations, and in trying to compute areas and volumes. Under GA, the “outer product”, mathematically dual to the cross product, is seen to be a bivector. This is a much better description because the behavior of vectors produced by cross products is quite different than other vectors, which means that considering them to be the same type of object results in a great deal of confusion.
- Geometric algebra has a natural relationship with projective geometry (homogeneous coordinates, Plücker coordinates). Such mathematics is used constantly in applications like computer graphics.
- Physicist David Hestenes has spent most of his career since the 1960s elaborating and advocating GA. In exploring various areas of physics, he found that in almost every case GA was at least as convenient as the alternative mathematical formulations, and often quite insightful. For instance, some quantum phenomena which were previously considered “counterintuitive” or strange become quite obviously necessary and natural conclusions of their mathematical context in GA.
- GA is still far from mainstream. As far as I can tell, it is taught to almost no undergraduates even though it is pedagogically and operationally superior to the alternatives. It mainly comes up in very technical advanced physics courses and extremely abstract mathematical proofs. Reading some of these, one would never realize how broadly applicable and powerfully connective its ideas are.
- Geometric Algebra is (I firmly believe) the future of geometric education. It should be taught to all students of mathematics, physics, engineering, computer graphics, computer vision, and similar fields, perhaps starting in high school. It simplifies both understanding and computation; for instance, many problems can be done coordinate free, making their solutions easier to tackle and more intuitively relevant.
Thanks for the linked article. I can't believe I never ran across this concept before, outside of a formal definition. The links given in the article to applications are convincing about its utility.
I am currently working my way through some geometric algebra (as well as algebraic geometry and topology) texts and building a [programming] library incorporating what I read to reinforce my mathematical intuition. I completely agree about how awesome Geometric Algebra is. In fact its emphasis of tying algebra with geometry makes it much more useful than for just teaching geometry. It makes a wonderful foundation for mathematical intuition. For self study I particularly like such subjects because they take a whole bunch of stuff and put them under one roof.
Grassmann's story is semi-tragic. He was a school teacher who created a powerful algebra of space where you could compute with not just points and vectors but with lines, planes, surfaces and other vector subspaces without being hampered by a particular coordinate system. Geometric Algebra can subsume complex numbers, quaternions and plane geometry - projections, rotations etc and easily extend them to higher surfaces. Matrices play less a role here. Hestens also puts forward that Geometric Calculus can contain matrix calculus, linear algebra, lie algebra and groups and Differential forms, see [1]. But because Grassmann was so ahead of his time and did himself no favours by writing a dense, super long text with lots of philosophy interwoven not many bothered to read or could understand it. Later, Clifford came and made extensions, cleaned it up and the subject gets named after him as Clifford Algebras, though not by his choice. But Clifford died early so the more geometrically unwieldy Vector Analysis won out. A setback for physics?...Angular momentum makes so much more sense as a bivector. From [2]:
Fearnley-Sander writes in [27]:-
All mathematicians stand, as Newton said he did, on the shoulders of giants, but few have come closer than Hermann Grassmann to creating, single-handedly, a new subject.
To read more about Grassmann I recommend: A History of Vector Anaylsis by Michael Crowe, well written, I couldn't put it down.
Perhaps the most interesting part, to me, was the goal of making geometric algebra operations computationally efficient. While I'm still reading (and enjoying) the original paper, I must confess an inability to see how it's more efficient than a standard vector representation. Would you have any pointers on that?
At the moment, the core of the gains are in the conceptually clean framework at the cost of speed. Kind of like going from C to Python. It will help your linear algebra by strengthening your geometric intuitions, stuff like how to figure out if a plane intersects a sphere or if a line intersects a plane or SLERP are much are easier for the novice. You solve it algebraically but the objects you are reasoning with are geometric entities so you can't help but build your intuition on both.
Computationally though, they take more operations and elements (2^dim). And there is an order of magnitude difference in speed - 100x if implemented naively. But there are optimizations that can get it to 2x - 7x [1]. But apparently for full generality, 3D geometry is best done in a 5D conformal geometry which requires 32 element representations but I haven't gotten that far yet. You can make it more efficient by taking advantage of the sparseness of most structures and being able to scale up to more information as required by filling more indices. Something that cannot be handled as cleanly in standard vector based methods. Strangely this method of conformal geometry is covered by a Patent so commercial usage in the US requires licensing arrangements [2].
Thanks, that's interesting, and seems more in line with what I figured: a lot easier to reason about, and a much cleverer coder than me to do it properly :)
(As an aside, one of the inventors listed on that patent happens to be the person who wrote this article.)
What advantage does this have over traditional differential geometry?
I scanned through the linked paper briefly, and didn't see much there, other than what seemed like a muddying of the concepts of a one-form and a vector, effectively always assuming the standard Euclidean metric tensor to flip from one to the other.
Having two directed concepts is very useful, especially once things start to get tricky. My background has a lot of general relativity and plasma in it, though, so my perceptions may be colored by the particulars of those fields.
I can't be of help because I only started studying this stuff on the side not long ago. But one of the first things I searched for was differential forms vs Geometric algebra. It appeared that the war between the two was old with both sides claiming subsumption. I decided to learn both but started with GA since it seemed easier to compute with. See:
The basic advantage is that it’s much more accessible to non-mathematicians, makes the connections between different mathematical abstractions more obvious and explicit, and of a bit more general use because it includes the concept of a “multivector” which combines multiple types of objects.
Cartan was directly inspired by Grassmann’s work when creating differential forms (indeed, the word form comes from Grassmann).
I see. It seems more natural for quantum mechanics than for general relativity, though. In GR, the metric is a living, breathing thing, and I don't see how to shoehorn that into GA. Similarly, differential geometry gives you a nice, mechanistic way to think about coordinate transformations, including accelerated/rotating coordinates.
Similarly, there's Littlejohn's work on applying "normal" differential geometry to plasma physics. I've not read nearly as much about that as I would like to, but it always seemed like fun stuff. I don't think this kind of notation helps with that.
As a former student of physics, this does appear quite interesting. The article feels concise and well-structured, and the math seems very elegant.
For physics and math students I recommend starting at page 9. The 8 pages of rationale seem a little unnecessary, with the math mostly speaking for itself.
I'm looking forward to giving this a more detailed read later.
As another former physics student, I'm excited about this too. I think my understanding of physics would benefit greatly if the various mathematical formalisms used to describe its different aspects were unified systematically.
I'm starting to read this article carefully right now.
This paper / talk was given in 2002, so of course it doesn't tell you what happened with the adoption of, or interest-level in Geometric Algebra (GA) since then. A quick search on Amazon.com turns up over a dozen recent books using GA in physics, computer science and engineering.
18 comments
[ 5.0 ms ] story [ 49.4 ms ] threadSome quick summary / commentary to motivate reading the link:
- By the 19th century, the complex numbers had become a generally acceptable concept, for example with lots of attempts to prove the fundamental theorem of algebra. The idea that the complex numbers could be thought of as 1–1 mapped to a two-dimensional plane was widely known. Complex numbers used as an algebraic proxy for points in the plane were useful for solving geometry problems that Cartesian analytic geometry couldn’t manage.
- Many thinkers were trying to extend these ideas to three-dimensional space. Hermann Grassmann, who invented GA in the 1840s, was one of many, but in many ways he got things “right”. Another thinker who worked extensively on this problem was Hamilton, who devoted decades to the quaternions (a sort of generalization of the complex numbers with three imaginary units i, j, k instead of one) which can be quite useful for characterizing rotations and are still used in niche physics applications even though they’ve been otherwise marginalized by matrix approaches.
- It turns out that there is a quite natural relationship between imaginary numbers, the vector part of a quaternion, and geometric algebra. As Hestenes describes in the linked essay, the imaginary unit, a “bivector”, can be thought of either as an operator which performs a 90° rotation on vectors in the plane, or as a “directed area” with unit magnitude. A quaternion’s vector part is just a combination of three unit bivector components of geometric algebra in 3-dimensional space. This gives complex numbers and quaternions a nice intuitive geometrical interpretation, and makes them a lot less foreign seeming to students.
- The traditional/mainstream conception of vector algebra (which ended up winning the 19th century fight among tools for the algebra of space) uses a dot product and a cross product, which are defined to be, respectively, a scalar and a vector. Cross products constantly come up in the physics of rotations, and in trying to compute areas and volumes. Under GA, the “outer product”, mathematically dual to the cross product, is seen to be a bivector. This is a much better description because the behavior of vectors produced by cross products is quite different than other vectors, which means that considering them to be the same type of object results in a great deal of confusion.
- Geometric algebra has a natural relationship with projective geometry (homogeneous coordinates, Plücker coordinates). Such mathematics is used constantly in applications like computer graphics.
- Physicist David Hestenes has spent most of his career since the 1960s elaborating and advocating GA. In exploring various areas of physics, he found that in almost every case GA was at least as convenient as the alternative mathematical formulations, and often quite insightful. For instance, some quantum phenomena which were previously considered “counterintuitive” or strange become quite obviously necessary and natural conclusions of their mathematical context in GA.
- GA is still far from mainstream. As far as I can tell, it is taught to almost no undergraduates even though it is pedagogically and operationally superior to the alternatives. It mainly comes up in very technical advanced physics courses and extremely abstract mathematical proofs. Reading some of these, one would never realize how broadly applicable and powerfully connective its ideas are.
- Geometric Algebra is (I firmly believe) the future of geometric education. It should be taught to all students of mathematics, physics, engineering, computer graphics, computer vision, and similar fields, perhaps starting in high school. It simplifies both understanding and computation; for instance, many problems can be done coordinate free, making their solutions easier to tackle and more intuitively relevant.
Grassmann's story is semi-tragic. He was a school teacher who created a powerful algebra of space where you could compute with not just points and vectors but with lines, planes, surfaces and other vector subspaces without being hampered by a particular coordinate system. Geometric Algebra can subsume complex numbers, quaternions and plane geometry - projections, rotations etc and easily extend them to higher surfaces. Matrices play less a role here. Hestens also puts forward that Geometric Calculus can contain matrix calculus, linear algebra, lie algebra and groups and Differential forms, see [1]. But because Grassmann was so ahead of his time and did himself no favours by writing a dense, super long text with lots of philosophy interwoven not many bothered to read or could understand it. Later, Clifford came and made extensions, cleaned it up and the subject gets named after him as Clifford Algebras, though not by his choice. But Clifford died early so the more geometrically unwieldy Vector Analysis won out. A setback for physics?...Angular momentum makes so much more sense as a bivector. From [2]:
Fearnley-Sander writes in [27]:-
All mathematicians stand, as Newton said he did, on the shoulders of giants, but few have come closer than Hermann Grassmann to creating, single-handedly, a new subject.
To read more about Grassmann I recommend: A History of Vector Anaylsis by Michael Crowe, well written, I couldn't put it down.
[1] http://geocalc.clas.asu.edu/pdf/DIF_FORM.pdf
[2] http://www.gap-system.org/~history/Biographies/Grassmann.htm...
Computationally though, they take more operations and elements (2^dim). And there is an order of magnitude difference in speed - 100x if implemented naively. But there are optimizations that can get it to 2x - 7x [1]. But apparently for full generality, 3D geometry is best done in a 5D conformal geometry which requires 32 element representations but I haven't gotten that far yet. You can make it more efficient by taking advantage of the sparseness of most structures and being able to scale up to more information as required by filling more indices. Something that cannot be handled as cleanly in standard vector based methods. Strangely this method of conformal geometry is covered by a Patent so commercial usage in the US requires licensing arrangements [2].
[1] http://staff.science.uva.nl/~fontijne/phd/fontijne_phd_compr...
[2]http://www.google.co.uk/patents?id=0CkWAAAAEBAJ&pg=PA7&#...
(As an aside, one of the inventors listed on that patent happens to be the person who wrote this article.)
http://mathdl.maa.org/images/upload_library/22/Ford/DesmondF...
I scanned through the linked paper briefly, and didn't see much there, other than what seemed like a muddying of the concepts of a one-form and a vector, effectively always assuming the standard Euclidean metric tensor to flip from one to the other.
Having two directed concepts is very useful, especially once things start to get tricky. My background has a lot of general relativity and plasma in it, though, so my perceptions may be colored by the particulars of those fields.
http://groups.google.com/group/sci.physics.research/browse_t...
http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node1...
http://geocalc.clas.asu.edu/pdf/DIF_FORM.pdf
Cartan was directly inspired by Grassmann’s work when creating differential forms (indeed, the word form comes from Grassmann).
Similarly, there's Littlejohn's work on applying "normal" differential geometry to plasma physics. I've not read nearly as much about that as I would like to, but it always seemed like fun stuff. I don't think this kind of notation helps with that.
For physics and math students I recommend starting at page 9. The 8 pages of rationale seem a little unnecessary, with the math mostly speaking for itself.
I'm looking forward to giving this a more detailed read later.
I'm starting to read this article carefully right now.
Also there is a differential geometry book coming out soon: http://www.amazon.com/Approach-Differential-Geometry-Cliffor...