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[Axiom-developer] Clifford Algebra

From: Martin Baker
Subject: [Axiom-developer] Clifford Algebra
Date: Thu, 8 Dec 2016 10:55:26 +0000
User-agent: Mozilla/5.0 (X11; Linux x86_64; rv:45.0) Gecko/20100101 Thunderbird/45.5.1


You recently mentioned Clifford Algebra a couple of times so I thought I would mention an idea that I had on the subject. The idea is still too vague and hand-wavy to turn into code but I would be interested to know if anyone thinks its viable?

The idea is to implement a 3-layer architecture in the CAS. The layers being:

1 - Physics
2 - Geometry
3 - Algebra

Each of these layers would have the ability to slot in different options, for instance:

Physics - classical, relativity or quantum.
Geometry - Euclidean, projective, conformal or Minkowski space.
Algebra - Clifford or matrix/tensor.

The general idea being that, when working at the higher level there is some independence from the layers below. For instance, if you are working on a physics problem and you have a issue like:
* It does not scale up.
* It has an awkward singularity.
* Need for different type of transform not supported by algebra.
Then you can slot on a different algebra or geometry to see if that fixes the problem without changing the physics code.

Unfortunately I can see some difficulties with this idea:
1) How to refer to literal values? Even when working at the physics level we may still need to refer to concrete values for fixed points, planes, transforms and so on. Is there a way to specify these literal values in a way that does not use actual Clifford or matrix values? Even if the answer is no then I think the model could give some independence between the layers. 2) The model may need 2 or more geometry layers. For instance, we may be working in 3 dimensions but we want to combine translate and rotate into a single transform, or we want to get rid of singularities, so we add extra dimensions but we still want to translate back to original coordinates so we need multiple coordinate systems.

As I say, this is just a vague idea but I just thought I would mention it.


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