complex not so complex

[Roquen]

@kram1032:  The notation portion here is mostly "much ado about nothing".  You need an expanded notation to formally construct, demonstrate a property, explain a concept, to look for "what I'm I missing here?" and to complete a derivation.  Otherwise you're going to be working purely symbolically.  The tuple notation simply fits my brain better...partly because I'm a programmer.  Still I'd do - standard 2D model: X = (s, v, b) = (s, (v_x,v_y,v_z), b).  Say ~X is grade reversal: ~(s,v,b) = (b,v,s).  Standard 3D model: (s, v, b, t) = (s, (v_x,v_y,v_z), (b_x,b_y,b_z), t).  Multi-vector algebra tend to have a fair number of null-vectors, so the form of say (s,0,b,0) jumps out at me faster.  If I were to start playing around with 3D conformal GA you couldn't pay be to write out all 32 terms.  Another potential benefit is when are are looking a different mathematical types, which are related:  say complex numbers and quaternions.  But seriously I can't see this sub-topic as being very important.

WRT: Quaternions - although they can be used to represent a 2D projective space, a 3D interpretation is much more natural (i.e. 3D bivector plus scalar).

[kram1032]

Rotation within vector algebra would typically be done via rotational matrices, right?
That's matrix multiplication and is undefined for equally shaped vectors.

[M Benesi]

I don't know what you mean. This fractal has obviously been created by iterating a non-conformal map of the Mandelbulb type. What looks like almost smooth rings are overly stretched versions of the pattern. Of course, this doesn't mean that there are no small scale detail. Just that they are stretched, some of them beyond recognition. This is not what happens in the M-set.

  You're right.  :p

[jehovajah]

Some very nice discussion here about the crucial role of symbolic notation in making the Algrbra more or less intuitive than the geometry.

If you have not read Hermann Grassmann's 1844 Ausdehnungslehre, in German, I highly recommend it for this part of the discussion. Die Ableitung der Begriffe I think is the chapter, excuse my poor German.