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 the 3rd dimension ? tommy1729 Ultimate Fellow     Posts: 1,358 Threads: 330 Joined: Feb 2009 04/06/2011, 04:20 PM till now we considered tetration for 2D complex numbers. what about 3D numbers ? are there methods for 3D numbers that are distinct from the 2D ? the problem might be that the Riemann Mapping Theorem does not apply in 3D. (only conformal mapping are moebius in 3D) i think there is no 3D solution and we will need to use the 2D solutions and apply them to get a 3D solution with is correct upto its " complex absolute value ". it should be noted that there are 2 types of 3D numbers. a + b P + c P^2 + d P^3 where 1 + P + P^2 + P^3 = 0 and P^4 = 1 and a , b , c , d are positive. ( group ring is the correct term here ) and the classical " 3D complex " a + b w + c w^2 where a , b , c are real and w^3 = 1 ( it is trivial to compute the " absolute complex value " , just replace P with i or w with the upper cube root of unity ) the advantage in 3D might be less fixpoints for exp^[r](z) = exp(z) and cycle detection / branch point understanding of the ordinary 2D sexp / slog. ( this relates to some threads like tid616 and tid499 amongst others ) so let me know what you think. regards tommy1729 bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 04/11/2011, 07:32 PM I never heard about 3 dimensional numbers, the possible finite dimensional division algebras must have dimension 1 (real), 2 (complex), 4 (quaternion) or 8 (octonion), see wikipedia. I remember that Hamilton tried to find to 3 dimensional numbers but failed, and came up in the end with quaternions. (04/06/2011, 04:20 PM)tommy1729 Wrote: it should be noted that there are 2 types of 3D numbers. a + b P + c P^2 + d P^3 where 1 + P + P^2 + P^3 = 0 and P^4 = 1 and a , b , c , d are positive. Isnt that 4 dimensional, a,b,c,d? Quote:and the classical " 3D complex " a + b w + c w^2 where a , b , c are real and w^3 = 1 How do you define division here? tommy1729 Ultimate Fellow     Posts: 1,358 Threads: 330 Joined: Feb 2009 04/11/2011, 09:56 PM (This post was last modified: 04/11/2011, 10:00 PM by tommy1729.) (04/11/2011, 07:32 PM)bo198214 Wrote: I never heard about 3 dimensional numbers, the possible finite dimensional division algebras must have dimension 1 (real), 2 (complex), 4 (quaternion) or 8 (octonion), see wikipedia. I remember that Hamilton tried to find to 3 dimensional numbers but failed, and came up in the end with quaternions. yes and no : we have zero-divisors in 3D. (04/06/2011, 04:20 PM)tommy1729 Wrote: it should be noted that there are 2 types of 3D numbers. a + b P + c P^2 + d P^3 where 1 + P + P^2 + P^3 = 0 and P^4 = 1 and a , b , c , d are positive.Quote:Isnt that 4 dimensional, a,b,c,d? no , since like i said : a , b , c , d are POSITIVE. the units are not orthogonal. Quote:and the classical " 3D complex " a + b w + c w^2 where a , b , c are real and w^3 = 1 Quote:How do you define division here? just as the multiplicative inverse. 1 = (a' + b' w + c' w^2)(a + b w + c w^2) if (a + b w + c w^2) is not a zero-divisor. --- in abstract algebra notation the 2 kinds of 3D numbers are RxRxR and RxC. matrix representation is a must , and they also satisfy the modified Cauchy-Riemann equations. ( they also extend the gaussian integers , which makes fun number theory but that is a bit off topic , also its possible to use them for rotations rather than euler angles and quaternions ) perhaps the following are illuminating : http://en.wikipedia.org/wiki/Group_ring ( amateur mathematician ) http://bandtech.com/PolySigned/PolySigned.html ( also check the links ) many papers by beresford or Silviu Olariu ( cant find them right now ) http://en.wikipedia.org/wiki/Tricomplex_number http://arxiv.org/PS_cache/math/pdf/0008/0008120v1.pdf http://arxiv.org/PS_cache/math/pdf/0011/0011044v1.pdf and many others. feel free to give more free references ! regards tommy1729 bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 04/12/2011, 07:42 AM Then go ahead, give us a taste how 3-complex numbers could have benefits for tetration. I guess these numbers are completely new for most of the forum members. (I always wonder why you are hiding most of the information that you seem to know in your posts) tommy1729 Ultimate Fellow     Posts: 1,358 Threads: 330 Joined: Feb 2009 04/12/2011, 12:23 PM i was just toying with the idea. « Next Oldest | Next Newest »

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