I believe so. Multiplying in the manner I described in my first post rotates everywhere, but in a sort of spiral-twist fashion. To rotate around y, you multiply any number of the form
A + By + Cz
by
(-C + By + Az)/(A + By + Cz)
and analogously for other axes or other directions.
possible alternative to quaternions?
It is not clear to me how your hypercomplex value is isomorphic to a quaternion or axis-angle pair. Nor do I undersand how a rotation is simply a multiplication by a particular hypercomplex quotient. How is the hypercomplex division defined?
It has been known since 1843 that you cannot come up with a three-component algebra representing 3D rotations -- it's impossible. You must use quaternions. If you're interested, you might also want to look at geometric algebra. The quaternions are actually a subset of a larger algebra known as the set of three-dimensional multivectors. (The imaginary units i, j, and k of the quaternions correspond to the three orthogonal unit bivectors.)
I just wanted rrrr to see that you cannot let some values be imaginary units, and then give them extra properties.
The whole idea behind the square of such a quantity being -1, is exactly what relates that quantity with rotation, since -consider a vector v on a plane through the origin-, -v = v*i^2, which is v rotated by π rad on the plane normal to i. Of course this lies in the fact that multiplication with i is known to rotate by π/2.
The whole thing behind Hamilton's basic rule of transformation of i,j,k (i^2 = j^2 = k^2 = -1, ij = k, ji = -k etc...) merely displays how multiplication with these quantities should affect a vector which is expressed via these quantities (like a set of basis vectors).
E.g., if you rotate i around j by π/2 you get k, which is concisely expressed as ij=k, or the other way around, ji=-k. This is no different than rotating the regular x,y,z axes of a cartesian system around themselves.
Quaternions arose naturally (I believe) by the study of the properties of vectors expressed with the i,j,k orthogonal unit vectors.
Try to multiply two vectors, of the form xi+yj+zk and you'll be left with a quaternion in no more than a couple of lines (considering the usual properties of course).
Divide them (you'll need inverse vectors), and you'll get a quotient quaternion, which when applied to the divisor vector, rotates and stretches it (for arbitrary non unit quaternions) into the dividend!
It's amazing what you can come up with if you start writing these things down.
If you lose one basis vector (k in your example) you're no longer assuming 3d space... or if you alter the fundamental property of i^2=-1, (like with y^2=z and z^2=-y) it's not a complex quantity anymore, and most likely it will not exhibit properties such as rotating vectors.
The whole idea behind the square of such a quantity being -1, is exactly what relates that quantity with rotation, since -consider a vector v on a plane through the origin-, -v = v*i^2, which is v rotated by π rad on the plane normal to i. Of course this lies in the fact that multiplication with i is known to rotate by π/2.
The whole thing behind Hamilton's basic rule of transformation of i,j,k (i^2 = j^2 = k^2 = -1, ij = k, ji = -k etc...) merely displays how multiplication with these quantities should affect a vector which is expressed via these quantities (like a set of basis vectors).
E.g., if you rotate i around j by π/2 you get k, which is concisely expressed as ij=k, or the other way around, ji=-k. This is no different than rotating the regular x,y,z axes of a cartesian system around themselves.
Quaternions arose naturally (I believe) by the study of the properties of vectors expressed with the i,j,k orthogonal unit vectors.
Try to multiply two vectors, of the form xi+yj+zk and you'll be left with a quaternion in no more than a couple of lines (considering the usual properties of course).
Divide them (you'll need inverse vectors), and you'll get a quotient quaternion, which when applied to the divisor vector, rotates and stretches it (for arbitrary non unit quaternions) into the dividend!
It's amazing what you can come up with if you start writing these things down.
If you lose one basis vector (k in your example) you're no longer assuming 3d space... or if you alter the fundamental property of i^2=-1, (like with y^2=z and z^2=-y) it's not a complex quantity anymore, and most likely it will not exhibit properties such as rotating vectors.
Just a quick note since I am so busy. The imaginaries were not arbitrarily posited: y and z actually represent another, underlying algebra that generates their properties--which I'd rather not go into at this time. However, the same algebra, I've observed, generates the complex number system.
The imaginaries rotate in 3D, not 2D, however--in an indescribable but completely orderly fashion.
EDIT: Will deal with other issues as time permits. (By the way, if it turns out four axes are REALLY needed, I have 4D, 5D, and 6D systems built on the same principles--but that's for another thread. But I am thinking three axes might be more efficient for representing space.)
[Edited by - rrrr on August 21, 2006 7:28:57 AM]
The imaginaries rotate in 3D, not 2D, however--in an indescribable but completely orderly fashion.
EDIT: Will deal with other issues as time permits. (By the way, if it turns out four axes are REALLY needed, I have 4D, 5D, and 6D systems built on the same principles--but that's for another thread. But I am thinking three axes might be more efficient for representing space.)
[Edited by - rrrr on August 21, 2006 7:28:57 AM]
250 years ago it was concluded after 300 years of investigation by many scores of people that you could not infact represent 3 dimensional space using number composed of triples.
1,2,4,8D are the only possible ever division algebras. You may wish to look into cayley algebras and their construction.
1,2,4,8D are the only possible ever division algebras. You may wish to look into cayley algebras and their construction.
I have the greatest respect for the geniuses of the past, and my intent is not to contradict them.
All I am saying is that, coming at it from another angle, with this particular hypercomplex system, the coefficients of the three terms, each having its own axis, do form a 3D space when put at right angles to each other, and when you multiply any number by, for example, the imaginary unit y, each of the axes rotates 90 degrees--try it--and also, with the operations of addition, one can mirror points in a plane, and do other transformations, so to that extent one might find the system useful for some things that quaternions are now used for.
I do accept that a 3D system cannot be a "division algebra"--from my own initial calculation, there are some numbers that lack a unique inverse (and that's what math theory predicts)--but this does not seem to make rotation impossible. The rotations happen! So my (tentative) conclusion is that some error in interpretation or application of math theory is being made, if one says this system has to be useless.
If a concrete example showing me that my rotations, though they happen, have no practical use, can be given, then I would of course accept that!
[Edited by - rrrr on August 21, 2006 4:24:56 PM]
All I am saying is that, coming at it from another angle, with this particular hypercomplex system, the coefficients of the three terms, each having its own axis, do form a 3D space when put at right angles to each other, and when you multiply any number by, for example, the imaginary unit y, each of the axes rotates 90 degrees--try it--and also, with the operations of addition, one can mirror points in a plane, and do other transformations, so to that extent one might find the system useful for some things that quaternions are now used for.
I do accept that a 3D system cannot be a "division algebra"--from my own initial calculation, there are some numbers that lack a unique inverse (and that's what math theory predicts)--but this does not seem to make rotation impossible. The rotations happen! So my (tentative) conclusion is that some error in interpretation or application of math theory is being made, if one says this system has to be useless.
If a concrete example showing me that my rotations, though they happen, have no practical use, can be given, then I would of course accept that!
[Edited by - rrrr on August 21, 2006 4:24:56 PM]
You still haven't answered my question about what number would represent a 90 degree rotation about the y axis.
Quote:Original post by SiCraneSorry, I should have drawn attention to this post above:
You still haven't answered my question about what number would represent a 90 degree rotation about the y axis.
To rotate around y, you multiply any number of the form
A + By + Cz
by
(-C + By + Az)/(A + By + Cz)
and analogously for other axes or other directions.
In effect, in the case of y, one switches the two non-y coeffients and changes the sign of one of them.
In general, to divide, one has to factor the numerator, then cancel if possible. Now, it appears that my system is not a "division algebra"--not every number has an inverse. Assuming that this is a problem, can we say that nevertheless, any number can be divided by itself to yield 1?
If there is some theoretical problem with this, is there any difficulty with telling the computer to simply switch coefficients and change a sign? I truly don't know and that's the sort of thing I needed feedback on.
I also just wanted to mention something that may be important here.
In the case of complex numbers (which turn out to be a special case of "my" systems ), the point multiplied by i does not rotate about an axis. Rather, it rotates about a point in 2D space (the origin).<br><br>It is the same here. The point multiplied by the imaginary rotates about the origin, not an axis–and touches both sides of all axes in 3D rather than 2D.
In the case of complex numbers (which turn out to be a special case of "my" systems ), the point multiplied by i does not rotate about an axis. Rather, it rotates about a point in 2D space (the origin).<br><br>It is the same here. The point multiplied by the imaginary rotates about the origin, not an axis–and touches both sides of all axes in 3D rather than 2D.
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