Re: are Real Numbers evil?

> Polysigned numbers can be used for 3D space.

I have spent some hours on that, but can't see
any practical case making it useful.

To give an example what I mean with useful:
Quaternions are abstract constructs, but they make
it possible to add rotations in 3D space and help to
make animate transitions from one orientation into
another look natural. If not looking behind the scenes
of operating on them it is easy to work with them, so
they are widely used replacing vector transformations
using matrices.

>From what I understand now, the polysigned numbers
are nothing else as a complicated way of speaking
about vector operations where the coordinate system
is not orthogonal but based on a regular triangle (2D)
and regular tetrahedron (3D). I suppose, that the core
of the fascination and the observation related to this
concept is, that going around a triangle or a tetrahedron
will allow to come back to same point with less turns as
needed in orthogonal systems (2D: two turnes for triangle,
three turns for square, 3D: max. four turns for tetrahedron,
six turns for cube).

Is there more behind the polysigned numbers
as pure theoretical fun to do some new math?

What idea was the initial spark which has lead to
creating this additional operators for real numbers?
What was the reason you have started to think about
such a construct and finally published it on Internet?

By the way: I suppose the core of the problem with
3D not occuring in 2D in the fact, that when in 2D it
is possible to cover a plane with regular triangles,
it is not possible to cover 3D space with regular
tetrahedrons (as it is possible using cubes).

Claudio



<tttpppggg@xxxxxxxxx> schrieb im Newsbeitrag
news:1118342475.019484.107700@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> Polysigned numbers can be used for 3D space.
> ( http://bandtechnology.com/PolySigned/PolySigned.html )
> 3D cartesian is equivalent to four-signed numbers.
> The graphical representation is four rays emanating from the center of
> a tetrahedron to its corners. These are unsigned components. There are
> four of them.
> They follow the rule:
> - x + x * x # x = 0. (where * and # are new signs)
> In other words the sum of vectors of magnitude x from each of those
> tetrahedral rays lands you back at the origin.
> This is equivalent to the two-signed numbers obeying:
> - x + x = 0.
> This is of course the real numbers. They have two signs. They are
> one-dimensional.
> In general a sum in two-signed will boil down to a single number but in
> four-signed general sums can only be reduced to three values, the
> fourth being zero. Reduction is not really a necessity. For example:
> - 1.32 + 1.24 * 0.24 # 3.24
> is a fine representation, even though the star (*) component could be
> reduced to zero, with the other values reduced by 0.24.
>
> A product also exists.
> Interestingly the three-signed numbers are equivalent to complex
> numbers under this product rule, as well as sums. The meaning of the
> four-signed product is more elusive however. The product does do some
> rotation. However, magnitudes are not preserved.
> Three-signed numbers do rotation as do complex numbers. perhgaps there
> is where your solution lies, since rotations are generally taken about
> an axis you can transform each plane normal to the axis, multiply by
> your theta equivalent, then transform back to get your original
> cartesian mess.
>
> -Tim
>


.

Relevant Pages

  • Re: are Real Numbers evil?
    ... I am not honestly that familiar with how you use them for rotations ... that going around a triangle or a tetrahedron ... I do not know of any other natural construction that can do this. ... Reals are two-signed. ...
    (comp.theory)
  • Re: are Real Numbers evil?
    ... The graphical representation is four rays emanating from the center of ... a tetrahedron to its corners. ... These are unsigned components. ... In other words the sum of vectors of magnitude x from each of those ...
    (comp.theory)
  • Re: Gauss-Lobatto Quadrature over a Tetrahedron
    ... > Fv is the sum of the function at the vertices ... > Fm the value at the center of the tetrahedron. ... tetrahedron and at the edge midpoints, ... order elements becomes a pain). ...
    (comp.lang.fortran)
  • Re: Gauss-Lobatto Quadrature over a Tetrahedron
    ... > circles, squares, triangles, hexagons, spheres, cubes, and ... > There are two solutions given for the tetrahedron: ... > Fv is the sum of the function at the vertices ...
    (comp.lang.fortran)