Re: are Real Numbers evil?

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?
    ... is not orthogonal but based on a regular triangle ... that going around a triangle or a tetrahedron ... > In other words the sum of vectors of magnitude x from each of those ... > is where your solution lies, since rotations are generally taken about ...
    (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)