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From: Clifford J. Nelson on 2 Jun 2010 14:46
> that doesn't really parse; what do you mean to say,
> and
> have you proven it -- if implimentation in
> Wolframatism and/
> or pseudocode constitutes a proof, say that's what
> y'got.
>

The vertices of the tetrahedron (a,b,c,d) in Synergetics coordinates are
(a, b, c, -a - b - c), 
 (a, b, -a - b - d, d), 
 (a, -a - c - d, c, d), 
 (-b - c - d, b, c, d)

The sum of the vertices divided by 4 is the center of volume of the tetrahedron (a,b,c,d)

(1/4 (3 a - b - c - d), 1/4 (-a + 3 b - c - d), 1/4 (-a - b + 3 c - d), 
 1/4 (-a - b - c + 3 d))

If a+b+c+d = 0 then the Cartesian coordinates of Synergetics coordinates (a,b,c,d) are

((1/2)*(-(2*b) - c - d), 
 -((3*c + d)/(2*Sqrt[3])), 
 (-Sqrt[2/3])*d)

Any mathematician can prove it.

Cliff Nelson
From: Clifford J. Nelson on 2 Jun 2010 14:54
> > that doesn't really parse; what do you mean to
> say,
> > and
> > have you proven it -- if implimentation in
> > Wolframatism and/
> > or pseudocode constitutes a proof, say that's what
> > y'got.
> >
>
> The vertices of the tetrahedron (a,b,c,d) in
> Synergetics coordinates are
> (a, b, c, -a - b - c), 
 (a, b, -a - b - d, d), 

> (a, -a - c - d, c, d), 
 (-b - c - d, b, c, d)
>
> The sum of the vertices divided by 4 is the center of
> volume of the tetrahedron (a,b,c,d)
>
> (1/4 (3 a - b - c - d), 1/4 (-a + 3 b - c - d), 1/4
> (-a - b + 3 c - d), 
 1/4 (-a - b - c + 3 d))
>
> If a+b+c+d = 0 then the Cartesian coordinates of
> Synergetics coordinates (a,b,c,d) are
>
> ((1/2)*(-(2*b) - c - d), 
 -((3*c +
> d)/(2*Sqrt[3])), 
 (-Sqrt[2/3])*d)
>
> Any mathematician can prove it.
>
> Cliff Nelson

Where did those question marks come from?

The vertices of the tetrahedron (a,b,c,d) in Synergetics coordinates are
(a, b, c, -a - b - c), (a, b, -a - b - d, d), (a, -a - c - d, c, d), (-b - c - d, b, c, d)

The sum of the vertices divided by 4 is the center of volume of the tetrahedron (a,b,c,d)

(1/4 (3 a - b - c - d), 1/4 (-a + 3 b - c - d), 1/4 (-a - b + 3 c - d), 1/4 (-a - b - c + 3 d))

If a+b+c+d = 0 then the Cartesian coordinates of Synergetics coordinates (a,b,c,d) are

((1/2)*(-(2*b) - c - d), -((3*c + d)/(2*Sqrt[3])), (-Sqrt[2/3])*d)

Any mathematician can prove it.

Cliff Nelson
From: spudnik on 2 Jun 2010 19:15
points & cells are dual in space, but I don't get your co-
ordination, where you take a tuplet, standing for the tetrahedron
-- how? -- and
then make tuplets composed of homogenous equations
of the entries of the original tuplet;
it seems correct, but what does it mean?

I'm sure that any mathematician can prove it, but can you describe it?

take a breather, though; I am going to.
From: Clifford J. Nelson on 2 Jun 2010 16:51
XYZ three-dimensional Cartesian coordinates (1,2,3) refer to the first plane perpendicular to the X direction, the second plane perpendicular to the Y direction the third plane perpendicular to the Z direction. The planes are numbered (labeled). The intersection of the three planes define a location.

ABCD four-dimensional Synergetics coordinates (1,2,3,4) refer to four planes perpendicular to the directions of A, B, C, D in a similar way. The planes are numbered. The intersections of the planes define a regular tetrahedron whose edge length is 10.

Cliff Nelson
From: Tim Golden BandTech.com on 3 Jun 2010 16:05
On Jun 2, 8:51 pm, "Clifford J. Nelson" <cjnels...(a)verizon.net> wrote:
> XYZ three-dimensional Cartesian coordinates (1,2,3) refer to the first plane perpendicular to the X direction, the second plane perpendicular to the Y direction the third plane perpendicular to the Z direction. The planes are numbered (labeled). The intersection of the three planes define a location.

This interpretation of the cartesian plane is unnecessarily
complicated.
http://bandtechnology.com/PolySigned/NonOrthogonal/index.html
Particularly because you are engaging a nonorthogonal system this
should be of interest.
Even within the orthogonal cartesian system the meaning of
( 1, 0, 0 )
is one unit along the first axis, which is the first coordinate. In an
(x,y,z) system the one is literally the x, and there is no need to
engage in perpendicular planes in order to execute the most general
description of these coordinates, which happens to be consistent with
Lorentz transformations, nonorthogonal coordinate systems, and
projections, where essentially a projection has to be nonorthogonal.
All of these can be applied in general dimension as well, whereas
reliance on the language 'plane' will not be a general dimensional
construct.

Simply consider that every drawing on a piece of paper of 3D systems
already engages the usage of parallels. The x,y, and z axes that are
drawn on the paper cannot be perpendicular to each other on the paper
itself, though within the projection we might indicate their
orthogonality with a 'L' near the origin connecting the axes. Each
paper rendition does not require computing perpendicular planes. It
merely requires running parallel to the axis as an offset, just as the
( a, b, c )
notation denotes. This is the simplest and most general description,
and I have bothered to put up this page on this topic precisely
because of these orthogonal misconceptions with reliance upon
perpendiculars. It happens that the axes of the traditional cartesian
system are perpendicular, but there need be no additional reliance
upon perpendiculars; especially in a nonorthogonal coordinate system.
This is perhaps why the conception to you Fullerites of the
tetrahedron as a 4D structure even exists. It is a 3D structure.

>
> ABCD four-dimensional Synergetics coordinates (1,2,3,4) refer to four planes perpendicular to the directions of A, B, C, D in a similar way. The planes are numbered. The intersections of the planes define a regular tetrahedron whose edge length is 10.
>
> Cliff Nelson

Is there such a thing as a conception of addition? Also are the
coordinates real valued? Is it trouble to write
( 1, 2, 3, 4 ) + ( - 4, -3, -2, -1 ) = ( -3, -1, 1, 3 ) ?
Thus two nonzero edge length tetrahedrons yield a null. What would be
your interpretation of the above?

Cliff, the language of the simplex is a general dimensional
phenomenon. Bucky was intimately entwined with 3D space, which I
understand is preached in some numeralogical position in your Bucky
bible. This is a very unfortunate circumstance that a definition of
such a strictly 3D basis could rely upon a 2D construct such as a
plane. This is a criticism that I lay sort of sideways, because this
is not at all my way of thinking, but is the Fuller way of thinking
under the layout you propose. This is a self contradiction.

Do you people have any notion of furtherance of BF's ideas? Do you
understand that to develop these new ideas sometimes means making a
few breaks with the existent language? Do you see your belief system
as a religious one? Will it ever evolve, or are you comitted to
preservation? No matter what variations you make BF's work will always
be available. Emergent spacetime is only barely around the corner that
you refuse to budge from. The definition of 3D space from the simplex
itself directly is completely acceptable. Not only that, but it is a
generalization of the real number and the complex number. The
Pythagoreans were wrong, but they still live on as an outstanding
bunch with some practical results.

Time is now a hot topic in physics, and the one-signed number is an
accurate portrayal of time. This requires the general dimensional
reasoning on the simplex to yield this zero dimensional geometry with
an instantiable algebra.

- Tim
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