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Jun 6, 2021, 10:22:08 AM6/6/21

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There is a matrix equation including a 3x3 matrix which, when solved for

its degenerate cases, gives orthonormal surfaces in the various confocal

ellipsoidal coordinate systems. There are 11 solutions if memory

serves. Adding the toroidal and helical systems, we have 13 systems and

so, at least 13 unit volumes. Since in Cartesian coordinates, the axes

are fully interchangeable, the unit volume (the unit cube) is fully

defined by its volume of 1 L^3. However, in all of the other systems,

the axes are not fully interchangeable.

Let's examine the system I consider next most challenging, the

cylindrical coordinate system. I believe it to be a degenerate case of

projected ellipsoidal coordinates. (That is not the most popular name

for that system)

With a surface of constant r=1, and surfaces at l (ell) = 0 and l = 1

and including a (alpha) from [-pi, ... pi] the unit "slug" has a volume

of pi.

We could normalize this volume to 1 by scaling each axis by pi. But that

won't work.

I am interesting in seeing and counting (enumerating) the most basic

unit volumes and their associated surfaces for inclusion in a computer

software library of "atomic" features from which "everything" (to second

order) may be designed, in a attempt to provide a reasonable and

nontrivial basis for para-universal constructors.

K. Eric Drexler leads the field in attempts to implement an atomic scale

additive universal constructor. Adrian Bowyer leads in attempts to

implement a human-scale universal constructor. Julian Leland Bell has

made significant progress in implementing a subtractive universal

constructor--his Swarthmore project was a self-reproducing externally

framed milling machine.

I built a four-axis mill with some self-reproducing features in 1997

and sold one of two copies to a hobbyist, advertising it in The Want AD

as a "self-reproducing milling machine" for $300. I wrote that up at

ESG at MIT. The writeup was mentioned in Kinetic Self-Replicating

Machines (KSRM) by Frietas and Merkle in 2004. I recovered the web site

mentioned in KSRM using the Wayback Machine maintain the page

first.replikon.net to this day, documenting that machine build.

It seems to me that including advanced math in CAD representations of

manufacturable objects would reduce file sizes and eliminate

digitization and tiling errors which are becoming a problem as the

resolution of additive and subtractive manufacturing machinery

increases, which is why I am writing about this here.

This post would go to sci.math were it not for the ubiquitous use of

change of coordinate system in solving the most advanced physics

problems. Briefly, when an initial, constraining, or terminal condition

of a physics problem is representable most effectively in a coordinate

system other than Cartesian, translating the entire problem into that

system can provide a solution where no other method will work. The

solution to the Navier-Stokes equations with viscosity for the case of

flow over a sphere is a famous example--after the change of system the

problem is thereby reduced from 3 dimensions to only 1, and is readily

solved.

Douglas Goncz

Replikon Research FCN 783774974

its degenerate cases, gives orthonormal surfaces in the various confocal

ellipsoidal coordinate systems. There are 11 solutions if memory

serves. Adding the toroidal and helical systems, we have 13 systems and

so, at least 13 unit volumes. Since in Cartesian coordinates, the axes

are fully interchangeable, the unit volume (the unit cube) is fully

defined by its volume of 1 L^3. However, in all of the other systems,

the axes are not fully interchangeable.

Let's examine the system I consider next most challenging, the

cylindrical coordinate system. I believe it to be a degenerate case of

projected ellipsoidal coordinates. (That is not the most popular name

for that system)

With a surface of constant r=1, and surfaces at l (ell) = 0 and l = 1

and including a (alpha) from [-pi, ... pi] the unit "slug" has a volume

of pi.

We could normalize this volume to 1 by scaling each axis by pi. But that

won't work.

I am interesting in seeing and counting (enumerating) the most basic

unit volumes and their associated surfaces for inclusion in a computer

software library of "atomic" features from which "everything" (to second

order) may be designed, in a attempt to provide a reasonable and

nontrivial basis for para-universal constructors.

K. Eric Drexler leads the field in attempts to implement an atomic scale

additive universal constructor. Adrian Bowyer leads in attempts to

implement a human-scale universal constructor. Julian Leland Bell has

made significant progress in implementing a subtractive universal

constructor--his Swarthmore project was a self-reproducing externally

framed milling machine.

I built a four-axis mill with some self-reproducing features in 1997

and sold one of two copies to a hobbyist, advertising it in The Want AD

as a "self-reproducing milling machine" for $300. I wrote that up at

ESG at MIT. The writeup was mentioned in Kinetic Self-Replicating

Machines (KSRM) by Frietas and Merkle in 2004. I recovered the web site

mentioned in KSRM using the Wayback Machine maintain the page

first.replikon.net to this day, documenting that machine build.

It seems to me that including advanced math in CAD representations of

manufacturable objects would reduce file sizes and eliminate

digitization and tiling errors which are becoming a problem as the

resolution of additive and subtractive manufacturing machinery

increases, which is why I am writing about this here.

This post would go to sci.math were it not for the ubiquitous use of

change of coordinate system in solving the most advanced physics

problems. Briefly, when an initial, constraining, or terminal condition

of a physics problem is representable most effectively in a coordinate

system other than Cartesian, translating the entire problem into that

system can provide a solution where no other method will work. The

solution to the Navier-Stokes equations with viscosity for the case of

flow over a sphere is a famous example--after the change of system the

problem is thereby reduced from 3 dimensions to only 1, and is readily

solved.

Douglas Goncz

Replikon Research FCN 783774974

Sep 25, 2021, 3:12:20 AM9/25/21

to

The Change of Variable Theorem is of interest as it generalizes the

transformations between coordinate systems.

I have found pictures of something like unit volumes for the Cartesian,

Cylindrical, and Spherical coordinate systems.

There are 16 orthogonal systems listed at Math World under Orthogonal

Coordinates and it is asserted on that page they are all degenerate

cases of elliptical coordinates, a mistake; Giankoplis gives the

derivation of the degenerate cases from the matrix equation.

I do not have a chart and may have to program in Mathcad. That's pretty

easy using the transformation equations to Cartesian coordinates tracing

along each edge of the coordinate system specific unit volume (not the

differential volume, but a substantial "chunk" of spaces, near the

origin, in each system). I think I can articulate some of those limits

here today before trying it:

For each coordinate axis with range from 0 to oo, apply limits of 1/2 to 1.

For each coordinate axis with range from 0 to 2pi, apply limits of pi/2 to 3pi/2.

For each coordinate axis with range from 0 to pi, apply limits of pi/4 to 3pi/4.

There are others, however, and there are inequalities.

Ideally the volume of each "chunk" would be 8 since the obvious chunk of

Cartesian space is:

x=[-1,1]; y=[=1,1]; z=[-1,1].

I am open to suggestions today and have not started programming yet.

Cheers,

Douglas Goncz

Replikon Research FCN 7837774974

On Sunday, June 6, 2021 at 10:22:08 AM UTC-4, Douglas Dana Edward^2 Parker-Goncz (fully) wrote (I wrote):

...

transformations between coordinate systems.

I have found pictures of something like unit volumes for the Cartesian,

Cylindrical, and Spherical coordinate systems.

There are 16 orthogonal systems listed at Math World under Orthogonal

Coordinates and it is asserted on that page they are all degenerate

cases of elliptical coordinates, a mistake; Giankoplis gives the

derivation of the degenerate cases from the matrix equation.

I do not have a chart and may have to program in Mathcad. That's pretty

easy using the transformation equations to Cartesian coordinates tracing

along each edge of the coordinate system specific unit volume (not the

differential volume, but a substantial "chunk" of spaces, near the

origin, in each system). I think I can articulate some of those limits

here today before trying it:

For each coordinate axis with range from 0 to oo, apply limits of 1/2 to 1.

For each coordinate axis with range from 0 to 2pi, apply limits of pi/2 to 3pi/2.

For each coordinate axis with range from 0 to pi, apply limits of pi/4 to 3pi/4.

There are others, however, and there are inequalities.

Ideally the volume of each "chunk" would be 8 since the obvious chunk of

Cartesian space is:

x=[-1,1]; y=[=1,1]; z=[-1,1].

I am open to suggestions today and have not started programming yet.

Cheers,

Douglas Goncz

Replikon Research FCN 7837774974

On Sunday, June 6, 2021 at 10:22:08 AM UTC-4, Douglas Dana Edward^2 Parker-Goncz (fully) wrote (I wrote):

...

>

> I am interesting in seeing and counting (enumerating) the most basic

> unit volumes and their associated surfaces for inclusion in a computer

> software library of "atomic" features from which "everything" (to second

> order) may be designed, in a attempt to provide a reasonable and

> nontrivial basis for para-universal constructors.

>

...
> I am interesting in seeing and counting (enumerating) the most basic

> unit volumes and their associated surfaces for inclusion in a computer

> software library of "atomic" features from which "everything" (to second

> order) may be designed, in a attempt to provide a reasonable and

> nontrivial basis for para-universal constructors.

>

Nov 28, 2021, 3:20:30 PM11/28/21

to

Well aside from 8 where 1 is better, it's what I meant.

On Saturday, September 25, 2021 at 3:12:20 AM UTC-4, Douglas Dana Edward^2 Parker-Goncz (fully) wrote:

> The Change of Variable Theorem is of interest as it generalizes the

> transformations between coordinate systems.

>

On Saturday, September 25, 2021 at 3:12:20 AM UTC-4, Douglas Dana Edward^2 Parker-Goncz (fully) wrote:

> The Change of Variable Theorem is of interest as it generalizes the

> transformations between coordinate systems.

>

> There are 16 orthogonal systems listed at Math World under Orthogonal

> Coordinates and it is asserted on that page they are all degenerate

> cases of elliptical coordinates, a mistake; Giankoplis gives the

> derivation of the degenerate cases from the matrix equation.

>

> Coordinates and it is asserted on that page they are all degenerate

> cases of elliptical coordinates, a mistake; Giankoplis gives the

> derivation of the degenerate cases from the matrix equation.

>

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