For what it's worth - An attempt at applying Hector's Method using Prolog

[Eric Marceau]

I've been trying to build my own C-language program that would replicate the Mexican Method.  I have tried to understand the various posts and spreadsheets that have been shared, and still, 3 years later, I have yet to understand enough to re-state the method using statements of method, incorporating equations and formulas that reflect the iterative process involved.

So ... this week, I've decided to "engage" Google AI to assist.

I've also decided that I will make one final attempt at learning Prolog.

Given those two decisions, here is what was offered to me, but it has not produced the degree of optimization that has been reported and posted on the Geodesic Help Group mailing list.

If someone else is more adept at Prolog, and can get the logic right, I think it would help the group by offering a tool that can be extrapolated to higher frequencies while still achieving the optimization desired.

From the study of the various coloured images posted to the Group, it is clear that optimization must be performed  using uniform/symmetrical adjustments over various "triangulated" vertex patterns.  If such patterns could be identified in advance, and the method for defining the pattern subsets that are to be simultaneously adjusted, the result could potentially offer the long-desired tool for generalization.

With that preamble, I offer the GNU Prolog source and the session log for the run.

Comments and feedback are very much welcome, but do keep in mind that I am very much a novice at Prolog.

Also note that the result is not resolving with the desired solution of strut count equal to frequency.

NOTE: The attached are released into the public domain, without restrictions of any kind.

Eric Marceau, Ottawa

[Bryan L]

Hello Eric,

Some years ago I also struggled to find a method to solve for struts in a Mexican Method dome.

It was Hector himself (the man who envisioned a dome with strut count equal to frequency) who put me onto a mathematical method to solve for a given frequency.

He showed me how to use Newton-Raphson to solve a set of equations.

I have some spreadsheets buried somewhere - I'll clean one up and post it later today...

Bryan

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[Ashok Mathur]

Instead of struggling with learning Prolog and not knowing if your optimisation failed due to the programming language or the optimisation algorithm , try a simpler approach. Ask Gemini to do it excel using optimise result changing values in one cell .

Gerry is a master if that approach though I also have used.

Regards

Ashok

Regards

Ashok

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[Bryan L]

Ashok,

except that after V4, there is more than one variable and the problem needs to be correctly defined with each equation in terms of relevant variables.

To view this discussion visit https://groups.google.com/d/msgid/geodesichelp/CAD-a3OgDbauH2xZd2EsRtBHvXrWfT2Aar1R5W%3DppmJJLc5j-kg%40mail.gmail.com.

[Bryan L]

Eric,

are you familiar with spherical trig and Newtons method for solving a system of equations?

[Eric Marceau]

I don't like spherical trig, but I am familiar with it and use it.

The only experience I have with "linear programming" is the solving of power 1 variables with coefficients solved simultaneously in the matrix form.  I don't know how to "level up" to handling of power 2 variables using that method.  Looking up the reference, there is mention of Jacobians, and I have no experience with those ... but if I find a good, clear reference with explicitly worked examples, I can definitely assimilate and apply the technique, seeing as it should all be just a matter of "math mechanics". 🙂

As a practicing engineer, I have never had occasion to use any of the "higher" math I learned for stress analysis and boundary value analysis, so, unfortunately, those memory cells are buried deep where I no longer have ready access to any indexes. 😳

The context for using macro-driven solvers in spreadsheets just never presented themselves for me where I would need to use those, so no experience.  Again, not an issue if I can get a good reference.  But for me, it would need to be worked in LibreOffice Calc.

As I said above, I have no issue diving in and learning, if necessary, but before I try to learn a tool, I want to learn the related algorithm and math, and would prefer to choose my own tool, which until this last posting based on Prolog, has always been C.

My latest attempt using C is attached, but I have not yet incorporated an algorithm for the optimization.  The full set of files, along with the requisite Geany project file, are attached, all having the file prefix string

• Spherical_Longitudinal

The intended next phase of this program's iteration would be to create two other complete sets of points by rotating the "array" of points around the centroid line thru the Origin, then create a start point for optimization by "averaging out" the non-edge "triple proximate points".

That was to be my next step, before I decided to take a stab with getting Google's "Artificial Dunce" to evolve a Prolog program based on all available information.  You saw where that got me, namely, not far! :-( 

At this point, I plan to re-focus on my C-program version.

   _____________________________________________________

Regarding the Geogebra files, a tool for handling of the C-program output to create Geogebra files is also offered in the tar file:

• CustomGeogebraModel.tar.gz

   _____________________________________________________

A parallel, but not fully related/integrated effort, is my attempt at creating an C-language API for 2D-/3D-matrix operations which avoid any pointer-related actions.  That is also attached as a compressed tar file:

• MatrixFunctionLibrary.tar.gz

   _____________________________________________________

Eric Marceau

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[Bryan L]

I don't like spherical trig, but I am familiar with it and use it.

That's good because it simplifies things a lot and everything I have done utilises it, 

The only experience I have with "linear programming" is the solving of power 1 variables with coefficients solved simultaneously in the matrix form.  I don't know how to "level up" to handling of power 2 variables using that method.  Looking up the reference, there is mention of Jacobians, and I have no experience with those ... but if I find a good, clear reference with explicitly worked examples, I can definitely assimilate and apply the technique, seeing as it should all be just a matter of "math mechanics". 🙂

My mathematics is at best at a backyard level. I did do maths to 2nd year at Uni but a lot has gone...

Just for starters, the Jacobian is just a matrix of derivatives of the equation's variables. Multiplied with the matrix of equation's coefficients gives a resultant that is (hopefully) a closer approximation to the solution. Rinse and repeat until the solution is within the desired tolerance.

 

I haven't followed what you have been doing but hopefully in the next day or two I will get you an example or two of what I have done previously. I need to clean it up and make it more easy to follow as my spreadsheets are not documented and contain some problem specific stuff that makes them appear obfuscated.

Below you mention creating a matrix library in C to avoid pointers. Do you struggle with pointers? I am sure there is a multitude of C libraries for matrix stuff but I get that the language can be unforgiving at times when trying to learn it. I stuck with Pascal because it's a lot more typesafe and a touch more verbose than C and my mind reads it a lot more easily. Perhaps Python could be a good alternative?

To view this discussion visit https://groups.google.com/d/msgid/geodesichelp/586f2ff9-82e3-4b26-a1ad-00a8066efc2b%40gmail.com.

[Ashok Mathur]

Linear programming handles only variables with single power ie it has no mechanism to handle higher level variables. 

Regards

Ashok

Regards

Ashok

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[Eric Marceau]

Thank you for replying again, Bryan.  Much appreciated!

I need to clean it up and make it more easy to follow as my spreadsheets are not documented and contain some problem specific stuff that makes them appear obfuscated.

That would be very much appreciated, and I am sure not just by myself!  🙂

Do you struggle with pointers?

I prefer all operations in a program to use a single approach to value assignment, by simple reference, and I do not appreciate the need to change "paradigms" depending on the function call.  Keeping those straight, for me, is a pain best avoided to avoid pulling out my hair when trying to debug programs.  😳

I stuck with Pascal ...

On that, I agree.  Never understood the attraction to C when Pascal was out there.  Ada kind of introduced a blurring and dilution of focus, for what is essentially the same language, in my mind, creating a whole other set of issues which turned potential users away, but maybe Ada has evolved to the point that it could now supplant Pascal "with enthusiasm". 😕  If you are a regular user of Pascal, you would be in a better position to offer an opinion on that.  I haven't touched Pascal in almost 30 years.  What little I did with Pascal modules with UCSD Pascal seemed to suggest Ada being a natural successor to Pascal.

Perhaps Python could ...

I've done some minimal, but "complicated" work with Python, and what made it "a bridge too far" is the lack of clear delimiting characters for logical code segments.  For me indentation has always been about making it clear to myself, as a programmer, things I wanted to stand out.  I can't do that in Python without it being blurred by language requirements imposed by dictated indentation rules.  I am forced to submit to those for the little I do with FreeCAD, but my full on-boarding with FreeCAD is severely hamstrung by this indentation "thing"!  Every couple of months, I try to go back to it, but I haven't been able to get my brain paths to "become one with Python" to make it my chosen vehicle for CAD work in general, let alone Geodesics, which is why I was getting back into CAD, 20 years after leaving that world but now in retirement, to see what I could possibly contribute under the OSS paradigm that Linux has offered to us all. 🙂 That was why I had previously looked at the Frame3DD, to see how much I could do with that.  I shared all I did with the Group back then.  If I can't make some headway with Hector's Method, I will regretfully be forced to re-direct my mental energies and limited skills somewhere that I would find less satisfying.

As a complementary aside, regarding some of my prior attempts, I have implemented elements that could eventually be incorporated into a fully functional, potentially interactive program not requiring a professional CAD system to both generate and visualize basic geometry:

• 3D display using OpenGL/glad libraries;
• localization for language sets (prototype with Spanish and English);
• exporting Points to CSV;
• exporting Points alone for Geogebra import;
• exporting Struts alone for Geogebra import;
• tool for combining export XML files into a valid Geogebra model file

My last email shared what I did in regards to the bottom four.  The prototypes for the first two have not been "segregated" the way I did for the Matrix operations functions.  If there is enough interest, I can take some time to do that and share here as well.

Bryan, I very much look forward to seeing your cleaned up procedural document.

Thank you again.

Eric Marceau

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[Eric Marceau]

Thank you, Ashok.  I didn't think it did, but I wasn't sure.  🙂

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[Bryan L]

Hi Eric,

Finally I have something that I hope explains the method and can be followed and applied to larger problems..

I chose the simplest problem (a V5) with just two equations in two unknowns.

Feel free to ask if something is not apparent.

I will move onto a V6 version. V6 and higher have more equations than variables but if formulated properly can still be solved with only small (~ 10^-5) discrepancies in some chords. It's also possible to use variable R of one or more vertices for square matrices with a similarly small deviation from unity.

Bryan

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[Bryan L]

On Thu, 8 Oct 2026 at 00:49, Ashok Mathur <ashokch...@gmail.com> wrote:

Linear programming handles only variables with single power ie it has no mechanism to handle higher level variables. 

 

We know that Ashok. 

Here we are dealing with Spherical Trig equations and the method to solve a set of non linear equations is provided.

Bryan

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