tets, "octets" and ivm - A Math Reference

[Eric Marceau]

The discussion of ivm vs xyz seemed to be going nowhere.

On google, I did an independent search for

"ivm coordinates" AND "cartesian coordinates" AND "converter"

and while the responses were almost nil, there appears to be a useful result:

1803.0263v1.pdf  (on vixra.org)

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwiX5La-0t-AAxX1jYkEHTetAGoQFnoECA4QAQ&url=https%3A%2F%2Fvixra.org%2Fpdf%2F1803.0263v1.pdf&usg=AOvVaw1bWPyitDWGnyCo75idvOIt&opi=89978449

That provides a 275 page PDF of a book entitled

QSO -- The Mathematics of Quasi-Spherical Orbits   by Robert G. Chester

The discussion of IVM starts on page 18 (PDF 31).

Others might be more apt to understand.  It seems a bit much for me.  But then again, I have no interest for other than spherical or cartesian.

[Eric Marceau]

There also appears to be a more generalized discussion of the QSO book's topic at the following link:

http://www.rwgrayprojects.com/synergetics/s10/toc10.html

where it is titled

OMNITOPOLOGY

For those who are more math/theory(/philosophy?) inclined.

FTR, it is not my cup of tea!  I am not in a position to discuss either this reference (omnitopology), nor the previous reference (qso).

Eric

[Eric Marceau]

One key image in the below-referenced QSO book might help explain visually the relationship between the two systems:

That is included in the attached "photocopy" of the pages where it is referenced.

But none of that clearly identifies any sense of "coordinates" within the IVM frame of reference.

Some references point to details in

"Synergetics - Explorations in the Geometry of Thinking", 1975
https://archive.org/details/buckminster-fuller-synergetics-explorations-in-the-geometry-of-thinking/mode/1up

and

"Synergetics 2 - Explorations in the Geometry of Thinking", 1979
https://archive.org/details/synergetics2expl00full/page/n7/mode/1up

Looking at the first of those two Synergetics books, I am unable to discern (possibly my failing) any outline of a mathematical methodology for the "mapping" (aka transformation matrix or whatever) from Cartesian to IVM.

From scanning all references of IVM, I only found this one short section that hinted at what was involved:

962.00 Powering in the Synergetics Coordinate System
962.01 In the operational conventions of the XYZ-c.g t .s. coordinate system of
mathematics, physics, and chemistry, exponential powering meant the
development of dimensions that require the introduction of successively new
perpendiculars to planes not yet acquired by the system.
962.02 In synergetics, powering means only the frequency modulation of the
system; i.e., subdivision of the system. In synergetics, we have only two
directions: radial and circumferential.
962.03 In the XYZ system, three planes interact at 90 degrees (three
dimensions). In synergetics, four planes interact at 60 degrees (four dimensions).
962.04 In synergetics there are four axial systems: ABCD. There is a maximum
set of four planes nonparallel to one another but omnisymmetrically mutually
intercepting. These are the four sets of the unique planes always comprising the
isotropic vector matrix. The four planes of the tetrahedron can never be parallel to
one another. The synergetics ABCD-four-dimensional and the conventional XYZ-
three-dimensional systems are symmetrically intercoordinate. XYZ coordinate
systems cannot rationally accommodate and directly articulate angular
acceleration; and they can only awkwardly, rectilinearly articulate linear
acceleration events.
962.05 Synergetic geometry discloses the rational fourth- and fifth-powering
modelability of nature's coordinate transformings as referenced to the 60-degree
equiangular isotropic-vector equilibrium.962.06 XYZ volumetric
coordination requires three times more volume to
accommodate its dimensional results than does the 60-degree coordination
calculating; therefore, XYZ 90-degree coordination cannot accommodate the
fourth and fifth powers in its experimental demonstrability, i.e., modelability.
962.07 In the coordinate vectorial topology of synergetics, exponential powers
and physical model dimensioning are identified with the number of vectors that
may intercept the system at a constant angle, while avoiding parallelism or
congruence with any other of the uniquely convergent vectors of the system.

That still doesn't illuminate the process of mapping

A Suggestion for a Method

If I go back to my early engineering days, I would suggest using a FEM (Finite-Element-Method) meshing engine that could generate a mesh where all elements are forced to be tetrahedra of uniform size (I think that is an option nowadays), with the exception of those on the boundary surfaces.  Then, from the mesh-generator, obtain a report of the count of all the "full" tetrahedra and a count of the "partial" tetrahedra (likely the count of those forming the vertical side-walls of the pool).   I would estimate the total volume to be

    count (full) + 1/2 * count (partial) = total

If the tetrahedra are sized with dimensions providing for a uniform 1 litre, then you would get a direct count of the volume required to fill the pool.

That would be my "algorithm", but I can't begin to ideate the method used for IVM.  It just doesn't make sense to me.

An Alternate Method using "2-Layer" Hex-Tiles

If you set one layer of 6 tetrahedrons, with apex up, arranged to form a hexagonal profile, then arrange a similar 6 tiles with the apex facing down, this dual-layer of tets forms the equivalent of 6 tiles (trapezoid-like) whose volume is the projection of the hexagonal profile projected straight to the plan of the opposing layer's flat face.

You could then attempt to calculate the volume by overlaying the map of best-fit hexagons, then determine the number of layers with thickness equal to the tetrahedron apex height, then multiply the two numbers to obtain your total water requirement. 

Tetrahedron volume = ( 20.3964890265551 cm) cubed / 6 * sqrt(2) = 1000 CCs = 1 litre

So number of hexagonal "tiles" * 6 = number of litres for the pool.

Hope the above helps in some way.

Eric

On 2023-08-15 19:42, Eric Marceau wrote:

[Curt McNamara]

Thanks Eric! 

One detail: tetrahedral mesh are used in software that models physical systems and electromagnetic waves, however they use irregular ones.

Numerical analysis

An irregular volume in space can be approximated by an irregular triangulated surface, and irregular tetrahedral volume elements.

In numerical analysis, complicated three-dimensional shapes are commonly broken down into, or approximated by, a polygonal mesh of irregular tetrahedra in the process of setting up the equations for finite element analysis especially in the numerical solution of partial differential equations. These methods have wide applications in practical applications in computational fluid dynamics, aerodynamics, electromagnetic fields, civil engineering, chemical engineering, naval architecture and engineering, and related fields.

https://en.m.wikipedia.org/wiki/Tetrahedron

      Curt

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[Eric Marceau]

Sorry, I forgot the PDFs.

Eric

On 2023-08-15 19:42, Eric Marceau wrote:

[Eric Marceau]

Thank you, Curt.

So ... is there no FEM tool that allows you to specify a starting point/surface for the first tetrahedron, then append/replicate in all directions, maintaining uniform ideal tetrahedral shape until reaching boundaries?  That's unfortunate.

I can see the "traditional" approach would be the preferred mode for actual stress analysis (or thermal expansion analysis), but for volumetric analysis, it seems excessively complicated, not lending itself to simplification by applying various rules for symmetrical shapes.  I had thought that the science/math would have evolved to encompass an approach using uniform tetrahedra (predominantly) for "special" cases to reduce the compute load and time.

A quick search on google does offer a number of such references but, to be honest, this side of engineering is a blind-spot for me, and I don't have the time or skills to push any further.

Eric

retired
https://stackoverflow.com/users/9716110/eric-marceau?tab=profile

On 2023-08-15 23:24, Curt McNamara wrote:

Thanks Eric! 

One detail: tetrahedral mesh are used in software that models physical systems and electromagnetic waves, however they use irregular ones.

Numerical analysis

An irregular volume in space can be approximated by an irregular triangulated surface, and irregular tetrahedral volume elements.

In numerical analysis, complicated three-dimensional shapes are commonly broken down into, or approximated by, a polygonal mesh of irregular tetrahedra in the process of setting up the equations for finite element analysis especially in the numerical solution of partial differential equations. These methods have wide applications in practical applications in computational fluid dynamics, aerodynamics, electromagnetic fields, civil engineering, chemical engineering, naval architecture and engineering, and related fields.

https://en.m.wikipedia.org/wiki/Tetrahedron

      Curt

On Tue, Aug 15, 2023, 10:14 PM Eric Marceau <eajma...@gmail.com> wrote:

...(snip)...

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[Eric Marceau]

If I can make a comment without being "blasted" ...

Given the "6-vector" description method for identifying any single point (see image and text in earlier posting),
I find the IVM approach far too impractical to even want to pursue trying to understand it.

I understand that tetrahedra are a "rigid" structure which cannot be deformed (under reasonable circumstances), and that it may be a structure that is frequently observed in the natural world, but to go from that to then attempt to define a referential-coordinate system based on a "four-plane" (non-orthogonal) dimensional conception of the 3-plane orthogonal space we live in, it seems to this "simplistic" human completely outlandish and, honestly, not worth the effort to try to understand it, especially since the most vocal of proponents, Buckminster Fuller, did not as far as I could tell, even in his published works, clearly spell out the mathematical relationship between the two spatial reference systems.

NO sarcasm intended, with all due respect to Dick Fischbeck, for myself, that makes evident that IVM may be a useful paradigm for conceptual/visual construction of inter-connected components, but has no demonstrable rigorous mathematics that can lend itself to being learned or understood, let alone applied via reproducible methodology which is defensible.  That became quite clear by the circuitous discussion in this group trying to get specifics about IVM.

Maybe I am completely wrong, but I couldn't find the proof to the contrary.

I invite others to do their own research to either confirm or disprove my conclusions on this matter.

Eric

[Chris Belcher]

Something I’ve been thinking about comparing the Cartesian and Synergetic Methods is that tetrahedrons are not all space filling, hence the IVM alternating tetrahedrons and octahedrons which seems impractical when constructing a basic grid describing all space.

This shows a basic difference in the two approaches. The Cartesian Method is the product of the Western metaphysical attempt (from Leibniz and perhaps before that the Greco-Roman wortd view) to re-present (and subjugate) the totality of the natural world. Besides not bearing much in common with Eintseinian space-time, where any parallel lines are inevitably warped by gravitation and space, I guess it works well for calculating volumes of hydrocarbons to be extracted, sold and consumed.

The Synergetic Method starts with tuning in the relevant elements to be considered:

963.13   Synergetics is a priori nuclear; it begins at the center, the center of the always centrally observing observer. The centrally observing observer asks progressively, "What goes on around here?"

Vertexes are the relevant events, inter-related to form a system of consideration, tuning out the macro and micro background. Moving around uses polar coordinates as opposed to grid addresses.

The pdf addresses the change of orientation but doesn’t appear to to account for Synergetics being based upon diagonal of the square as the unit of measure, instead of the edge used in the metric (Cartesian) system.

Kirby Urner has a comparison of the two methods:

https://github.com/4dsolutions/tetravolumes/blob/master/Computing%20Volumes.ipynb

If you want to convert back to cubic terms, Fuller does provide a formula, the synergetics constant, 1.06066:

982.30   Diagonal of Cube as Control Length: We have learned elsewhere that the sum of the second powers of the two edges of a right triangle equals the second power of the right triangle's hypotenuse; and since the hypotenuse of the two similar equiedged right triangles formed on the square face of the cube by the sphere-center-connecting diagonal has a value of two, its second power is four; therefore, half of that four is the second power of each of the equi-edges of the right triangle of the cube's diagonaled face: half of four is two.

982.31   The square root of 2 = 1.414214, ergo, the length of each of the cube's edges is 1.414214. The sqrt(2)happens to be one of those extraordinary relationships of Universe discovered by mathematics. The relationship is: the number one is to the second root of two as the second root of two is to two: 1:sqrt(2) = sqrt(2):2, which, solved, reads out as 1 : 1.414214 = 1.414214 : 2.

982.32   The cube formed by a uniform width, breadth, and height of sqrt(2) is sqrt(23), which = 2.828428. Therefore, the cube occurring in nature with the isotropic vector matrix, when conventionally calculated, has a volume of 2.828428.

982.45   Humanity's conventional mensuration cube with a volume of one turns out in energetic reality to have a conventionally calculated volume of 2.828428, but this same cube in the relative-energy volume hierarchy of synergetics has a volume of 3.

3 --------- 2.828428

= 1.06066

982.46   To correct 2.828428 to read 3, we multiply 2.828428 by the synergetics conversion constant 1.06066. (See Chart 963.10.)

I’m not sure I understand where this constant comes from, besides the floating decimal point.

I found it interesting that even Antiprism’s native OFF format maps polyhedra based upon the Cartesian coordinates. I wonder if it would be possible to create a toolbench for FreeCad or something similar that generates polyhedra based upon the IVM.

To Chris Kittrick:

Thank you for identifying the Las Vegas structure as a lamella. I see there are some beautiful examples, though the Vegas Sphere isn’t really one of them. I guess it does do its job.

And to Dick Fischbeck, thanks, the Wanger Flange is interesting but I am going in a different direction.

Cheers,

-Chris

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[Adrian Rossiter]

Hi Chris

On Wed, 16 Aug 2023, Chris Belcher wrote:
> I found it interesting that even Antiprism’s native OFF format maps
> polyhedra based upon the Cartesian coordinates. I wonder if it would be

OFF format is a simple text format that can be written by hand and
employs a coordinate system that many people learn at school.

One of Antiprism's goals is to provide components that support the
modelling that people do. If you start modelling with your own
coordinate system then let me know what format you are using and how
you would like to use Antiprism to process your models. A converter
from your format to OFF would likely be very easy to write, and
it would also likely be trivial to modify a program that outputs your
format to simply output OFF as an alternative.

Similarly, internally, Antiprism uses a number of coordinate systems,
as different coordinate systems are useful for solving different
problems.

Regarding polyhedra, the IVM vertices can have integer Cartesian
coordinates, and many polyhedra which can be created in the IVM can
be created in Antiprism with these integer coordinates by prefixing
the model name by 'std_', for example

off_util tetrahedron | off2crds

0.3535533905932737 0.3535533905932737 0.3535533905932737
0.3535533905932737 -0.3535533905932737 -0.3535533905932737
-0.3535533905932737 0.3535533905932737 -0.3535533905932737
-0.3535533905932737 -0.3535533905932737 0.3535533905932737

off_util std_tetrahedron | off2crds

1 1 1
1 -1 -1
-1 1 -1
-1 -1 1

Adrian.
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adr...@antiprism.com
http://antiprism.com/adrian

[lemondealc]

Has anyone asked ChatGPT to clarify the ideas?

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[Chris Belcher]

Thanks Adrian,

Every time you post a reply here it opens up new dimensions of your program.

Reporting out the data on the default tetrahedron model:

[general]

num_verts = 4

num_faces = 4

num_edges = 6

vertex_centroid = (0 0 0)

volume_centroid = (0 0 0)

oriented = yes

orientable = yes

connectivity = polyhedron, closed, even, known

num_parts = 1

convex = yes (strict)

genus = 0

area = 1.732050807568877

volume = 0.11785113019775789

...

[edges]
num_edges = 6
perimeter = 5.9999999999999991
edge_length_max = 0.99999999999999989 (0,1)
edge_length_min = 0.99999999999999989 (0,1)
edge_length_avg = 0.99999999999999989

The edge lengths are 1 (just about) and I am guessing that the area and volume are calculated as square and cubic units? I did see that the polyhedral models do align with IVM values in most ways, especially in figuring angles, frequency and strut lengths.

Thanks!

-Chris

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[Adrian Rossiter]

Hi Chris

On Fri, 18 Aug 2023, Chris Belcher wrote:
> Reporting out the data on the default tetrahedron model:

...

> area = 1.732050807568877
> volume = 0.11785113019775789
...

> edge_length_avg = 0.99999999999999989
...

> The edge lengths are 1 (just about) and I am guessing that the area and
> volume are calculated as square and cubic units? I did see that the
> polyhedral models do align with IVM values in most ways, especially in
> figuring angles, frequency and strut lengths.

The area and volume are reported by off_report in square units and cubic
units.

The bare "tetrahedron" model has edge length 1, but is nicely algined with
the Cartesian coordinate axes.

However, if you switch to the std_ models then they are sized and aligned
in a way that makes thm easy to use for making models that appear in
Synergetics, e.g.

off_util std_tet std_cube std_oct std_cubo std_rd | off_color -e K | antiview -x f

[image attached]

[Chris Belcher]

Hi Adrian,

Very interesting to see the nested polyhedra, thanks. I found it easier to not use the std_ prefix to generate the intersecting flipped tetrahedrons defining the edges of a cube.

Here is the readout from off_report:

[edges]
num_edges = 24
perimeter = 40.970567999999993
edge_length_max = 1.9999999999999996 (8,9)
edge_length_min = 1.4142140000000001 (0,1)
edge_length_avg = 1.7071069999999997

As you can see the tetrahedral edges are 2, and the cube edges are 1.414214 as defined in Synergetics. It would be nice to also see an exploded drawing with the 1/8th (if I remember correctly) octahedra  pieces - something to work on.

I guess the next step would be to lay out side by side the cubic and tetrahedral volume formulae (?) to see how they work and how the synergetics conversion constant comes in to play. 

-Chris

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[Adrian Rossiter]

Hi Chris

On Tue, 22 Aug 2023, Chris Belcher wrote:
> As you can see the tetrahedral edges are 2, and the cube edges are 1.414214 as
> defined in Synergetics. It would be nice to also see an exploded drawing
> with the 1/8th (if I remember correctly) octahedra pieces - something to
> work on.
>
> I guess the next step would be to lay out side by side the cubic and
> tetrahedral volume formulae (?) to see how they work and how the
> synergetics conversion constant comes in to play.

I believe the key is that there is also a conversion of length units:

982.45 Humanity's conventional mensuration cube with a volume

of one turns out in energetic reality to have a conventionally
calculated volume of 2.828428, but this same cube in the
relative-energy volume hierarchy of synergetics has a volume of 3.

I.e. the cube edges in your model, that you have calulated as length
root 2, are length 1 in "Humanity's conventional mensuration" system.

[Chris Belcher]

Hi Adrian,

Yes, I am working from that section. On my model the tet edges (cube diagonals) are 2, while the cube edges are the square root of 2.

Incidentally I had an issue with running off_report on the combined model. When I tried to generate a text file it was empty. I ran it again without specifying output and it printed the report in the terminal window, so just copied and pasted that into the blank text file.

Next work on volume formulae.

Cheers,
-Chris

Sent from my iPhone

> On Aug 22, 2023, at 4:39 AM, Adrian Rossiter <adr...@antiprism.com> wrote:
>
> Hi Chris

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