checking

[Dick Fischbeck]

I'm checking what you get when you search  n=10f^2+2.

Anything?

[RC]

It's the formula for close packing of equal spheres in a cuboctahedron.  This was mentioned in a previous post.

More information can be found on the following website:  https://grunch.net/archives/56

[Dick Fischbeck]

Yes, and also, perhaps more importantly, the icosahedron.

My question pretty much proves no one is think of a spherical geometry!

[Dick Fischbeck]

Also, since a sphere is a cluster of vertexes approximately equidistant from a center, we can analyze any sphere by its number of vertexes, or spheres,

its frequency (radius), its area and its volume, not just for 12, 42, 92, etc.

[RC]

True.  A sphere is smooth and has no vertices.  A convex polyhedron with near infinite vertices is very similar to a sphere and can be defined by existing volume and surface area formulas that define a true sphere.  Most geodesic domes we are interested in here will have edges of varying lengths and triangular face surfaces of different areas.  Other than Euler's formula for the relationship of vertices, edges, and faces, I don't know of any universal formulas to calculate surface area and volume based solely on number of random vertices equidistant from a central point.  Is this the type of solution you are seeking?  Please give us a more in-depth, multi-paragraph explanation of what you are after.  Illustrations would help.  Thanks.

[Dick Fischbeck]

http://www.rwgrayprojects.com/SynergeticsDictionary/SDCards.php?cn=16597&tp=1

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[Dick Fischbeck]

But that's not real.

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[RC]

What is not real?  Please explain more in depth so I might know what you are getting at.

[Dick Fischbeck]

I have next week off, winter break. I will attempt to clarify.

Have you watched this short video by Ed Applewwhite? About bypassing xyz?

https://www.youtube.com/watch?v=eVb9uoUdFo0&t=1471s

On Thu, Feb 16, 2023 at 7:18 PM RC <clark.rob...@gmail.com> wrote:

--

[Dick Fischbeck]

It is impossible to have a continuous surface.

To view this discussion on the web visit https://groups.google.com/d/msgid/geodesichelp/15786aa2-6837-4ce6-bedd-46ad1df605c5n%40googlegroups.com.

[Dick Fischbeck]

Thinking of a sphere  discreetly, Is where I am.

On Thu, Feb 16, 2023 at 7:30 PM RC <clark.rob...@gmail.com> wrote:

What s not real?  Please explain more in depth so I might know what you are getting at.

[RC]

I see I'm getting sucked into this.

So, Fuller had a preference for a spherical coordinate system rather than an xyz cartesian system just as astronomers and astrophysicists do.  And, that makes sense when we are talking about spherical bodies and spacecraft moving and interacting in space.  On Earth, we use the xyz system for designing and building and we for all practical matters ignore the curvature of the Earth's surface.  We live and function in a physical, cartesian environment.  Architectural plans, machine part fabrication drawings and geodesic dome plans all are based in Cartesian coordinates.

If you are going to deal with things at an atomic level, then yes, it's true there are no continuous surfaces.  But, you seem to be switching back and forth between theoretical and real world.  You're tying together the mathematical definition of a sphere with the properties of materials at an atomic level.  They have nothing to do with each other.

It's like saying because at an atomic level, we are mostly space, then nothing really exists.

[Dick Fischbeck]

Wait. Are there continuous surfaces? Can we begin there?

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[Dick Fischbeck]

He had a preference for the world as it is. His spherical coordinates are not the same as the xyz spherical coordinates.

On Thu, Feb 16, 2023 at 8:03 PM RC <clark.rob...@gmail.com> wrote:

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[RC]

I'm going to wait till after your winter break when you can give us a a more lengthy explanation.  I wish you luck on whatever conjecture it is you are attempting to prove.

[Dick Fischbeck]

Whether I have a conjecture I can articulate or not, the question remains, are there continuous anythings in nature? I'm not sure I can explain anything without an answer about that. 

[RC]

Please assume the answer is "no".

[Dick Fischbeck]

Then, if there are no continuums, we have common ground and can discuss stuff. Looking forward to exploring this. 

[Dx G]

I will also look forward to hearing more.   In the meantime, if you haven't already considered items like a Mobius Strip, you might save that for when you really think you've got the issue nailed...and maybe, think again.   Somebody once told me that Einstein said, if you twist something in space, it loses one dimension, and showed a Mobius Strip to illustrate.  Don't know if any of that is true, but if you want to contemplate the cosmos, that's a good one to keep on the list.

https://brilliant.org/wiki/mobius-strips/

DxG

[Dx G]

Additional note - on the fusion of theoretical with "what's real", you might enjoy this example (Mobius Strip);

https://www.dezeen.com/2020/08/03/chengdu-infinity-loop-wuchazi-bridge/

DxG

[Dick Fischbeck]

A mobius strip is a torus, no?

[Dx G]

I don't believe so.  If you look at the definition of a torus, and compare that to what you will see on the Mobius web site, I believe you will see the differences.

DxG

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[Dick Fischbeck]

Paper has thickness. That's why a Moebius strip has to be considered as a torus.

831.00  Sheet of Paper as a System

 If it is a rectilinear sheet of typewriter paper, we recognize that it has four minor faces and two major faces. The major faces we call "this side" and "the other side," but we must go operationally further in our consideration of what the "piece of paper" is. Looking at its edges with a magnifying glass, we find that those surfaces round over rather brokenly, like the shoulders of a hillside leading to a plateau. - RBF

Also, https://demonstrations.wolfram.com/MoebiusStripAsAHalfTwistedSquareTorus/

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[Dick Fischbeck]

Attachment from Cosmography

[RC]

A moebius has a cross section of a line.  A twisted torus is not a moebius.  Unless of course you're going to play word games and describe the cross section of a moebius having an infinitely short length and the cross section of a torus being an infinitely small circle till the two cross sections are just identical points.  Isn't it fun to split hairs over pseudo-intellectual triviality.  Buckminster Fuller was a master at this.

[Dx G]

Its always interesting to explore concepts that, at first, seem like obscurities, as at times they lead to useful new understanding.  However, at this point, its not obvious to me why the distinction between a torus and a mobius strip is important.    I do find it interesting that a mobius, once twisted, goes from being a two surface to a "single surface" object.  Its even more interesting when one begins cutting the strip in half, and then cutting those strips in half.  

  So if addressing the question has useful or important ramifications, those of us with more limited vision on the issue would enjoy hearing about that.

DxG

[Dick Fischbeck]

I'm trying to find common ground. If you don't think paper has thickness, I don't know how to move on. I like topology. I pretty much think Fuller was right on most of the time.

To view this discussion on the web visit https://groups.google.com/d/msgid/geodesichelp/c88fe9c5-5bb9-4131-86fc-9e29b06b2d25n%40googlegroups.com.

[RC]

Why do you need agreement on every breadcrumb you throw out here?  Why not just say what's on your mind in complete clear detail and explain to everyone it's significance?  You obviously have something greater in mind.  I feel as though your playing a game of "can you guess what I've discovered?".  If you have a point, put it out there with full explanation.  Back it up with some reasoning we can try to follow.  Please.

[jhausman]

If you're not happy with only 2 sides, start chaining tetrahedra together into a ring and you'll see that there are many ways for a mobius strip to live on a torus. 
In this album, click through the photos until you see colored straws. 
https://www.facebook.com/media/set/?set=a.230145187078823&type=3 

[Dick Fischbeck]

I mean, how can we talk without a basic agreement on geometry.

[Dick Fischbeck]

I'm saying the line that represents the Mobius strip has 3 dimensions and has an inside and an outside.

[RC]

What?  You're saying the line that forms the mobius has 3 dimensions because you've decided it's going to be made of paper now, and it's no longer going to be just a mathematical construct. You keep switching back and forth from theoretical to real world.  The mobius by definition is a line rotated around a center with a half twist.  If you wish to create a representation of it in the real world then you might create it from a strip of paper with thickness.  You can't point at a crude physical model like that and conclude that a mobius is really a torus just because their is a thickness to the paper. I can hardly wait to hear your take on the Klein bottle.

You might need to lay off the Buckminster indoctrination.  The guy was a promoter, a showman, a narcissist with a need to impress through pseudo-intellectual babble word salad.  He claimed to have invented the octet truss - the same one invented by Alexander Graham Bell 50 years earlier.  He claimed to have invented the geodesic dome - same design built by  Dr. Walther Bauersfeld in Germany in 1927 years before him.  He claimed to have invented tensegrity structures - it was actually his student Kenneth Snelson.

He like to give lectures that no one could really quite follow or understand.  So, he must have been brilliant, right?  For example, a quote of his from a lecture:

"The entire regenerative hierarchy of major, intermediate, and minor constellations of component patterns-within-component-patterns of universe are continual processes of synchronous, yet independent and unique, transformative patternings. That is, all components of universe are in continually accommodative, associative-disassociative motion reciprocity, and all the moving components of universe continuously affect all the other moving components--in varying degrees, ranging between high and low tide reciprocities of critically intense to critically negligible. All of these inter-effects of all the motional components upon one another are precessional, and precession always produces transformative resultants in vectorial patterns which always articulate angular accelerations in directions other than the 'straight' lines of directions between the inter-effective components"

Nonsense.

[Dick Fischbeck]

This is what Fuller calls operational mathematics. For example, a continuum is a mathematical construct but no such thing exists. No one can point to it. It can't be demonstrated. 

Synergetics and cartesian math are different. Synergetics is what we can show to be true and what we experience in the world. There are no points with zero dimensions. No lines with one dimension. No planes with only two dimensions. And no solids.

[Dick Fischbeck]

Okay. I get it now. You think Fuller was a fraud through and through. I guess I'm done. I can't talk to you about geometry. Maybe somebody else will pick up this tread. Someone who thinks Fuller had something important and serious to consider. That would be cool. We have had 10F^2+2=n since 1948. It was true then and it is true now. Cheers, Robert.

On Sat, Feb 18, 2023 at 7:06 PM RC <clark.rob...@gmail.com> wrote:

[Gerry in Quebec]

Dick, Robert & others,

Replying to Dick, Robert wrote, "Other than Euler's formula for the relationship of vertices, edges, and faces, I don't know of any universal formulas to calculate surface area and volume based solely on number of random vertices equidistant from a central point.  Is this the type of solution you are seeking?"

I think I may be stating the obvious, but two polyhedra with the same number of vertices may have differing areas and volumes. There is no way to precisely calculate surface area and volume of a convex, constant-radius polyhedron based solely on the number of vertices.... And even if you know the number of edges and faces as well, this is still not enough information to precisely determine area and volume. The simplest example is that of two tetrahedra. The volume and area of a regular tetrahedron differ from those of an irregular tetrahedron. Yet both have the same number of faces, vertices and edges.

The images below give another simple example: two distinct polyhedra, each with the same number of faces, vertices, and edges. Their areas and volumes differ. 

- Gerry in Québec

[Dx G]

Nice work Gerry.  Always refreshing to see your clear commentary and illustrations.

One thought which is likely no news to you, but something I keep in mind...  Even if there are constraints on a given formula or approach, they may have some real value if they have been overlooked in the past. So if some of these math assets can only be used with tessellations of regular polyhedra, or only certain ones, they may still be worthwhile additions to the collective wisdom, especially if they were not on the group radar screen before.  So I recognize what you point out, but still keep in mind there may be some mathematical assets that have escaped us in the past.  As is said, you can't do the impossible unless you can see the invisible  :-)

DxG

[Dick Fischbeck]

Yes, thanks Gerry. I'm trying to be clear describing my conjecture, So far, I'm not.

First, the icosa has 12 vertexes and 20 faces. But that doesn't really matter,

Then, in topology, the shape of a thing can change without changing its frequency, area or volume. It is the study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures.

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[Dick Fischbeck]

As to the belief that, "There is no way to precisely calculate surface area and volume of a convex, constant-radius polyhedron based solely on the number of vertices."

However, Fuller has moved in that direction by using the tetrahedron, rather than the cube as the unit, to measure the frequency, area and volume of the regular cuboctahedron. Volume = 20F^3.  So I have stretched that same theme to apply to all other spherical polyhedrons with the caveat that the vertexes have some minimal slack in their positions, They all tend toward a resolution, and equilibrium position. Edges and angles vary slightly. This is something similar to the jitterbug on a larger scale than just to the concentric hierarchy, (tet, octa, icosa and cubocta)

[Gerry in Quebec]

Hi Dick & Dan,

Thanks for the replies. 

Dan: Yes, we should always keep our eyes and minds open to alternative views or approaches to doing things, as simple but useful innovations/concepts often fall through the cracks.

Dick ... In the volume formula you mention for the cuboctahedron, what does the F stand for? Also, if the regular tetrahedron can be used as a unit of volume, what is the length of its edges (regardless of the selected linear unit such as metres or feet)? Or does this matter? I have not plunged into Synergetics partly because of the opacity of Bucky's writing, as illustrated by the excerpt Robert cited. So it's hard for me to see where you're going with this idea of using an alternative, hopefully useful geometry. A few clarifications and maybe some drawings would be helpful.

About the drawings I posted.... The ones on the left are not regular icosahedrons. Each has 24 isosceles faces rather than 20 equilateral faces, so I guess they can be called icositetrahedrons. The ones on the right also have 24 faces. They are class II, frequency 2 octahedral "spheres". In each case, all vertices are equidistant from the spherical centre. 

- Gerry in Québec

[Dick Fischbeck]

Dick ... In the volume formula you mention for the cuboctahedron, what does the F stand for?

Frequency

 

Also, if the regular tetrahedron can be used as a unit of volume, what is the length of its edges (regardless of the selected linear unit such as metres or feet)? Or does this matter?  

A tet with a certain volume can change shape some and with some asymmetry but if it is relatively round*, its area will change very little, and its frequency not at all. 

 

I have not plunged into Synergetics partly because of the opacity of Bucky's writing, as illustrated by the excerpt Robert cited. So it's hard for me to see where you're going with this idea of using an alternative, hopefully useful geometry. A few clarifications and maybe some drawings would be helpful.

Thanks for playing! 😊 (i was thinking of translating that quote into more common language, i still might)

*Roundness is defined as the ratio of the surface area of an object to the area of the circle whose diameter is equal to the maximum diameter of the object

[Curt McNamara]

Bucky was very clear on distinguishing between the mathematical abstractions we use and the real world. You are correct that a lot of useful stuff can be built using the abstractions!

All actual mobius structures have two sides. Stick a pin through them and it enters one side and exits the other.

From what I know, Bucky always said he (and all of us) discover things rather then invent anything. If you spend a few minutes with the Froebel pea work (lumps of clay and sticks) you will also discover the octet truss.

Buckys work in tension and compression allowed Kenneth Snelson to develop the first model. It is true that Bucky didn't credit Kenneth properly.

Check my other message with reference to the Pattern Thinking book to get an idea of how comprehensive Bucky's work with domes was.

Agree, it is tough to follow Bucky's language.

     Curt

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[Curt McNamara]

Working backwards towards how the geodesic list serve got created:

- there is a lot of interest in geodesic structures

- many folks have built their own domes or are interested in building one

- some are intrigued by geodesic math

- the major reason for this interest was the widespread adoption of domes and the amount of information about them

- the best way to communicate a design idea widely is to publish a patent

- other good strategies are to publish books, and give public workshops

Bucky is the reason for all of this. I recommend Pattern Thinking by Lopez-Perez for a good overview.

You can see pages from the book and pages from Bucky's journals here:

https://store.aia.org/products/r-buckminster-fuller-pattern-thinking?variant=31363729129517

      Curt

On Mon, Mar 20, 2023, 7:42 AM RC <clark.rob...@gmail.com> wrote:

It's ironic how Bucky said we discover things rather than invent anything, because he was notorious for patenting and trademarking things discovered by others and slapping his name on it as the original inventor.  Aside from that he seemed like a likeable, imaginative, bright but quirky fella somewhere on the spectrum.

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[Dx G]

Well, I could be glib and vague with respect to a mobius strip.  A circle has two sides, an inside and an outside, but it has 2 surfaces.  If you trace the outside of a circle with a marker, you eventually return to where you started.  However, you will notice no marker on the inside.  In contrast, you could say that a mobius strip has two sides (as per a pin prick), but it only has one surface. That is to say, if you begin making a line on it with a marker, and continue, you will return where you started, but the marker will appear on its entire, single surface - unlike the circle. Consider the difference.

 With respect to Bucky, his communications, and items posted on lists/forums, this is how I see it:

Sometimes when someone uses terminology few understand, describing concepts that make no sense to many listeners, there are those in the audience who are impressed, and think the speaker must be very smart. Others in the audience simply dismiss what they don't understand.  When the speaker is mistaken, and is wrong, there are few real consequences, if any, from this failure to communicate. Whether either side of that audience is right or wrong, if the ravings of the speaker remain a mystery to the audience, their lack of understanding does little harm.
  In contrast, what if the speaker is actually right, and the ideas being presented are correct and important. Society experiences a loss when such things are ignored.  As such, although it may not seem fair to some people, the reality is that a speaker in possession of important new knowledge also has a responsibility to present it in terms that people will understand so that others will recognize the nature and value of this new knowledge. In that case, if the speaker is actually right, a failure to communicate really does matter. Sure, I'm not saying there are no boneheads in the audience who are only enamored with their own ideas and those of no one else. However, I also have seen people who do not want to be burdened with the task of communicating in understandable terms simply excuse themselves from that task by blaming the failings of the audience. 

    Thus, when you read Bucky, or post here, consider - when a speaker is wrong, a failure to communicate is not such a tragedy as when the speaker actually has it right. In the latter case when it fails to get across because the speaker is unable or unwilling,  this leaving new, important knowledge to languish in the pages of history, and never reach its potential. 

-DxG

[Dick Fischbeck]

Is it RC or Robert? Are there 2 RC's?

Anyway, as to your statement,  "Fuller had a preference for a spherical coordinate system rather than an xyz cartesian system just as astronomers and astrophysicists do."

But spherical coordinates are derived from xyz, not ivm, which was Fuller's main interest (although he didn't get too far in the immortal words of CJ)

On Thursday, February 16, 2023 at 7:18:21 PM UTC-5 RC wrote:

[Gerry in Quebec]

Hi Dick,

Spherical coordinates aren't "derived" from XYZ (Cartesian) coordinates. The two systems of defining points in 3D space differ, but you can easily convert coordinates from one system to the other -- a bit like switching between metric and imperial measure. 

- Gerry

[Dick Fischbeck]

No, spherical coordinates is all cubes, pretty sure!  Your answers are in cubes.

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[Dick Fischbeck]

https://keisan.casio.com/exec/system/1359534351

[Gerry Toomey]

Yes. That diagram shows the relation between the two systems. One set of coordinates readily converts to the other. One little confusion is that the labels of the phi and theta angles in the spherical system are sometimes interchanged (physics versus mathematics)... a simple problem of symbolic conventions.

As long as you have all your "ducks" lined up correctly, you can translate "spherical to Cartesian" and vice versa.

He

Virus-free.www.avast.com

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[Dick Fischbeck]

So my goal is exclusively to use the ivm. Of course, tetrahedral values can be converted to cartesian. 

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[Curt McNamara]

Maybe this is interesting:

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/32%3A_Math_Chapters/32.04%3A_Spherical_Coordinates

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[Dick Fischbeck]

Right. But think in terms of triangular steradians.

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[Dick Fischbeck]

And, if you use triangles instead of squares, you no longer need angles theta and phi because it is easy to count triangles on the surface of a icosahedral polyhedron, which is 2(n-2), where n is the number of vertexes.