Exploring the opposition of Lobel frames

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Re: Exploring the opposition of Lobel frames

Postby Eric » Sun Feb 26, 2023 8:36 pm

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Re: Exploring the opposition of Lobel frames

Postby Eric » Sun Feb 26, 2023 8:58 pm

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Re: Exploring the opposition of Lobel frames

Postby Eric » Sun Feb 26, 2023 9:00 pm

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Re: Exploring the opposition of Lobel frames

Postby Wagyx » Sun Feb 26, 2023 9:27 pm

Yeah, what you are dealing with when having the pentagonal pyramids point inwards or outwards is close to what happens in chemistry with molecular configuration like stereoisomerism.
Basically, the connectivity between the rods and balls is the same. In graph theory, people use the terms of edges and vertices respectively for these elements and isomorphism to denote two graphs with the same connectivity.

So you have found two geometric configurations of the same structure which is a very common feature for almost all models, especially when having vertices of degree 5 or 6
(the degree of the vertex is the number of edges connected to it).
I mean that you have decided to have all the pentagonal pyramids pointing inwards or outwards but you could also have chosen to have some pointing inwards and other pointing outwards and do the same of any vertex of degree 6.
This would generate many alternative valid configurations though the symmetry of the model would be affected.
Actually, this is a problem when I numerically constrain the 3d model and it outputs unwanted configurations.
I have also obtained the flat pancakes at first when making the 3d models based on your constructions. A little bit of fiddling to change the initial condition for the solver gave me the desired result.

When building the model, I find it quite pleasing to be able to distort the model and find a new configuration, I get a feeling that everything gets put into order (it clicks) and it can turn a somehow smooth and bland model into a more concave and complex shape like sculpting a piece of clay.
I think that the second configuration of the model would also be a part of the Lobel Frames.

Regarding the strip of triangles between the white lines, it is deeply engrained in the making of the Lobel Frames.
You can think of it as some kind of antiprism that has been inserted into the shape like the meat in a burger.
Once you have found one it can be used to either elongate (or gyroelongate) the model in the same way it was done in the Johnson solids, turning the model into a tube or reducing it by removing the strip.
A simpler thing you could test to get variants is to identify a closed line (like each of the white lines). They separate the model into two parts that you can rotate relative to each other.
For instance, if the line is 8 vertices long, it means that you can rotate one half 8 times and get potentially 8 different models.
This does not guarantee that the model is valid though unless the line already makes a regular polygon.
This operation also changes the connectivity most of the time.
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Re: Exploring the opposition of Lobel frames

Postby Wagyx » Sun Feb 26, 2023 10:10 pm

I am playing around with the exporting features of Stella.
As you may see on the hexagonal model, the 6 vertices of degree 6 are pointing inwards.
Nudging them from one configuration to the other is very easy on the physical model.
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lobel-frame-opposite6-inwards.gif
lobel-frame-opposite6-inwards.gif (297.8 KiB) Viewed 1424 times
lobel-frame-opposite5-inwards.gif
lobel-frame-opposite5-inwards.gif (290.48 KiB) Viewed 1424 times
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Re: Exploring the opposition of Lobel frames

Postby Eric » Sun Feb 26, 2023 11:03 pm

Wagyx wrote:Regarding the strip of triangles between the white lines, it is deeply engrained in the making of the Lobel Frames.
You can think of it as some kind of antiprism that has been inserted into the shape like the meat in a burger.
Once you have found one it can be used to either elongate (or gyroelongate) the model in the same way it was done in the Johnson solids, turning the model into a tube or reducing it by removing the strip.
A simpler thing you could test to get variants is to identify a closed line (like each of the white lines). They separate the model into two parts that you can rotate relative to each other.
For instance, if the line is 8 vertices long, it means that you can rotate one half 8 times and get potentially 8 different models.
This does not guarantee that the model is valid though unless the line already makes a regular polygon.
This operation also changes the connectivity most of the time.


Understood!
Find a closed loop that forms a flat regular polygon, and you can go from this:
t-mcdonalds-cheeseburger-BB-1 1-3-product-tile-desktop.jpg

to this:
10-meter-high-hamburger_redim.jpg

:lol:

Thank you for these great explanations!
It's very clear, interesting and it opens up perspectives. :)
Once you have found this flat regular polygon that separates your shape in two, you can rotate the 2 halves of your structure in relation to it, or even transform this polygon into a prism or an anti-prism (even or odd number of rows of triangles) to lengthen the shape.
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