general method for Kruschke domes

[David Zhang]

After working on this for about a week, i finally figured out a method for generating Kruschke domes of any frequency, working off of a sketch up model and spreadsheet i found in this group from Gerry.

Here are the steps:

1.  Create an insphere of the base polyhedra  - i.e. a sphere that fits perfectly inside an icosahedron or octahedron.   it almost works with a tetrahedron but the corners get cut off.  

2.  subdivide the primary triangles on the polyhedra into the desired frequency.

3.  now we need to cut the insphere with parallel slices perpendicular to the 'up' direction to create the flat layers.  we can do this by using a different axis for each side of the primary triangle on the polyhedra.  on the polyhedra, we rotate a point at the center of the base of the primary triangle up the triangle's height plus half its side length to get the 'axis point'.  this axis point represent the 'north pole' if the triangle is on the equator, pointing up, and parallel with a vector that goes from the center of the polyhedra to the axis point.

4.   Now we apply this magic formula to get points on the insphere that corresponds with a parallel slice projected from the polyhedra.   Take the dot product of the axis point in step 3 and a point that corresponds with the subdivisions created in step 2.  Multiply this dot product by the ratio of the polyhedra's  insphere to its circumsphere side length, then take the arccosine of this number.     Now we have an angle that corresponds with the radius of a small circle on the surface of the insphere centered on the corresponding axis point .  this small circle is a parallel slice. 

5.  Now do this for all three sides of the triangle for each frequency subdivision  to get the intersecting small circle slices we see in the sketchup model.    Above 4v, these slices don't quite intersect - they form a 'window' like in the equal arc subdivision, so you need a rounding method to get the flat layers.  

Anyway, here's an animated gif demonstrating  my method creating 12v Kruschke sphere.

[Christopher Jones]

David,

As I recebtky misreplied to another of  your posts:-

In geodesign you label 8v chord factors as Kruschke. As I understand it, there should then be 7 flat circular layers which divide each ring of 10 faces of the original icosahedron. The ones adjacent to an apex do not look very flat in your geodesign app

I download your .stl model for this subdivision in the app and asked ChatGPT to look for these 7 flat circular layers.

This is what it said:-

Layer                                            Absolute spread        Relative to radius
Very flat layers                            0.00001                     1×10−7
Good flat layers                           0.00070                     7×10−6
Outer 40-vertex merged band  1.642                          1.64×10−2 (≈1.6% of the radius)

The first two are essentially flat. The last one is not—a variation of about 1.6% of the sphere radius is far too large to represent a genuine flat cutoff.

This actually tells us something important about the Geodesign model. If Kruschke's true 8v construction has seven genuine flat cutoff rings of 40 vertices each, then this STL is not an exact implementation of that construction. It is extremely close, but it appears to have undergone some optimization (or uses a different construction method) that produces near-flat rings rather than mathematically exact planes, especially towards the top and bottom.

Can your general method be fine-tuned to improve these outer layers? It is already the best method I have found online to generate Kruschke domes.

Christopher Jones

[Christopher Jones]

David, 

ChatGPT has coded for me a Python script, partly based  on your geo3d.js, which creates more accurate Kruschke domes of any subdivision.

Here is an 8v example,

Chris Jones

[Levente Likhanecz]

hi, iam trying to understand (even with chatgpt) the manual method first. but no success.
i can get to the early point to make an icosahedron, find the insphere.

but then nothing more.
i get it, that any arbitrary 10 triangle belt or ring i slice up paralel to its bottom and top pyramid cap. 
i may have different patterns of understanding that pole thing.

i collided into AI hallucination, where once we agreed on something, then one more answer and no agreement again on previous things.
like a cheating car dealer. give up, because after end of credits becomes very stupid machine.

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[Levente Likhanecz]

i guess i found it later, i have some other understanding of the design / symmetries.

these are the raw paralel flats of a 8V, intersected with the insphere.
i think this situation you optimized with your python script.
but this is lot of deviation.

[Ashok Mathur]

All verticies are four strut ones but the triangular shapes vary hugely and there a few pentagons.?

Is it good worry, I wonder?

Regards

Ashok

To view this discussion visit https://groups.google.com/d/msgid/geodesichelp/CAOkvEDk5otkL0nvQBb2rM30EagE%3D6MYLAgKVNbidqtFrr9FG2g%40mail.gmail.com.

[Christopher Jones]

Yes, ChatGPT is giving me the runaround too. That script is not so good for 16v, I asked for an improvement, it promised me a radical new design but just tweaked the existing script.

[Levente Likhanecz]

i am fighting with chat gpt, to create a 2017make plugin. it slowly understands how to populate lesser circles around the sphere, with icosahedron symmetry:

the AI said, that he understands already the optimalization theory with it. it is just the initial distribution of the beginner sets of paralels.

To view this discussion visit https://groups.google.com/d/msgid/geodesichelp/38372aca-b778-4621-bcbb-2209df81a6d2n%40googlegroups.com.

[Levente Likhanecz]

i made a small helper plugin for "make 2017". it creates a smooth sphere of chosen radius / segment number.
after installation click on it on the extension menu, a small popup window asks for radius /segment count.
then it will create a sphere, that can be placed inside dome frame, to cover out backside.
like the sphere above.