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[ta0]

An interesting twist arises when the polygon under consideration is a non-equilateral isoceles triangle. Then the CM of the wire frame never coincides with that of the filled triangle.

Although this is true, the derivations don't change as long as you use the correct center of mass.
I think it would even work for non-uniform mass distributions.

[Bill Wells]

Bary-center literally means (from Greek) weight-center. When I was in school (in Spanish) we called the centroid of a triangle the barycenter.

I don't mean to be picky, just want to understand terms. Iacopo also uses "barycenter".

[Bill Wells]

Good to see you at this party Bill!

Thank you for that sentiment Ortwin. I appreciate it.
I would be delighted to make a project on my lathe. You just need to describe it c-l-e-a-r-l-y. Coincidently, I just bought a supply of 1/4" bearing balls today for a top I will reveal soon - video included.

[Bill Wells]

Hm, I have some doubts that we are talking about the same thing now.

I'm glad to see that reply from you Ortwin. The thought had crossed my mind also...

[Jeremy McCreary]

An interesting twist arises when the polygon under consideration is a non-equilateral isoceles triangle. Then the CM of the wire frame never coincides with that of the filled triangle.

Although this is true, the derivations don't change as long as you use the correct center of mass.
I think it would even work for non-uniform mass distributions.

Probably true.

This fact becomes relevant if you go for a different configuration than the one you diagrammed -- say, with the overall CM at the apex of the smaller isoceles triangle. Still working on the solution to that one.

[Jeremy McCreary]

Thank you for that sentiment Ortwin. I appreciate it.
I would be delighted to make a project on my lathe. You just need to describe it c-l-e-a-r-l-y.

Be careful. ortwin will have no end of assignments for you.

[ortwin]

...
Be careful. ortwin will have no end of assignments for you.

@Bill: That is probably true, but you can always refuse to do them or postpone them week after week until I forget about them. That is what Jeremy is doing! 
 
Edit: I am not so sure any longer that a lathe is the best tool to make an "ex-centric" hole into a disk.....

(@ Jeremy: How is Miss Skimpy in chains  doing? Do I really need to remind you of the powers of Mistress von Karmann?)

[ta0]

This fact becomes relevant if you go for a different configuration than the one you diagrammed -- say, with the overall CM at the apex of the smaller isoceles triangle. Still working on the solution to that one.

I don't understand. The general result from reply #96 covers this case.

[Jeremy McCreary]

This fact becomes relevant if you go for a different configuration than the one you diagrammed -- say, with the overall CM at the apex of the smaller isoceles triangle. Still working on the solution to that one.

I don't understand. The general result from reply #96 covers this case.

Yes, it does. I didn't read carefully enough. Scheming to make one of these tops in LEGO.

[Jeremy McCreary]

@Bill: That is probably true, but you can always refuse to do them or postpone them week after week until I forget about them. That is what Jeremy is doing! 
 ...
(@ Jeremy: How is Miss Skimpy in chains  doing? Do I really need to remind you of the powers of Mistress von Karmann?)

Darn, that strategy isn't working as well as I'd hoped.

BTW, my checkered relationship with Mistress von Karman has entered a new phase of mutual support and respect. Now we enjoy the dungeon together.

[ortwin]

...

Words like "self similarity", "fractal geometry" and "Sierpinski Gasket" come to mind.
I have the feeling that cool looking top can be found here, or maybe a series of tops....
Can't quite see it clearly, something with circles inside circles, or like a crescend moon... something where a large part of the reduced smaller version coincides with a part of the original version..... 
I will go and have some sleep now, maybe when I check again tomorrow morning one of you will have found and drawn that cool top....

[Jeremy McCreary]

My best shot at a square double-ring offset top in LEGO...



Got the least wobble with studded LEGO construction, but that meant square rings far from wire frames. The squares' median diameter ratio (halfway between inner and outer edges) might arguably have been the one to "goldenize" in that case, but keeping the top to a reasonable size and stiffness ruled that out, too.

So I goldenized the squares' outer diameter ratio as best I could, coming as close as 16/10 = 1.60 in the top above. That meant inner and median diameter ratios of 12/6 = 2.00 and 14/8 = 1.75, respectively -- both far from golden.

Balancing trade-off

Before paving the rings with studless decorative tiles, I'd all but eliminated visible wobble by adding extra parts to the bottom of the rotor with the help of a knife-edge static balancing rig. And this despite the couple unbalance introduced in the process.




Unexpectedly, adding the tiles made wobble reduction much, much harder. As with my stab at a Cylon raider top, all attempts to improve static balance just made the couple unbalance worse, and vice versa. The trade-off above was the best I could find.

Also tried some studless triangular solutions, but the necessary couple unbalance and resulting wobble were much worse than above.

[ortwin]

Nice!
Did you also check if a version is feasible where the two squares are both rotated by 45 degrees, and the tip sits at the corner of the smaller square?

[Jeremy McCreary]

Nice!
Did you also check if a version is feasible where the two squares are both rotated by 45 degrees, and the tip sits at the corner of the smaller square?

Might well be doable, but I'd probably  run into the same time-consuming trade-off between static and couple unbalance. So I'll pass.

[ortwin]

Alright, so I go and check what I can find in my boys' LEGO.