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[ortwin]

... a tribute to ortwin for leading us to the offset top concept, which I've not seen elsewhere. Via beer no less!
...

Thank you Jeremy, but as said in the starting post of this topic, it was the paper with the elephant top that you recommended that gave me the idea, just the most simple version of that concept I could think of.

And that idea had nothing to do with beer, the beer just brought the solution!

[Jeremy McCreary]

Thank you Jeremy, but as said in the starting post of this topic, it was the paper with the elephant top that you recommended that gave me the idea, just the most simple version of that concept I could think of.

Well, I've read and reread that article dozens of times and never thought to apply it an off-center-looking top with a more or less planar rotor doable in LEGO. You triggered that.

BTW, their focus on "spinnability" is worth taking to heart. I prefer a slightly different figure of merit for my own building, but I see why they chose the one they did.

And that idea had nothing to do with beer, the beer just brought the solution!

Life works in mysterious ways.

[ortwin]

...thanks ortwin for starting to reference the reply numbers on a thread: I had never paid attention to them).
...

That is  something I saw in Jeremy's posts, I'll forward the credits to him.
On the colors of the Lego parts: You should also consider no color contrast at all! The asymmetry itself should be able to deliver most of the surprise effect!
At least you could buy the smaller parts in two different colors: one of them being the same as the large plate, the other one to contrast with the large plate. That way you could still decide, upon judging by seeing, which one you prefer to be on top.

[Jeremy McCreary]

...thanks ortwin for starting to reference the reply numbers on a thread: I had never paid attention to them).
...

That is  something I saw in Jeremy's posts, I'll forward the credits to him.

Come to think of it, the 1st time I recall seeing that was when Iacopo did it here.

On the colors of the Lego parts: You should also consider no color contrast at all! The asymmetry itself should be able to deliver most of the surprise effect!

This is a matter of taste, of course, but I like lots of contrast in my offset tops. To me, it only enhances the surprise the offset sets up.

[Jeremy McCreary]

Finally, the promised LEGO parts list for the square non-Ansatz offset top called "Top 1" in the video in Reply #65. Same version also seen in Reply #28, where I called it "White"...

Photo 1 (on the left). Ignore red stem holder.


Photo 2 (on the left).


Photo 3. Two of 4 possible tip options with parts below.


And a slightly different version in Reply #12...

Photo 4. This yellow stem holder is the one you'll be making with the parts below.


Photo 5. This yellow stem holder is the one you'll be making with the parts below.


Photo 6. Another tip option.


Parts list (with direct links to the parts on BrickLink)

1. One 16x16 plate, the main structural element (dark gray in all photos).

2. Four 4x8 plates, to thicken the smaller sub-rectangle (white in all photos).

3. One 6L axle (black in Photos 1-2 and 4-5) or 5.5L axle with stop (not shown) for the stem.

4. Five 2x2 round plates — 3 to mount the stem (as in Photos 4 and 5 in yellow) and 2 to provide a low-profile mount for the tip contact (as in Photo 3 on the right in red).

5. One minifig microphone (black in Photo 3) or bar with tow ball (dark gray in Photo 6) to provide a spherical contact for the tip.

Whether you use the mic or bar with tow ball, you'll have to trim the mic handle or bar to mount it securely in the low profile holder seen in Photo 3 on the right in red. If you prefer not to trim these parts, use a 2x2 dome bottom with studs to mount them as shown in Photo 3 (left) or in Photo 6. This will raise the top's CM ~6 mm.

[ortwin]

The discussion with Jeremy in the "Battlestar Galactica tops" topic, got me thinking of offset tops again. This time I don't want to use material of different densities to move the center of gravity off the geometrical center of a disk like the first time around:


 
 This time I want to to make a circular hole in the disk so that one rim of that hole coincides with the rim of the original disk:



As I make the radius (x) of that hole larger, the center of gravity shifts off center. But eventually the rim of the hole catches up with the center of gravity. That is when I need to stop making the hole larger and that is the case where the offset reaches its maximum if I want to place the stem directly on the disk without the use of spokes. The picture above shows that situation. The black dot symbolizes the stem/tip.The calculations that are not too hard lead to the cubic equation:
 x³ - 2x + 1 = 0
Where x is the radius of the hole if the radius of the original disk is 1.
Since I forgot how to solve these cubic equations, I consulted an online calculator:




The offset is not very impressive, but the solution is independent of the densities of the used material (of course).
I used cutting compasses, a piece of cardboard and a toothpick to build a physical version. The diameter of the disk is 10 cm, that of the hole about 6 cm. You need a lot of fantasy though, to see the "Cylon raider" in that top.
 


It does not work great, but it proves well enough for me that the calculations where correct.

[Jeremy McCreary]

Another interesting exercise in static balance from you! I like your solution.

The similarity to the Cylon raider shape in plan view (2nd view from top below) is clear.



The challenge in turning the raider shape into a stand-alone top is the inherent couple unbalance you run into when you choose a vertical spin axis in the uppermost side view.

I see this view as showing the raider flying at 0° pitch, and the frontal view as flying at 0° roll. A vertical spin axis in these 2 views would seem to be the best choice on both esthetic and aerodynamic grounds.

To simulate that challenge in your test top, add some mass below each arm and above the body aft of the arms while keeping the CM on your spin axis to maintain static balance. Then see how it spins.

[ortwin]

... interesting exercise in static balance ...

 
Correct, I considered only static balance. The couple unbalance which is the problem if you want the spin axis to be a nice one, I totally ignored by flattening out everything. 
Aerodynamic reasons don't count if those vehicles are supposed operate some hundreds of kilometers  away from a planet : no atmosphere to bother with, out there.

[ta0]

I like the design. Very elegant. And it reminds me some flat tippe tops if you spin it on the edge.

Do you realize that your solution for the radius is the inverse of the Golden Ratio? 
Proof: After you determine by inspection that x₁ = 1 is a solution, you can write the cubic equation as the product x³ -2x +1 = (x-1) (x² + x-1) = 0
and solve the quadratic:  x₂= (-1+sqrt(5))/2 = 1/φ  and x₃= -(1+sqrt(5))/2 = - φ

[Jeremy McCreary]

Correct, I considered only static balance. The couple unbalance which is the problem if you want the spin axis to be a nice one, I totally ignored by flattening out everything. 
Aerodynamic reasons don't count if those vehicles are supposed operate some hundreds of kilometers  away from a planet : no atmosphere to bother with, out there.

Like most Battlestar Galactica ships, the writers had the raiders flying around in both space and planetary atmospheres. At least the raiders kinda looked like they could pull it off. The rest looked like they'd burn up right away.

But the aerodynamics I was thinking of were those of the top.

[Jeremy McCreary]

Do you realize that your solution for the radius is the inverse of the Golden Ratio? 

Good pickup!

[ortwin]

I like the design. Very elegant. And it reminds me some flat tippe tops if you spin it on the edge. ...

Thank you ta0 !
And although I have two self made tippe tops of the kind you mention, I did not see the similarity until you mentioned it.
Now I want a metal offset stemless top that works as a tippe top if spun on the edge! I don't think it could be a good Euler's disk at the same time.
 

...After you determine by inspection that x₁ = 1 is a solution, you can write the cubic equation as the product (x-1) (x²+x-1) = 0
..

Yes, I saw the x₁ = 1 solution, and I knew that the equation could be written as some kind of product with that solution, but I did not instantly find the second factor, the quadratic equation, I got impatient and turned to the online calculator.
That is why I missed the sqrt(5) showing up. That would have given it away to me.

...
Do you realize that your solution for the radius is the inverse of the Golden Ratio? 

So no, I missed this point that makes me now even happier with this top, thank you.
The numbers I saw in the solution looked remotely familiar, and also the fact that the two solutions x₂ and x₃ differed exactly by 1 raised my suspicion a bit, alas not enough to stop and think before rushing to build the physical version.
Now if you look at the ratio of the circumference of the disk and the circumference of the hole, you arrive at the Golden Ratio, no need for the inverse or the radius or tau....

Now ta0 please show us how you fold an offset top of this kind with the Golden Ratio from an Indian (or was it Japanese?) newspaper! 

  Edit: Can I make it a challenge between Jeremy and ta0 to explain fast and clear  how to arrive at that cubic equation? Or is that too straight forward and easy a problem for you guys?
 

[ta0]

And although I have two self made tippe tops of the kind you mention, I did not see the similarity until you mentioned it.
Now I want a metal offset stemless top that works as a tippe top if spun on the edge! I don't think it could be a good Euler's disk at the same time.

You could start a Kickstarter project . . .

Yes, I saw the x₁ = 1 solution, and I knew that the equation could be written as some kind of product with that solution, but I did not instantly find the second factor, the quadratic equation, I got impatient and turned to the online calculator.

To factorize a polynomial when you already now a factor, you just divide the polynomials exactly as you would divide two large numbers by hand. Try it and you will be surprised how easy it is.

  Edit: Can I make it a challenge between Jeremy and ta0 to explain fast and clear  how to arrive at that cubic equation? Or is that too straight forward and easy a problem for you guys?
 

When you add the crescent to the circle you removed, you get back the big circle with the center of mass on its center. So you equate the two levers: mass of crescent times distance from its center of mass to the center of big circle equal to mass of the small circle times distance from its center to the center of the big circle. The mass of the crescent is the area of the big circle minus the area of the small circle, that is proportional to (1-x²). The two distances are 2x-1 and 1-x. So you get:
(1-x²)(2x-1) = x²(1-x).

[ortwin]

You could start a Kickstarter project . . .

You have suggested that with another thing I proposed before. But I think I am somehow missing a business or entrepreneur gene.
Ah, okay maybe I look into it what it actually takes to start a Kickstarter project. 

...
When you add the crescent to the circle you removed, you get back the big circle with the center of mass on its center. So you equate the two levers: mass of crescent times distance from its center of mass to the center of big circle equal to mass of the small circle times distance from its center to the center of the big circle. The mass of the crescent is the area of the big circle minus the area of the small circle, ....

Yes, that was exactly my thinking!
From there on your way of thinking seems more elegant and direct, it took me a few more lines to get to the cubic equation.
So congratulations on winning the challenge! Should ever something come out of a Kickstarter project that I might or might not start, you won yourself a free sample!

[ortwin]

And yet another theoretical exercise in static balance !

It is related to the idea of reply #81.

But here I use two rings, instead of a disk I make a hole into.

I imagine something like the material the flywheel of my brass band tops is made of.

I want the center of mass to be on the ring with the smaller radius. For that smaller ring I calculate in that case a radius of (sqrt(2) -1) times the radius of the large ring. The offset is even less than in the case with the hole in the disk, only 18% or so. It could be increased if one would use two small rings: one above and one below the large ring, that would also avoid couple unbalance, but I did not calculate that case.

The idea with one small ring and one big ring could also be made from one single continuous "wire" that makes a small and a big loop before it meets its starting end again........

 

Edit: This time I am aware that it could be a reversing top if spun on the side. It has some resemblance of "curtain ring top Nr. 9"