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[ortwin]

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Now I'm seeing my photo attachments! I see my diagram pasted everywhere now. You fixed it, thanks!

Glad to hear those problems are solved!
I hope you could still start on that offset top?
Anxious to see a first pic of it.
Don't worry that we think it would be too simple: Have you seen my "curtain ring top nr.0" ??

But I would not trust any calculations on the center of mass to be very close to the real one in your top. Especially with the non-uniformity in density distribution to be expected in wood. I think I would go for the "balancing on needle" method before drilling a hole into the disk.

[Jeremy McCreary]

The code tag doesn't transmit. Tried [X]paste your failed image code here[/X]

Sorry, you have to replace the "X" above with the word "code" to get the code tag to do its thing. And you have to replace the "paste your failed image code here" with the image link that didn't work as you'd hoped.

The code tag isn't supposed to make your image links work. Just the opposite. It's there to allow you to show us the text of the link as it appeared in the message text sent out when you hit the "Post" button. That makes it easier for us to help you fix it.

So you are all seeing my photo attachments but I cannot?

I saw your diagram with a colored circle in Replies #144 and #145. No other images and no code box like this...

Code: [Select]

This is what a code box looks like. Use the Quote button to see how I made it appear.
The forum server will ignore all text inside here.

[ta0]

On this diagram the original figure and the hole have the same shape and orientation and are in the Golden Ratio proportion (φ):



The hole also touches the edge of the original figure, in the direction the figure was displaced, but is otherwise completely enclosed.

C_H = centroid of hole
C_T = centroid of total figure
C_C = centroid of carved figure
a = distance between C_C and C_T
From the Golden Hole rule (reply 132) we know that the distance between C_T and C_H is: aφ
b = distance between C_H and outside edge (along line through centroids)
therefore, the distance between C_T and the same edge is: bφ
x = distance from C_C to the edge

We have:
aφ + b = bφ => a = b (φ-1)/φ = b (1-1/φ) = b (1 - (φ -1)) => a = b (2 - φ)

Also:
x = a + bφ

Combining them:
x = b (2 - φ) + b φ = 2b

Therefore the new center of mass, C_c is at the same distance as the edge from C_H, but on the opposite side.

Conclusion: if the centroid of the figure is halfway from the edges, along the line the figure was displaced, the new center of mass is at the edge of the carved figure.

[Bill Wells]

I think I would go for the "balancing on needle" method before drilling a hole into the disk.

Exactly. I always compare reality to theory wherever possible.

[ortwin]

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This looks almost like your forum avatar and its enlarged shadow! Maybe you should consider 3-D printing such a thing.

[ta0]

This looks almost like your forum avatar and its enlarged shadow! Maybe you should consider 3-D printing such a thing.

I will 3D print some shape, but I have not decided which yet.

Some quick examples of figures with "golden cutouts" that have the center of mass on the edge, using the above rule for construction:




In reality, the figure does not have to be symmetric. Just the "diameter" used (direction in which it's displaced to touch the edge) has to have the centroid at it's center.
Of course, the hole has to be completely contained in the figure when they touch at the edge, what together with the above eliminates many shapes.

[Bill Wells]

Final specs for my wood offset top:
Design: two semi-circles of different densities
Diameter = 116mm
Thickness = 29mm
Wood species: Jatoba, density 0.94 g/cc^3 and pine, density 0.47 g/cc^3
Calculated offset of centroid from center of disc = 8.2mm

Will post video after I get it polished up.

[Jeremy McCreary]

Exactly. I always compare reality to theory wherever possible.

Spoken like a true engineer. Test, test, test!

Looking forward to your offset top. And any others you'd like to show us.

[ortwin]

@ta0: task completed! Proof plus generalization plus construction principle! Congratulations.

But of course you would not believe its me who is saying that,  if there wouldn't be an additional thing I would ask for:
I think it would be a tiny little bit nicer if you wouldn't  have to plug in the Golden Ratio by hand into the proof, but instead the Golden Ratio would pop out during the proof by itself.
Actually I am not asking you to do that. I have some idea how that can work, just give me some time, with all the results you gave us already it should be possible to be plugged in there.
 

This looks almost like your forum avatar and its enlarged shadow! Maybe you should consider 3-D printing such a thing.

I will 3D print some shape, but I have not decided which yet.

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You could make a poll and we would help you decide. I would vote for your avatar and its shadow. Or maybe a shape like one of the tops in the many medals in your signature.

[Jeremy McCreary]

I will 3D print some shape, but I have not decided which yet.

A tour de force of mathematical top dedign!

Given the self-referential nature of your figures, and of the golden ratio itself, would love to see an offset cut-out top in the (smoothed) shape of a famous fractal like the Mandelbrot set.

[ortwin]

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So I am trying to get to the golden ratio from the condition that C_C should be on the edge of the carved figure and the other edge of the carved figure coincides with the original figure. I will make use of the property that "the centroid of the figure is halfway from the edges".
 The scaling factor I will call k to begin with, it should turn out in the end to be φ as in the figure above.
(1)      ak = b_k - b_k/k      = b_k (1 - 1/k)  (this should be clear from the figure)
(2)      a  = 2 b_k/k - b_k   =b_k (2/k - 1)    (this makes sure that C_C is on the edge of the carved figure using the fact that "the centroid of the figure is halfway from the edges". )
 If we devide (1) by (2) we get:  k = (1 - 1/k) / (2/k - 1) after two more lines of rearranging we have k-1 = 1/k . Since this is a very characteristic equation for the Golden Ration we can now identify k with φ .
So, can this be  the way how we can make the Golden Ratio pop up in our proof?

[ortwin]

....

Given the self-referential nature of your figures, and of the golden ratio itself, ...

How about the Penrose tiles? The kites and darts?

[Jeremy McCreary]

Quasi Golden Ratio LEGO top spinning for a minute plus

Very nice! And it did spin for over a minute. But my own tests on a flat surface with a much finer tip (17 vs. 1.5 mm radius of curvature) indicate that a top like this spins down to critical speed in 10-15 s at most.

True, the rest of the time, the top was artificially held up at the tip. But balance would have to be pretty good for support that meager to actually keep the top from falling over.

[Jeremy McCreary]

Well done! Off to find some LEGO-friendly figure and hole shapes that will really blow people's minds when the corresponding offset top spins without wobble.

I thought my stealth bomber from reply #124 does just that!   

Funny, that's exactly the name I had in mind for the one I made (video coming). Would be cool to make these tops look even more like the stealth bomber.

But I was thinking of less rectangular shapes.

[ta0]

A tour de force of mathematical top dedign!
Given the self-referential nature of your figures, and of the golden ratio itself, would love to see an offset cut-out top in the (smoothed) shape of a famous fractal like the

I was surprised that it generalized like that, but very happy.
I agree, some self-referencing drawing would be great.  The Madlebrot set is a very good idea. I need to check if the centroid falls close to the center of the horizontal line.

@ta0: task completed! Proof plus generalization plus construction principle! Congratulations.

You started this thread, found the first 2 examples and lead the way. By the way, the thread is on the 12th page and 167th reply! 

[/i]So, can this be  the way how we can make the Golden Ratio pop up in our proof?

I see that you named b the length on the original figure while I used it for the scaled figure (the hole). That is confusing as you referenced my diagram.
For the general case, it's not true that you get "ak" for the distance between C_T and C_H. It should be "a(k²-1)" (see reply 135).
A little more work to be done . . .