iTopSpin

Please login or register.

Login with username, password and session length
Advanced search  

News:

Author Topic: offset top  (Read 23394 times)

Jeremy McCreary

  • ITSA
  • Demigod member
  • **********
  • Posts: 3786
    • MOCpages
Re: offset top
« Reply #180 on: September 10, 2021, 01:19:28 PM »

Anyway, it's now proven by assuming the symmetry of the diameter and not the golden ratio.  ;)

Could you elaborate on this symmetry? Not quite sure how it's defined or what it preserves.
Logged
Art is how we decorate space, music is how we decorate time ... and with spinning tops, we decorate both.
—after Jean-Michel Basquiat, 1960-1988

Everything in the world is strange and marvelous to well-open eyes.
—Jose Ortega y Gasset, 1883-1955

ta0

  • Administrator
  • Olympus member
  • *****
  • Posts: 14366
    • www.ta0.com
Re: offset top
« Reply #181 on: September 10, 2021, 03:09:14 PM »

Anyway, it's now proven by assuming the symmetry of the diameter and not the golden ratio.  ;)

Could you elaborate on this symmetry? Not quite sure how it's defined or what it preserves.
Just saying that the centroid is in the middle of the segment between the point of contact and the opposite side of the figure. The golden ratio scaling then appears as a condition for the new centroid to be on the edge (i.e., on the end of the segment.)

Some observations from looking for golden ratio tops:
From a square you can create an infinite number of different tops: any position of the scaled square along the edge would work.
Only stars with an even number of points can be made golden.
Any triangle can be carved in 3 different ways, by placing the scaled triangle on each side in precise locations. But none of the resulting shapes has symmetry (non even for the equilateral triangle).

I tried to make a series of fish eating each other, but it was more difficult than I expected. I'm not very happy with what I have obtained so far:



The cool thing is that all the golden fish tops can be stacked to form a complete fish. Something like a Matryoshka doll.
« Last Edit: September 10, 2021, 03:16:33 PM by ta0 »
Logged

ortwin

  • ITSA
  • Hyperhero member
  • ********
  • Posts: 1493
Re: offset top
« Reply #182 on: September 10, 2021, 05:02:32 PM »

...
Only stars with an even number of points can be made golden.
..

Why is that? In any "normal" shape there is a line through the centroid  in which the centroid sits in the middle with respect to the edges. Do the points of the "hole star" cut through the the edges of the original star?
Logged

In the broader world of tops, nothing's everything!  —  Jeremy McCreary

ta0

  • Administrator
  • Olympus member
  • *****
  • Posts: 14366
    • www.ta0.com
Re: offset top
« Reply #183 on: September 10, 2021, 05:28:59 PM »

...
Only stars with an even number of points can be made golden.
..

Why is that? In any "normal" shape there is a line through the centroid  in which the centroid sits in the middle with respect to the edges. Do the points of the "hole star" cut through the the edges of the original star?
Yes, it seems to me that the some parts of the small star will always fall outside the large star, for an odd number of points. But I haven't proven it rigorously.
Logged

ortwin

  • ITSA
  • Hyperhero member
  • ********
  • Posts: 1493
Re: offset top
« Reply #184 on: September 10, 2021, 05:45:47 PM »

Yes there seems to be a problem. In my five star example I just made, not by calculating but just by eye judgement and some functions of a drawing program, the large shape is  cut by the hole into two separate parts. That is probably why I felt like babbling something about "convex outline" some posts above.


Edit: looking at this, the colors remind me again of your avatar. Why are you not going for that? The shadow theme is also self referential, kind of ...
« Last Edit: September 10, 2021, 06:21:16 PM by ortwin »
Logged

ta0

  • Administrator
  • Olympus member
  • *****
  • Posts: 14366
    • www.ta0.com
Re: offset top
« Reply #185 on: September 10, 2021, 07:46:04 PM »

Ortwin: You need to move the star further along the segment with the centroid at the center, until it touches the other star. But then they will be crossing each other.

I'll look at the logo. But you give tough homework.  ;D

Logged

Jeremy McCreary

  • ITSA
  • Demigod member
  • **********
  • Posts: 3786
    • MOCpages
Re: offset top
« Reply #186 on: September 10, 2021, 08:25:19 PM »

Just saying that the centroid is in the middle of the segment between the point of contact and the opposite side of the figure. The golden ratio scaling then appears as a condition for the new centroid to be on the edge (i.e., on the end of the segment.)

So the golden ratio emerges when...
a. Three special points are colinear.
b. The final CM  falls on the inner edge of the hole.

Sorry I'm not getting this. Which 3 special points again? Where are they in the diagram from Reply #152 below...


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.

CH = centroid of hole
CT = centroid of total figure
CC = centroid of carved figure :) :)
Logged

ta0

  • Administrator
  • Olympus member
  • *****
  • Posts: 14366
    • www.ta0.com
Re: offset top
« Reply #187 on: September 10, 2021, 08:47:25 PM »

Sorry I'm not getting this. Which 3 special points again? Where are they in the diagram from Reply #152 below...

Sorry, I should have drawn the complete segment of interest and indicated all the points on that diagram. Here it is again:



We are interested in making Cc coincide with Q'. The conclusion was that CT has to be in the middle of QP (i.e. CH in the middle of Q'P') and the scaling k has to be the Golden Ratio. From the levers you first find that y = k2-1. Then if you assume x = 2 b, you get k = φ, or alternatively, if you assume k = φ you get x = 2 b.
« Last Edit: September 10, 2021, 09:10:36 PM by ta0 »
Logged

Jeremy McCreary

  • ITSA
  • Demigod member
  • **********
  • Posts: 3786
    • MOCpages
Re: offset top
« Reply #188 on: September 10, 2021, 09:44:18 PM »

Thanks! Now I get it. Very nice!

To form the diagram in Reply #152, you might...
Step 1. Copy the outline of the blank (original shape, no hole)
Step 2. Scale down the copy by area so that Acopy = Ablank / k² , k > 1, while keeping the 2 centroids surperimposed.
Step 3. Translate the copy outward to kiss the blank's outline without rotation.
Step 4. Cut out the copy to form the hole.

Guessing that your derivation also needs the lack of rotation in Step 3. Is it also a requirement that corresponding points of the blank and copy kiss?
« Last Edit: September 10, 2021, 09:55:41 PM by Jeremy McCreary »
Logged

ta0

  • Administrator
  • Olympus member
  • *****
  • Posts: 14366
    • www.ta0.com
Re: offset top
« Reply #189 on: September 10, 2021, 10:54:19 PM »

Yes, those are the steps. If you want the centroid to end up on the edge, the direction you select has to have the centroid in the center and should use k = φ.
Yes, no rotation. As you are moving the smaller one in a radial direction it will automatically kiss the other one in the corresponding place. This is critical for this construction and derivation.   This does not mean that it's impossible to find other cases that do not follow this construction.
« Last Edit: September 10, 2021, 11:01:12 PM by ta0 »
Logged

Bill Wells

  • Hero Member
  • *****
  • Posts: 263
Re: offset top
« Reply #190 on: September 11, 2021, 12:29:53 AM »

When not inverted, right? (Resting vertically on the body, not the stem.)
Yes, when not inverted. I have seen a tippe top invert and almost come to rest balanced upon the flat stem. That would be cool.
Logged
It's not worth doing if you're not obsessed about it.

ortwin

  • ITSA
  • Hyperhero member
  • ********
  • Posts: 1493
Re: offset top
« Reply #191 on: September 11, 2021, 05:25:00 AM »

Ortwin: You need to move the star further along the segment with the centroid at the center, until it touches the other star. But then they will be crossing each other.

I'll look at the logo. But you give tough homework.  ;D


I don't give homework, only ideas you can work on- if you want- in your spare time. >:D
Logged

Jeremy McCreary

  • ITSA
  • Demigod member
  • **********
  • Posts: 3786
    • MOCpages
Re: offset top
« Reply #192 on: September 11, 2021, 10:16:33 AM »

When not inverted, right? (Resting vertically on the body, not the stem.)
Yes, when not inverted. I have seen a tippe top invert and almost come to rest balanced upon the flat stem. That would be cool.

I've found that it doesn't take much of a flat if the top's balanced well enough. The white/gray offset top at left (first up in the video) often comes to rest without falling. The less well-balanced one on the right never does.

https://youtu.be/sIbKwPYFpXY

It takes a 14x loupe to see the flats (tiny mold scars) on their identical tips! See Reply #28 for details.
« Last Edit: September 11, 2021, 10:39:31 AM by Jeremy McCreary »
Logged

ortwin

  • ITSA
  • Hyperhero member
  • ********
  • Posts: 1493
Re: offset top
« Reply #193 on: September 11, 2021, 11:08:05 AM »

Ortwin: You need to move the star further along the segment with the centroid at the center, until it touches the other star. But then they will be crossing each other....
of course you are right again, I just stopped when the first thing  touched the outer edge. This should be a little bit closer to what you would get doing it the correct way:

 

Logged

ortwin

  • ITSA
  • Hyperhero member
  • ********
  • Posts: 1493
Re: offset top
« Reply #194 on: September 11, 2021, 11:30:36 AM »

When not inverted, right? (Resting vertically on the body, not the stem.)
Yes, when not inverted. I have seen a tippe top invert and almost come to rest balanced upon the flat stem. That would be cool.
Maybe something like the balls wee discussed here would have a good chance to remain balanced on that hole when inverted? - In the not multiple flip version of course.
But then again an inverted ball is not as  impressive as a top with stem that reverses and stays balanced on that stem even without spinning. 

How about your flipping disks you mention here Jeremy? Any chance they stay balanced after inversion?
Logged
Pages: « 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 »   Go Up