Balancing things, and centres

[Jon Corby]

Can any static object theoretically be balanced, if you were able to remove all air disturbance?

We used to have a standard lamp in our lounge next to the couch, and during programmes I was forever trying to tilt it on its circular base and get it to balance perfectly at an angle.

If I take something like a tissue box, in theory is it possible to balance it on any corner or any edge? It must be, mustn't it? If you push it too far one way it falls away from you, too far the other and it falls towards you, so there must be a point where it's balanced, isn't there?

Related, but not directly, is does every shape (no matter how irregular) have a definite calculable centre? If I take a circle, the centre is easy to find. If I stretch it into an oval (extending the same amount at the top and bottom), it makes sense that the centre stays the same. But what if I just pinch it out a bit at the top (so it looks like a snowman), where does the centre move to then. It makes sense that it would move down very slightly, but where specifically? Why can we easily locate the centre of some shapes but not others? I guess it's to do with symmetry, but does a really irregular shape still have an absolute centre? Is it that if you cut a shape in a straight line through its centre, you should end up with two equal amounts of that shape, regardless of which direction you cut in?

[Jon O'Neill]

Yeah I like questions like this.

[Michael Wallace]

Is it that if you cut a shape in a straight line through its centre, you should end up with two equal amounts of that shape, regardless of which direction you cut in?

I'd imagine most people would take it as the centre of mass of a uniform plane lamina of that shape (which is something I remember having to do as part of an OU course several years ago). So imagine balancing it on a pencil or something (although obviously that doesn't really work if the centre of mass is then not inside the shape, such as for a ring).

[Peter Mabey]

The answer is no to the final question. Think of a stick with equal weights on each end, one compact and the other broad. The balance point is in the middle, but a cut at an angle through it could miss the compact one entirely, while including most of the broad one

[Jon Corby]

The answer is no to the final question. Think of a stick with equal weights on each end, one compact and the other broad. The balance point is in the middle, but a cut at an angle through it could miss the compact one entirely, while including most of the broad one

I think you've fused the two questions together there, or I've got confused. The second part (to do with centres) isn't to do with balancing anything. The bit to do with centres can just be about 2D shapes. Maybe I should have done this in two threads.

[Charlie Reams]

If I take something like a tissue box, in theory is it possible to balance it on any corner or any edge? It must be, mustn't it? If you push it too far one way it falls away from you, too far the other and it falls towards you, so there must be a point where it's balanced, isn't there?

There will be a centre of gravity and if you align the object so that the centre of gravity is over the part of the object that's touching the ground then it will balance. But in most cases (e.g. a pencil on the tip) it's an unstable equilibrium, meaning that the slightest disturbance will magnify up and the whole thing will fall over. And since there are constant air currents and so on, it won't take long for that to happen, so it depends whether you include those in your theory. The usual example is a pendulum, which has a stable equilibrium when it's hanging down (if you push it gently, it'll just move back to where it was) and an unstable equilibrium when the bob is directly over the pivot (if you push that gently, it'll fall over). Here's a cool video on a related topic (I think I posted this before but maybe it'll be new to someone).

Related, but not directly, is does every shape (no matter how irregular) have a definite calculable centre? If I take a circle, the centre is easy to find. If I stretch it into an oval (extending the same amount at the top and bottom), it makes sense that the centre stays the same. But what if I just pinch it out a bit at the top (so it looks like a snowman), where does the centre move to then. It makes sense that it would move up very slightly, but where specifically? Why can we easily locate the centre of some shapes but not others? I guess it's to do with symmetry, but does a really irregular shape still have an absolute centre? Is it that if you cut a shape in a straight line through its centre, you should end up with two equal amounts of that shape, regardless of which direction you cut in?

What do you mean by centre? If you mean centre of gravity then yes, the centre certainly exists. Here's Wikipedia to explain it probably better than me. Potentially the relevant integral might be analytically insoluble so you wouldn't be able to say precisely where it was as a fraction or whatever, but you could find it numerically to any degree of accuracy that you wanted.

[Mark James]

I don't know if it has much to do with the topic but I've been trying to find, to no avail, a the name of a thing that was featured on an episode of QI that would always revert to its original position no matter what side it was placed on. It would do this by design only and not because it was weighted or anything. Anyone know what I'm talking about?

[Michael Wallace]

I don't know if it has much to do with the topic but I've been trying to find, to no avail, a the name of a thing that was featured on an episode of QI that would always revert to its original position no matter what side it was placed on. It would do this by design only and not because it was weighted or anything. Anyone know what I'm talking about?

Gömböc. (Which I found by googling your description of it.)

[Gavin Chipper]

I don't know if it has much to do with the topic but I've been trying to find, to no avail, a the name of a thing that was featured on an episode of QI that would always revert to its original position no matter what side it was placed on. It would do this by design only and not because it was weighted or anything. Anyone know what I'm talking about?

Gömböc. (Which I found by googling your description of it.)

That is sick. I want one.

[Gavin Chipper]

What do you mean by centre? If you mean centre of gravity then yes, the centre certainly exists. Here's Wikipedia to explain it probably better than me. Potentially the relevant integral might be analytically insoluble so you wouldn't be able to say precisely where it was as a fraction or whatever, but you could find it numerically to any degree of accuracy that you wanted.

I think a simple way of looking at it is to imagine dividing the shape up into lots of tiny cubes or some other shape (you'd get a good approximation of any object if your little cubes are small enough) and then just find the mean of all the centre points of the cubes. That's if we're just looking at shape-centre. Centre of mass can presumably just be done with different weights of cubes and do a weighted average.

[Gavin Chipper]

Is it that if you cut a shape in a straight line through its centre, you should end up with two equal amounts of that shape, regardless of which direction you cut in?

I'm not sure if this has been answered in a way that makes sense, but the answer is definitely "no". It's a bit like mean vs median I suppose. The mean of 1, 2, 3, 4 and 10 is 4 but the median is 3. So you could have ball bearings at different spacings along a line - 1cm, 2cm, 3cm, 4cm and 10cm. You can turn it into a "shape" by connecting the ball bearings with a really thin light wire. The "centre" is at 4cm but there are two ball bearings on one side and one on the other. It would only be the same where the median is the mean.

[Jon Corby]

I'm not sure if this has been answered in a way that makes sense, but the answer is definitely "no".

Yeah, I think I should have put this in a new thread because it's getting confused with the balancing thing. What I meant was, if I draw a simple shape, like a square, rectangle or circle, it's fairly easy to find the centre. If I pull the circle at the top to make a sort of egg shape, it makes sense to me (visually speaking) that the centre sort of "goes down" a bit, so it's slightly nearer the fat end of the egg than the thin end. So is this just the way I look at things, or is there actually a true "centre" to my egg shape? If so, does every 2D shape have a centre? And if so, where is it? (I think what I said earlier about being able to cut a straight line through the centre in any direction and split it into two parts of equal size might be correct, but I haven't really thought about enough and I'm not sure) If not, how can we determine which shapes have a centre, and which don't?

(Raccoon made an interesting point too that the centre might not even be within the bounds of the shape itself, which certainly appears to be true)

[Gavin Chipper]

Yeah, I think I should have put this in a new thread because it's getting confused with the balancing thing. What I meant was, if I draw a simple shape, like a square, rectangle or circle, it's fairly easy to find the centre. If I pull the circle at the top to make a sort of egg shape, it makes sense to me (visually speaking) that the centre sort of "goes down" a bit, so it's slightly nearer the fat end of the egg than the thin end. So is this just the way I look at things, or is there actually a true "centre" to my egg shape? If so, does every 2D shape have a centre?

There is always a centre of mass, and a centre of shape can simply be seen as the centre of mass for a shape with uniform density, which I think Raccoon said already. So the answer is "yes" - there is a true centre. But this would be the mean centre, like in my example with the ball bearings. The centre of mass would not divide an object into two parts of equal size in all cases.

And if so, where is it? (I think what I said earlier about being able to cut a straight line through the centre in any direction and split it into two parts of equal size might be correct, but I haven't really thought about enough and I'm not sure) If not, how can we determine which shapes have a centre, and which don't?

(Raccoon made an interesting point too that the centre might not even be within the bounds of the shape itself, which certainly appears to be true)

There is also the geometric median, which is probably more like what you seem to be after. In statistics, the median is the point that divides the numbers in half, so half of the numbers are one side of it and half are the other side. The geometric median is a sort of shape version of that. But it has to be defined in a different way.

As it happens, the median of a set of numbers is also the number that minimises the mean difference to the numbers in the set. So with 1, 2, 8, the median is 2. The mean to the numbers in the set is (1+0+6)/3, right? Try it with any number - 2 will be the lowest. With a set with an even number of numbers, the entire range between the two centre-points minimises this. So with 1, 2, 8, 10, any number from 2 to 8 gives the same mean difference to the numbers. Normally people take the median as halfway between the two most central numbers, but this is arbitrary anyway. (The mean minimises the squared difference.)

The geometric median is defined in this mean distance minimising way, because doing the "half the area/volume one side and half the other" doesn't work. Basically, because I think there isn't always a point that does this. I'll think up an example. The geometric median is probably the closest you'll get to your sort of centre.

[Brian Moore]

I need a pretty picture in this thread while I read it....

I'd imagine most people would take it as the centre of mass of a uniform plane lamina of that shape (which is something I remember having to do as part of an OU course several years ago). So imagine balancing it on a pencil or something (although obviously that doesn't really work if the centre of mass is then not inside the shape, such as for a ring).

I was thinking about this idea exactly a couple of days ago as a way of finding 'the centre of Devon', as I was cycling thorough one place that claims to be that (but probably isn't, but is just trying to flog a few more cream teas.) So it'll be a cardboard cut-out of Devon, and a pencil then.

[Gavin Chipper]

Corby, I've been drawing some lines in an equilateral triangle. Put points a third of the way along each line and basically join them all up. It seems to me that you end up with nine smaller equilateral triangles. There is a definite centre to the original triangle, and you'll see that five of the nine smaller triangles (which are all the same size) are below it and four are above. So I don't think your centre would work here.

Edit - The centre is the point where the three lines of symmetry cross. But if you decide to your own centre a bit lower down so that you have equal area above and below, you'll move it off two of the three lines of symmetry, so you won't have half the area each side of these.

[Gavin Chipper]

I was thinking about this idea exactly a couple of days ago as a way of finding 'the centre of Devon', as I was cycling thorough one place that claims to be that (but probably isn't, but is just trying to flog a few more cream teas.) So it'll be a cardboard cut-out of Devon, and a pencil then.

Of course, you open a whole new can of worms here. The centre of inhabited places can also be defined with population, so areas with a higher population density have more effect on the "average" point.

[Brian Moore]

I was thinking about this idea exactly a couple of days ago as a way of finding 'the centre of Devon', as I was cycling thorough one place that claims to be that (but probably isn't, but is just trying to flog a few more cream teas.) So it'll be a cardboard cut-out of Devon, and a pencil then.

Of course, you open a whole new can of worms here. The centre of inhabited places can also be defined with population, so areas with a higher population density have more effect on the "average" point.

Right, so to include the population density, it'll have to be hundreds&thousands stuck on in appropriate ratios. This could end up looking like a Blue Peter Christmas decoration at this rate.

My unscientific feeling is that for irregular 2D shapes (in the raccoonish uniform plane lamina sense) the 'centre' and 'centre of mass' (or gravity? - A level physics was a long time ago) will be the same thing - in other words, there will be equal area either side of any line bisecting the plane. (I think.) EDIT - which is pretty much what Gavin said earlier, I guess.

[Charlie Reams]

is there actually a true "centre" to my egg shape?

You haven't answered my earlier question about what sort of centre you're talking about. There are many ways of defining a centre even for something as simple as a triangle. Last time I wrote a long post explaining something you'd asked about you tl;dr'd me, so make sure you do that again you utterly ungrateful stupid useless cunt.

[Jon Corby]

is there actually a true "centre" to my egg shape?

You haven't answered my earlier question about what sort of centre you're talking about. There are many ways of defining a centre even for something as simple as a triangle. Last time I wrote a long post explaining something you'd asked about you tl;dr'd me, so make sure you do that again you utterly ungrateful stupid useless cunt.

I think the answer to the question appears to be "no" then. That's a cool link. It looks like I was thinking of centroids.

I was going to then move on to "centring one shape within another". Sometimes I might want to "centre" a shape (possibly even in a complex logo) in another shape (perhaps a coloured circle or square background), and it's always just a case of "shifting it around slightly and seeing what looks best" and I was wondering whether there was a true way, and then I wondered if every shape had a true centre, you might just put the centre of one on the centre of the other.

But I won't bother now, since there isn't

(PS. Are you referring to your Angry Birds defence for the tl;dr thing? I can't imagine I'd ask a question like this and then be so rude at a sensible answer)

[Jon Corby]

PPS. Having read a bit more, it seems that I'm thinking of the "centroid" (when I'm talking about the centre of a shape), and it does indeed always produce two pieces of equal size when cut in a straight line across the centroid. I has a happy.

[Liam Tiernan]

Googling "centroid" I found this:

The center of mass of a body does not generally coincide with its geometric center, and this property can be exploited...... When high jumpers perform a "Fosbury Flop", they bend their body in such a way that it is possible for the jumper to clear the bar while his or her center of mass does not.

which sounded a bit odd to me at first, but reading further it seems that the trick to the Fosbury flop is that the jumpers body manoeuvres in such a way that the center of mass passes underneath the bar, meaning less lift is required. Clever, eh?

[Gavin Chipper]

PPS. Having read a bit more, it seems that I'm thinking of the "centroid" (when I'm talking about the centre of a shape), and it does indeed always produce two pieces of equal size when cut in a straight line across the centroid. I has a happy.

You shouldn't have a happy. If you'd read and fully understood all the posts on here, your worldview would have been shattered.

The centroid is the same as the centre of mass for a uniformly dense object, and a line going through it definitely does not always produce two pieces of equal size. See my other post:

The mean of 1, 2, 3, 4 and 10 is 4 but the median is 3. So you could have ball bearings at different spacings along a line - 1cm, 2cm, 3cm, 4cm and 10cm. You can turn it into a "shape" by connecting the ball bearings with a really thin light wire. The "centre" is at 4cm but there are two ball bearings on one side and one on the other. It would only be the same where the median is the mean.

The centroid and centre of mass would be at the ball bearing 4cm along. A line through it would have three ball bearings one side and one the other. (Think of these ball bearings as like points masses). So the centroid does not do what you want it to.

But there is still the median at 3cm which would divide the object into two equal sizes. But as I pointed out in another post (did you read it?) it is not always possible to have such a convenient median for a shape. Read what I wrote about the geometric median and equilateral triangles. With an equilateral triangle there is no point that you can draw a line through it at any angle and make two bits of equal size.

[Jon Corby]

You shouldn't have a happy. If you'd read and fully understood all the posts on here, your worldview would have been shattered.

Oh yeah. Fuck. I'm going to stop thinking about this now, I'm clearly out of my depth. Thanks to all those that have contributed, even if I clearly didn't understand much of it (just to stop Charlie throwing another strop).

[Gavin Chipper]

So if you got a shape and found the point along the x axis that half of the shape was to the left of and half to the right, and did the same for the y axis, and used these coordinates as your centre, you'd find for most shapes that this point would be in a different place in the shape depending on its orientation. Which is a bit weird when you think about it. But then you could rotate the object round and draw a line round where the centre points are. Because you end up where you started, presumably you'd end up with an enclosed area. So you could define your Corby-centre as the centre of this area. But then if it's a funny shape you've still got the same problem again. But as this would be a smaller shape, you can repeat the process and you'd end up with an even smaller area. So your Corby-centre can be the limit of this.