Congruent potato paths

[bonanova]

This guy walks up to you and says, give me any two potatoes,
and I bet you that I can draw a closed curve on one of them,
a curve that divides the potato in half, and then draw
exactly the same curve on the other one.

Do you take the bet?

By "same curve" he means it has exactly the same
three-dimensional size and shape.

[grey cells]

the curve divide the second potato also into half?

[bonanova]

the curve divide the second potato also into half?

Nope.

[Guest]

I don't remember much topology. Too much dope at uni. But my instinct tells me that if he finds the shortest curve to bisect the smallest potato, then he will be able to cut open the biggest potato and draw the same curve on in. So I won't take the bet.

[kingofpain]

To Steve:

I am assuming he HAS to draw on the surface of both potatoes (and no cutting).

To Bona:

Does he pick which potato the first curve is drawn on?

Cheers!

--

Vig

[bonanova]

To Steve:

I am assuming he HAS to draw on the surface of both potatoes (and no cutting).

To Bona:

Does he pick which potato the first curve is drawn on?

Cheers!

--

Vig

Surface.

No cuts.

Yes, he picks which one to draw the first curve.

All good questions...

[Guest]

Would I be right in thinking:

1) The 1st potato is divided into halves of equal size (as opposed to just 2 pieces)

2) The curve has to be drawn on the surface of the 2nd potato (not on the inside by cutting it 1st)

If so I'm pretty sure you can't do it, unless you resort to this sort of thing:

The thing that divides a potato is really a 2-dimensional plane or a warped version of one, and the curve is merely the set of points where that intersects with the potato surface. So I suppose you could create some sort of bubble shape inside the potato that only meets the surface at some tiny point, but includes half the potato. Then you only have to replicate that tiny point on the other potato...

...but I don't think that's what you're driving at. If not maybe we need to clarify what you mean by a curve "dividing a potato in half".

[kingofpain]

I wouldn't take the bet. I think it is always possible to find a surface on one of the potatoes that intersects the other potato dividing it in half. Imagine a superposition of the 3D images of two potatoes. The larger one can have the smaller one intersecting it lengthwise across the large one's side. We then simply move the smaller one back and forth so that the volume outside is half the potato's volume and voila, we have the intersection as the curve which can be drawn on both potatoes. It would be interesting to design 2 potatoes for which this is not possible, but I am not sure if that is possible mathematically.

[Guest]

It may be playing with words, but he says "Draw the curve on the second potato", not specifying that the same curve would cut the potato in half.

Suppose you chose two potatoes, one extraordinarily large and one extraordinarily small. Something like a walnut compared to a softball. No matter what curve you draw on the outside of, say, the softball-sized one, it would be impossible to draw it on the walnut-sized one, since the walnut would always pass through any curve large enough to cut the softball in half. However, if he chose the walnut-sized one, cut it in half anywhere, then cut a flat surface on the softball-sized one and traced one half of the walnut-sized one.. that would be "drawing the exact same curve".

Again, probably just playing with words ^.^;

[Guest]

Since the curve is on the surface, I assume that "cutting a potato in half" means that the surface area is split in two, rather than the volume.

In that case ...

Take the smaller potato and imagine a surface matching the larger potato intersecting with it.

Consider a function calculating the (amount of the smaller potato's skin to the right of this surface) - (amount of the skin to the left of the surface).

Assuming smooth potato skins (no worm holes, no cuts from the fork) the function will be a continuous function of the position of the potato

At one extreme, the small potato is entirely to the right of this surface and so the function is positive.

At the opposite extreme, the small potato is on the left and so the function is negative.

At some position, the function must be 0 (Intermediate Value Theorem, or common sense as it's also known). At this point the smaller potato's surface is cut in half by a curve which is drawn on the larger potato.

Theory says that this is possible so I shouldn't take the bet.

Practice says that this solution requires passing one potato through another so it's not going to be possible in real life. Therefore I do take the bet. I can always quibble about it being "exactly" the same curve.

Edited June 10, 2008 by alchymist

[Guest]

I wouldn't take the bet. I think it is always possible to find a surface on one of the potatoes that intersects the other potato dividing it in half. Imagine a superposition of the 3D images of two potatoes. The larger one can have the smaller one intersecting it lengthwise across the large one's side. We then simply move the smaller one back and forth so that the volume outside is half the potato's volume and voila, we have the intersection as the curve which can be drawn on both potatoes. It would be interesting to design 2 potatoes for which this is not possible, but I am not sure if that is possible mathematically.

I think I agree with kingofpain. The only possible exception I can think of would be if the smaller one you were trying to cut in half were a toroid, and the larger one were not big enough to stretch across the whole shape to cut it in two places. But then again, I've never seen a donut shaped potato so even if that is a topographic exception it's not really applicable.

[Guest]

Since the curve is on the surface, I assume that "cutting a potato in half" means that the surface area is split in two, rather than the volume.

In that case ...

Take the smaller potato and imagine a surface matching the larger potato intersecting with it.

Consider a function calculating the (amount of the smaller potato's skin to the right of this surface) - (amount of the skin to the left of the surface).

Assuming smooth potato skins (no worm holes, no cuts from the fork) the function will be a continuous function of the position of the potato

At one extreme, the small potato is entirely to the right of this surface and so the function is positive.

At the opposite extreme, the small potato is on the left and so the function is negative.

At some position, the function must be 0 (Intermediate Value Theorem, or common sense as it's also known). At this point the smaller potato's surface is cut in half by a curve which is drawn on the larger potato.

Theory says that this is possible so I shouldn't take the bet.

Practice says that this solution requires passing one potato through another so it's not going to be possible in real life. Therefore I do take the bet. I can always quibble about it being "exactly" the same curve.

Ah yes but what about those long helical potatoes they use to make curly fries? Intersect that with, say, a long straight potato, and I dare say the intersection would never amount to half.

[Guest]

Ah yes but what about those long helical potatoes they use to make curly fries? Intersect that with, say, a long straight potato, and I dare say the intersection would never amount to half.

It doesn't matter if half of the potatoes thesmselves intersect, just that the amount of smaller potato skin to the left and to the right of the larger potato surface is equal. Specifically you only need to intersect a single loop of the helix to make that possible

However, you've made me realise that there is one weak point in my theory. It must be possible to draw the cross-section of the 'smaller' potato completely within the cross-section of the 'larger' potato. A cross-shaped potato and a spherical potato larger than the intersection of the arms but smaller than the length of the arms would make that impossible.

Back to the topological drawing board.

[Guest]

Maybe if the first drawn potato is small enough, you could draw the line so that you can place that potato on the bigger one and trace it?

[Guest]

It is always possible, provided you start with the smaller potato. The intersection of any two closed surfaces (or "potatoes") is bounded by a finite number of closed curves - provided the surfaces of the potatoes themselves are continuous (and they probably also have to be integrable... How should I know? - ask a mathematician!!).

Once you have found a surface intersection (and any one will do), you can do what you want with the interior of the potato. It doesn't say you have to make a planar surface with your curve, or even a "tight" surface (i.e. minimum area). You can meander around inside the potato until you have exactly half on each side of the surface. I think, therefore, there are an infinite number of solutions for any two potatoes.

Now if you got your potatoes from Fractal Farms, you may have a problem...

D

[bonanova]

Three nevers:

1. Never eat at a place called Mom's
   
2. Never play cards with a guy named Doc
   
3. Never bet against a guy with two potatoes.
   

Yeah, you all basically have it http://brainden.com/forum/uploads/emoticons/default_wink.png' alt=';)'> KoP may have been the first.

Imagine the ghosts of the potatoes.

"Insert" the smaller one into the other until half its volume is inside.

The closed curve intersection of the surfaces lies on both potatoes.

[Guest]

Three nevers:

1. Never eat at a place called Mom's
   
2. Never play cards with a guy named Doc
   
3. Never bet against a guy with two potatoes.
   

Yeah, you all basically have it http://brainden.com/forum/uploads/emoticons/default_wink.png' alt=';)'> KoP may have been the first.

Imagine the ghosts of the potatoes.

"Insert" the smaller one into the other until half its volume is inside.

The closed curve intersection of the surfaces lies on both potatoes.

I beg to differ. That only works if there is an intersection that could swallow half of one of the potatoes. If they are of sufficiently wierd shapes (say, a helix and a very long thin one) then no such intersection exists. You just need to employ some creative farming techniques!

"It is a mistake to think you can solve any major problems just with potatoes."

Douglas Adams

[Guest]

This guy walks up to you and says, give me any two potatoes,

and I bet you that I can draw a closed curve on one of them,

a curve that divides the potato in half, and then draw

exactly the same curve on the other one.

Do you take the bet?

By "same curve" he means it has exactly the same

three-dimensional size and shape.

Question is, working with two ethereal potatoes, can you always push one into another such that at least one of the potatoes is bisected?

This is a mathematical given if I recall, no matter the shape of the relative potatoes it is always possible to have them intersect bisecting one or the other and sometimes both. Note that it may be possible to bisect the larger potato with the smaller one given strange enough shapes.

But if i'm going to be a stickler about it I would still take the bet as these are physical potatoes that cannot be pushed into each other so any attempt to do this manually will almost certainly have some errors in it. IE this is mathematically possible but in practice extremely difficult to do.

[Guest]

Question is, working with two ethereal potatoes, can you always push one into another such that at least one of the potatoes is bisected?

This is a mathematical given if I recall, no matter the shape of the relative potatoes it is always possible to have them intersect bisecting one or the other and sometimes both. Note that it may be possible to bisect the larger potato with the smaller one given strange enough shapes.

But if i'm going to be a stickler about it I would still take the bet as these are physical potatoes that cannot be pushed into each other so any attempt to do this manually will almost certainly have some errors in it. IE this is mathematically possible but in practice extremely difficult to do.

This has really got me wondering. I doubt that it is a mathematical given, though my earlier example of the helix and the long cylinder may be flawed. Although the intersection between the two volumes would never be half the volume of either, you could use the thicker of the two shapes to create an intersection which you slide along the other shape until it bisects.

On the other hand, if you used something like a torus (major radius 10, minor radius 1) and a sphere (radius 5) then it certainly wouldn't work. Maybe the principle works but only if you stick to solids with the topology of a sphere. But I'm far from convinced of that. Trying to think of a counterexample...

[Guest]

Aha!!! I think I have a counterexample! Lets say one "potato" consists of a small sphere (radius 3) with three long arms (say radius 2, length 100), and the other is a sphere radius 10. The one with the arms cannot bisect the sphere. The sphere's surface can divide the other into 2 parts. But most of the volume is in the arms, and one part will always have 1 arm and the other part 2 arms so it will always be an uneven bisection.

[Guest]

This has really got me wondering. I doubt that it is a mathematical given, though my earlier example of the helix and the long cylinder may be flawed. Although the intersection between the two volumes would never be half the volume of either, you could use the thicker of the two shapes to create an intersection which you slide along the other shape until it bisects.

On the other hand, if you used something like a torus (major radius 10, minor radius 1) and a sphere (radius 5) then it certainly wouldn't work. Maybe the principle works but only if you stick to solids with the topology of a sphere. But I'm far from convinced of that. Trying to think of a counterexample...

Yes that's right. So it wouldn't work with a torus and a sphere where the sphere is smaller than the major radius of the torus. The solution of course is to ban the torus from geometry altogether. If a shape cannot conform to such simple mathematical concepts then it doesn't deserve to exist period, as that's just... unamerican.

Seriously though, I suppose it's possible to have a torial potato, so make sure you give him one if you take the bet.

[Guest]

I'm pretty sure "The Toroidal Potatoes" would be an awesome name for a band.

[Guest]

Aha!!! I think I have a counterexample! Lets say one "potato" consists of a small sphere (radius 3) with three long arms (say radius 2, length 100), and the other is a sphere radius 10. The one with the arms cannot bisect the sphere. The sphere's surface can divide the other into 2 parts. But most of the volume is in the arms, and one part will always have 1 arm and the other part 2 arms so it will always be an uneven bisection.

Sounds like a more concrete example of my cross-shaped vs spherical potato. Still, I like your potato. Excellent for chips (fries if you're across the pond).

[Guest]

Yes that's right. So it wouldn't work with a torus and a sphere where the sphere is smaller than the major radius of the torus. The solution of course is to ban the torus from geometry altogether. If a shape cannot conform to such simple mathematical concepts then it doesn't deserve to exist period, as that's just... unamerican.

Seriously though, I suppose it's possible to have a torial potato, so make sure you give him one if you take the bet.

Surely a torus makes geometry a hole lot more interesting?

[kingofpain]

I'm pretty sure "The Toroidal Potatoes" would be an awesome name for a band.

Potatoroid or Potatorus perhaps?

Cheers!

--

Vig

[Guest]

Sounds like a more concrete example of my cross-shaped vs spherical potato. Still, I like your potato. Excellent for chips (fries if you're across the pond).

Refined problem

If you can only offer convex potatoes (no indentations or holes), can the man win his bet?

I'm going to be offline for a while, so here's my expected answer

A tetrahedral potato and a spherical one whose cross-section is larger than the in-circle but smaller than the circumcircle of one face of the tetrahedron have the same problem. The original problem requires that the man can draw *a* closed curve on each potato but the intersection of the sphere and the tetrahedron will break into multiple curves.