Maths/Physics trolls

[austere]

It's pretty clever, see if you can figure out what's going on.

[Ex-Cyber]

Kinda flying by the seat of my pants here rather than properly analyzing, but I'd guess that it converges to an octagon in which the circle is inscribed rather than converging to the circle itself.

[Magic Knight]

No, it converges to the same square essentially, just turned on its edge. Like this: <>. The circumference of this circle is equal to pi. The perimeter of the fractal shape described will always be 4, and if you "zoomed in" to any part of the fractal you would see that the difference in length between the two points on any two corners and the arc between them is constant, in other words, the difference between them would never approach 0.

It's a bit tricky to explain in words.

[E. Randy Dupre]

There's a fairly basic error of logic there. Extending this idea to an infinite number of smaller corners doesn't make a curve. It makes an infinite number of smaller corners.

Or, to put it another way, a circle is a one-sided shape. It's not a circle if it has corners, regardless of how many there are or how small they are.

[Ex-Cyber]

There's a fairly basic error of logic there. Extending this idea to an infinite number of smaller corners doesn't make a curve. It makes an infinite number of smaller corners.

Or, to put it another way, a circle is a one-sided shape. It's not a circle if it has corners, regardless of how many there are or how small they are.

That's true in terms of the usual definition of a circle, but from an analytic standpoint a circle can be described as the limit of an N-sided regular polygon as N approaches infinity (holding the distance from the center to a corner constant, which becomes the radius). The given figure is clearly not a regular polygon, though.

No, it converges to the same square essentially, just turned on its edge. Like this: <>. The circumference of this circle is equal to pi. The perimeter of the fractal shape described will always be 4, and if you "zoomed in" to any part of the fractal you would see that the difference in length between the two points on any two corners and the arc between them is constant, in other words, the difference between them would never approach 0.

It's a bit tricky to explain in words.

It sounds like you're saying that it never really converges, not that it converges to a square. I'm probably misinterpreting something, though.

[Magic Knight]

No, it converges to the same square essentially, just turned on its edge. Like this: <>. The circumference of this circle is equal to pi. The perimeter of the fractal shape described will always be 4, and if you "zoomed in" to any part of the fractal you would see that the difference in length between the two points on any two corners and the arc between them is constant, in other words, the difference between them would never approach 0.

It's a bit tricky to explain in words.

It sounds like you're saying that it never really converges, not that it converges to a square. I'm probably misinterpreting something, though.

Yes, it never really converges, but the shape would resemble a square.

Basically you can break down right angles as much as you like, they'll never become a curve. In mathematics a curve is simply not the same thing as a series of corners, no matter how small they are.

This guy describes it far better than I did:
http://mathisdermaler.wordpress.com/201 ... %E2%80%9D/

[jonny5]

this thread made me dizzy

[PC Engine Fan X!]

For the mathematically not-so inclined, it sure is. It's quite heady stuff indeed.

PC Engine Fan X! ^_~

[Ganelon]

Anybody should be able to understand this; no mathematics necessary. Just notice that when you're cutting the area, you're not cutting the perimeter at all.

The easiest way to see it is to compare the dotted lines of the last iteration's corner to the solid lines of the newly retracted corner. Top matches bottom. Left matches right. No perimeter is lost; it's just the lines are inverted. You can keep cutting corners to infinity and that same property holds. It's just the 5th diagram is drawn in a misleading way.

Nice logic puzzle though.

[KindGrind]

There are some other very nice/funny "troll physics" and "troll math" going around.

A favourite of mine: http://redux.com/f/1760540/Troll-Physic ... t-downhill

Many more involve magnets and windmills to generate infinite energy and stuff. Those are intended to "troll" your science/math teacher and are pretty funny.

[neorichieb1971]

If you zoom into infinity you will always see that circle one is not the same as circle two.

Perception is rarely accurate.

[Ruldra]

Alright, I want you guys to explain this one:

[TrevHead (TVR)]

Im crap at maths but all this talk about the 1st puzzle makes me want to go and look at some mandlebrots

[neorichieb1971]

The stick doesn't travel though does it?

Thats like saying the road made it to the end before the fastest car in the world.

[Magic Knight]

Just read this: http://math.ucr.edu/home/baez/physics/R ... FTL.html#4

[Udderdude]

Your physics troll-fu is weak.

[VorpalEdge]

Alright, I want you guys to explain this one:

Nope! Kinetic motion is not instantaneous. If you take a stick and thrust it forwards, the thrust would 'travel' down the stick at the speed of sound through the stick.

Your physics troll-fu is weak.

Nope! Even if there's lots of empty space, there's still electrical repulsion and shit going on. Isn't even theoretically possible (at least not with that logic; supposedly quantum mechanics fucks this up but I have no idea how so w/e).

Dunno about the first one though.

[austere]

I'd guess that it converges to an octagon

No, it converges to the same square essentially, just turned on its edge. Like this: <>.

Nope.

It's not a circle if it has corners, regardless of how many there are or how small they are.

Yes, however that has no bearing on the argument of iterating the steps until you reach infinitesimal increments. As a counter-example to your argument, taking tangential line segments and linking them up to construct a circle then taking the limit nets you a continuous and differentiable "circle". (Big hint here btw!)

In mathematics a curve is simply not the same thing as a series of corners, no matter how small they are.

It's not the "same thing" (i.e. not all curves are a series of corners), however a series of corners can be a curve. See: a parametrised square.

This guy describes it far better than I did:

Cheater. Anyway, you will have to demonstrate that you actually understood what this dude said. Unfortunately, his proof is flawed (not wrong, just lacking in rigor):

However by assumption since B is a straight line it has the shortest possible length, and since R is a curve distinct from B it must have length greater than B. We have two contradictory statements: l(B) = l(R) and l(B) < l(R).

Two distinct curves with the same starting point and ending point can indeed have the same length, even if one of the curves is a straight line. It all depends on your metric. Another big hint here.

Just notice that when you're cutting the area, you're not cutting the perimeter at all. ...

You've described why the shape converges to a length of 4, not why it isn't correct. You are correct that the 5th diagram is misleading, try and figure out why.

Shall I give it away? I've yet to see anyone on the internet answer this in a simple and yet rigorously correct way. Also, perhaps we should change the title of the thread to "Maths/Physics trolls" ... keep them coming they're very amusing.

[Udderdude]

ROUND 2 GO

[ED-057]

Ganelon wrote:

Just notice that when you're cutting the area, you're not cutting the perimeter at all. ...

You've described why the shape converges to a length of 4, not why it isn't correct. You are correct that the 5th diagram is misleading, try and figure out why.

Maybe it converges to a circle that is larger than d=1? I'm not clear on the "repeat to infinity" part. Exactly what steps are being repeated? In other words, when a corner is "removed" what determines where the new corner is placed?

[ED-057]

Here is what I would call a (very successful) physics troll:
http://tech.slashdot.org/story/10/11/06 ... wered-Cart

[austere]

Maybe it converges to a circle that is larger than d=1?

Nope, it converges to a circle that has a diameter of 1.

I still don't get why people are surprised at the wind cart, it's just a simple demonstration of mechanical resonance, the end speed will be inversely proportional to the losses in the system. Start condition is pretty simple since static friction is greater than kinetic friction in this system. Woopdido.

[neorichieb1971]

Plane is on an imaginery escalator. As the plane thrusts the escalator counters the thrust with the exact speed in the opposite direction. If the imaginery escalator always counters the thrust exactly will the plane take off?

[ED-057]

Nope, it converges to a circle that has a diameter of 1.

OK... but all those line segments (no matter how small) that make up the pseudo-circle are necessarily going to originate on the circle and extend to someplace outside the circle right? (never inside the circle?) That on its own means the jagged version of the path will always be longer.

[austere]

The key is to realise that if we take his successive approximations of the circle and extend it to infinity, you will end up with all the points on a circle, nothing more nothing less. I think I may as well give it away so that we can move on to some new ones.

Theorem: The fifth and sixth panel of the first image are misleading.

Proof:
First we should start with the formal (but somewhat simplified for our purposes) definition of the length of a simple (i.e. it never crosses itself) closed (i.e. the end reaches the beginning) curve. Lets say omega denotes a parametrisation of a curve, such that it omega:[0,1) -> R^2. Now let Y be the set of all finite monotonically strictly increasing sequences on [0,1). Then the length is given by:

Length(omega) = sup_Y {sum}

Where distance() is defined by the distance between two points. Remember this one, we'll get back to it soon. So, for those not mathematically inclined (and even then, not familiar with "pure mathematics") what does this mean? It means that we take ALL subdivisions of a curve, that is, all finite sets of points on a curve, and string together a bunch of distances to compute the length of a curve. Then, the subdivision which nets you the maximum length (that is what sup, i.e. supremum) is the length of the curve.

Now, taking this definition and looking at each iteration of the troll curve and ignoring the infinite case for now, the minimal but supremum (i.e. the one that nets the maximum length) element of Y is actually the set containing all corners of the troll curve. Indeed, if you do the maths, at all points it will have a distance of 4. Thus, as you step through the iterations, it will tend to 4.

Consider the circle, however, the length will approach the conventional value of pi as you increase the length and density of the set Y. See: Archimedes (note, there is no circular (lol) reasoning here, just a reference to a previous demonstration).

So, what gives? The trick is actually in the final step, that is to say, when you "reach" the infinite iteration, the parametrisation of the troll curve will have all the points on the circle, nothing more nothing less. Now, even though we may have established an inductive step such that every finite iteration of the troll curve has a length 4, this is not a sufficient demonstration for the infinite iteration. Indeed, since the image claims the circle and the troll curve are the same curve and each point on the troll curve corresponds to the exact same point on a circle, the supreme will be the same as that of a circle.

QED□

There you go, no need for hand waving, differential/integral calculus, hallucination, mockery, fractals (this isn't a fractal btw) and so on. Just a little bit of logic and rigorous use of definitions. They are formulated precisely to avoid silly situations like this. Funnily enough, we can go about it another way. All this time we have assumed the distance between two points is given by the euclidian metric, that is using pythagoras theorem on the right angle triangle formed by an origin-aligned axis and the points in question. That is distance(a,b) = sqrt((bx-ax)^2+(by-ay)^2). What if we used something different? How about the so called Manhattan distance, distance(a,b) = |bx-ax|+|by-ay|? Well, in the metric space formed by this metric, it turns out that BOTH the circle and the troll curve have a length of 4. That is to say, 'pi' is equal to 4 in the manhattan metric space. Anyway, that's just an interesting aside.

So yeah, feel free to pick holes in this proof, ask for a clarification, more in-depth explanation etc. I will be away for a while but I'll try to answer questions when I have the time. Thanks for playing.

[Ed Oscuro]

Plane is on an imaginery escalator. As the plane thrusts the escalator counters the thrust with the exact speed in the opposite direction. If the imaginery escalator always counters the thrust exactly will the plane take off?

horrible

[Specineff]

^^^ NeoRichie: Mythbusters already proved the plane will take off. Aerodynamics FTW.

In regards to the one about aligning the empty space between atoms to pass through solid matter: Don't Fsck with atomic bonds and Heisenberg's Principle of Uncertainty. In order to align atoms to pass through (Provided you could ascertain the position of every single electron that allows atoms to form bonds at a specific moment in order to achieve such alignment) one would have to be rent (rendered?) apart at the molecular level.

If I'm mistaken, blame it on not being able to sleep at this time of the night. (I hate my body)

[Ed Oscuro]

Also, if that were a valid method for determining the perimeter of a circle, then the hypotenuse of a 45-45-90 right triangle would just be adding together the two shorter sides!

Looking at it that way, I think it starts to become obvious why the number is too large. In the triangle example, you would want the corners to run through the middle of the hypotenuse; compare this with Archimedes' Pi algorithm which alternates polygons inside and outside the circle. In the bad hypotenuse calculation, the error is too large because the approximation is all fit to one side - biasing it too large. The same when approximating Pi. (You can also compare to method #4 here.) Of course I know this does not seem to be admitted by the language of the bad example; it's not that you could assert the starting square and the iterations could be fit inside the circle, in some kind of extraplanar pocket - the result will always be biased too large - but what's troubling is that the steps seem to work (even though removing some corners do not take a square chunk away, the perimeter from removing a rectangular chunk still doesn't alter the perimeter of the square being approximated to the circle). I will be interested to see an answer for this one...

^^^ NeoRichie: Mythbusters already proved the plane will take off. Aerodynamics FTW.

From the plane's perspective, all that changes is that the air coming in the front is a bit less forceful - but assuming the engine isn't so powerful that it can evacuate the space in front of it (like a nuclear weapon or some other very large bomb!) it will again still take off.

And it's comforting to know that running a conveyor belt (or escalator) in reverse against a plane coming in to land will also not bring it immediately to a stop.

[Ex-Cyber]

He clearly said that it's an imaginary escalator. An imaginary escalator can't stop anything, so the plane obviously takes off.

More generally, the problem with "plane on a treadmill" scenarios is that the engines/propeller push/pull the air backward (Newton's third law => plane is pushed/pulled forward), while the conveyor belt pushes/pulls on the bottoms of the wheels (which, um, rotate). So where is the force opposing the thrust of the engines?

Also, the "sit on an electron one" is great, because it's arguably true (though you don't need to sit on an electron or close your eyes; it happens anyway). Hey, no one (or at least no one worth listening to) said that "random" implies a uniform distribution.

[neorichieb1971]

Well I disagree the plane will take off. Its the wings which make the plane take off and there is no wind going under them. The more likely scenario is that the jet engines would burn out.

The only case where the plane will take off in such a scenario is if the jets were quicker than the escalator and the plane went forward anyway.

In reverse, I agree that if a 250mph wind was hitting the plane head on, the plane would take off with no thrust at all.