Making Waves

[Qaanol]

You find yourself face to face with an Evil Mathematician™. Behind him there is a treasure chest, and an infinite cat hotel with a kitten in every plush-lined cubbyhole. There is also a little spray bottle of water pointing toward every kitten. The Evil Mathematician tells you, “I am about to pose a puzzle. If you have not solved it in 12 minutes, then I will have all of these kitten spritzed once with water, which will mildly bother each of them for a few seconds”

You stagger back in shock, “But—but that would be—how can you—you’re threatening to do an infinite amount of harm to kittens! That’s—”

The mathematician finishes your sentence for you, “Infinitely evil. I know. By the way, you only have 11½ minutes left to solve my puzzle.”

“Well get on with it then, what’s the puzzle?”

The mathematician smiles evilly and speaks slowly, “You must draw a perfect sine wave, using only the tools in this treasure chest. And now you only have 11 minutes remaining.”

“Wait a second,” you object, “we live in a world of quantum indeterminacy. How perfect is ‘perfect’ for these purposes?”

“Perfect enough that the limitations come from the physical characteristics of the drawing utensils and your own fine motor control, not from any flaws in your method. You’re down to 10½ minutes.”

You open the treasure chest and find the following items:

1 sheet of posterboard, white
1 pencil, number 2
1 compass, the circle-drawing kind
1 compass, the magnetic kind
1 slab of plywood, smooth
1 circular saw, cordless
2 sawhorses
1 straight-edge, unmarked
1 gallon of milk, expired
4 gallons of paint, green
1 paintbrush, wide
1 paintbrush, narrow
1 teddy bear, missing a button
1 bag of beef jerky, half eaten
1 laser pointer, red
1 bag of iron filings
1 paint roller
1 paint roller tray
1 bucket of water, 4 liters
1 stainless steel bowl, large
8 apples, Macoun
1 jackknife
3 dollar bills, Canadian
1 $20 bill, Monopoly
1 box of thumbtacks
1 ball of string
1 pair of scissors

The Evil Mathematician glances at the kittens then taps his watch, saying “Ten minutes left. How’s that sine wave coming?”

[jestingrabbit]

Preliminary thoughts.

Spoiler:

You obviously want to do something like this. The question is "how?"

[WarDaft]

First, we stab the Evil Wizard Mathematician with the jack knife. (It won't save the kittens [or else this is a terrible logic puzzle...] but he deserves it, and the puzzle doesn't say we can't)
The rest of the steps, being relevant, will be spoiled. Like the milk.

Spoiler:

Since we only need draw a curve as accurate as our drawing utensils and motor skills allow, all we need is a method to stochastically approximate a sine curve with the box of thumbtacks. Then our method is perfect, it is only the finiteness of the thumbtacks preventing us from achieving a perfect sine wave. Either that, or we must draw one as accurately as any possible combination of these tools allow, and that does not quite seem to be the intent.

Roll the posterboard into a cylinder, use the spoiled milk as a sort of make-shift glue to hold it together.

Dunk the cordless circular saw in the green paint, set it aside. You'll need it later.

Jab a thumbtack into the plywood somewhere at random.

Eat some beef jerky (you'll need the energy for this next bit!)

Take the stainless steel bowl, and dig a notch into the rim with the jack knife.

Woah... that was tough! Also, the jack knife rather looks like it won't be able to do much cutting now. Good thing you already
stabbed the wizard mathematician, eh?

Using the circle drawing compass and straight edge, draw lines parallel to the edges of the plywood through your thumbtack.

Starting with the notch in the bowl at the edge of one of the lines you drew, roll the bowl to the thumbtack, turn it 90 degrees, and then roll it to the edge. Take note of where on the bowl the edge of the plywood is touching.

Eat one of the apples, because, really, what else are they good for?

On the floor, using your circle drawing compass and straightedge, draw a graph to the best of your abilities.

Use the magnetic compass to figure out where north is, in case you get lost.

Stop and think for a moment.

Have a drink from the bucket of water, since you've been doing so much work and you're thirsty.

Now you silly, all that puttering around you did and you've knocked over the bowl, and you lost your measurement, you'll have to start again.

Duplicate your graph using your circle drawing compass and straightedge. (Yes, on the floor again.)

Drop a thumb-tack on the duplicate graph.

Draw horizontal and vertical lines through the thumbtack with your circle drawing compass.

Roll the bowl along the line as if it were a wheel, starting with the notch at the edge of your graph and stopping when it's on the thumbtack. Lie it down when you get there so that it's touching but not crossing the line. (You don't want crossed lines, they hold grudges for a very long time)

Make a mark at the notch.

Using your circle drawing compass and straightedge, copy the horizontal line form the edge of your duplicate graph that you started rolling from to the thumbtack, to the bottom of your original graph.

Copy a vertical line from your duplicate graph (again, using your circle drawing compass and straightedge) from the bottom of your duplicate graph to the edge of the line you just drew. Make sure it is again vertical.

Have another apple, because really, why else would you have been given a freaking apple? Does it have a PhD in math? I DOUBT IT! Eat the blasted thing.

Put a thumbtack on the top end point of the line you drew. You have completed the first point on your curve!

Pat yourself on the back for successfully demonstrating that your method will eventually approximate a sine curve to precision limited only by your tools and skills. (Specifically, how good your eyesight is to see that the bowl is indeed on top of the thumb tack, how round the bowl really is, how large the radius of the thumbtacks are, how carefully you can roll the bowl and lie it down, and some other totally unimportant factors considering the potential for measurement errors already... but these are all merely physical limitations! Your method itself was superb!)

Now, take your freshly painted circular saw, and deal with the evil wizard mathematician once and for all!

Oh, and have some more beef jerkey, mmm, tasty!

[mfb]

Easy:

Spoiler:

Draw a coordinate system, construct the bisection of one of the angles (you have compass and straight-edge, so this is no problem), draw it. Label the origin with 0 and the axes (at the same distance) with 10^(-10) each. Clearly the width of your pencil line is too big to see any deviation between x and sin(x) for x~10^(-10).

What a poor mathematician, that he did not give any interval to draw!

[legend]

Spoiler:

Using the compass and the straight-edge you can:

• Draw a coordinate grid
• Draw a circle at the origin
• Split the circle in n equal section, for a sufficiently large n. This gives you n points a_j on the edge of the circle.
• Construct n new points b_j=(a_j\cdot\hat{y},j)
• Interpolate between the points.

[Gwydion]

Trying a more physical solution than a mathematical one:

Spoiler:

Mark two points (A, B) on the posterboard, and with the compass and straightedge mark a third point halfway between them (C) with the pencil. Place two thumbtacks at A and B. Tie a length of string between the thumbtacks, as taut as possible. Spread the iron filings over the whole mess. Lightly touch the string at point C, then pluck the string. The displacement caused by the vibrating string (with the harmonic added by your finger over the midpoint) should move the iron filings into one period of a perfect sinusoid. Repeat the plucking procedure as needed for accuracy.

Alternatively, you could coat the string in paint and achieve much the same effect, but this might limit the string's capacity to vibrate.

[jestingrabbit]

Trying a more physical solution than a mathematical one:

Spoiler:

Mark two points (A, B) on the posterboard, and with the compass and straightedge mark a third point halfway between them (C) with the pencil. Place two thumbtacks at A and B. Tie a length of string between the thumbtacks, as taut as possible. Spread the iron filings over the whole mess. Lightly touch the string at point C, then pluck the string. The displacement caused by the vibrating string (with the harmonic added by your finger over the midpoint) should move the iron filings into one period of a perfect sinusoid. Repeat the plucking procedure as needed for accuracy.

Alternatively, you could coat the string in paint and achieve much the same effect, but this might limit the string's capacity to vibrate.

That's not going to work imo.

Spoiler:

And this guy's opinion too.
http://www.acs.psu.edu/drussell/Demos/string/Fixed.html
You'll always get harmonics that you don't intend making up some of the standing wave.

[Macbi]

Spoiler:

1)Take the posterboard and wrap it around a cylindrical thing like the paint roller.
2)Saw through it at an angle with the circular saw.
3)Unroll it.
4)Use the pencil to trace the edge of the sine wave template you've just made.

[Qaanol]

Spoiler:

1)Take the posterboard and wrap it around a cylindrical thing like the paint roller.
2)Saw through it at an angle with the circular saw.
3)Unroll it.
4)Use the pencil to trace the edge of the sine wave template you've just made.

Spoiler:

This particular posterboard is rather too stiff to bend into a circle, as it tends to crease into ridges when you try. And even if it weren’t, it is thick enough that when it overlaps itself at the beginning and end, it would wind up being more of a spiral and not a perfect circle.

All that notwithstanding, the mathematician leans in close to see what angle you make the cut, then stifles an evil laugh when the saw blade jams upon striking the metal inner part of the paint roller.

Although,

Spoiler:

The tubular part of the paint roller is indeed a perfect right circular cylinder, and it’s the only one you have.

[jestingrabbit]

So

Spoiler:

Make decently large 45, 45, 90 triangle (circle compass, circular saw, plywood, saw horses), use that to dip half the paint roller into the paint at a 45 degree angle, roll the roller on the poster board.

I'd prefer a bizarre clockwork contraption that did the job of the gif I posted, but its a nice construction I guess.

[Qaanol]

So

Spoiler:

Make decently large 45, 45, 90 triangle (circle compass, circular saw, plywood, saw horses), use that to dip half the paint roller into the paint at a 45 degree angle, roll the roller on the poster board.

I'd prefer a bizarre clockwork contraption that did the job of the gif I posted, but its a nice construction I guess.

Spoiler:

The mathematician watches where you place the plywood triangle, to see what it makes a 45º angle relative to.

[mfb]

Spoiler:

Any angle is fine, as long as the border between "has paint" and "has not paint" fits onto the roller.

[jestingrabbit]

It think its clear from context that the sine wave has to be such that the verticle and horizontal axes are such that a length of 1 on one axis is a length of 1 on the other (else why does the mathematician care about the angle. So, a slight revision.

Spoiler:

Make a set square as before, attach a plumbob (string, apple and tack) to the set square, with a line on it that is parallel to one of the square sides. Hold the setsquare so that the plumbob is vertical (ie so that the string lines up with the line on it, and is not kinked by, or hanging away from, the set square), slide the roller into the paint along the hypotenuse, and continue as before.

There, I think that covers every possible nitpick.

[HonoreDB]

First, we stab the Evil Wizard Mathematician with the jack knife.

Surely the only truly poetic way to kill the Evil Mathematician would be to dance the dot of the laser pointer on him, thus causing him to be pounced on by infinitely many kittens.

Spoiler:

Wrap the string around the paint roller at a 45 degree angle, dip the roller in paint, remove the string, then paint a negative-space sine wave?

[Xias]

Spoiler:

If you follow a point on the circumference of a circle as it rolls, doesn't it draw the absolute value of a sine wave?

[jaap]

Spoiler:

If you follow a point on the circumference of a circle as it rolls, doesn't it draw the absolute value of a sine wave?

No, that makes a cycloid. You can see that is not a sin wave because the slope is vertical at the dips, whereas with a sine wave it would be at 45 degrees (assuming equal scaling on the axes).

[legend]

Spoiler:

no that would be a cycloid
http://en.wikipedia.org/wiki/Cycloid

EDIT: sorry, haven't seen jaap's post

[RFLS]

Easy peasy.

Spoiler:

Define a polar coordinate system, with the units equal to one of your circular objects. Use radians. Draw a circle, and enjoy your perfect sin wave.

[Qaanol]

HonoreDB: I think you’re going to have a difficult time actually doing that in practice.

It think its clear from context that the sine wave has to be such that the verticle and horizontal axes are such that a length of 1 on one axis is a length of 1 on the other (else why does the mathematician care about the angle. So, a slight revision.

Spoiler:

Make a set square as before, attach a plumbob (string, apple and tack) to the set square, with a line on it that is parallel to one of the square sides. Hold the setsquare so that the plumbob is vertical (ie so that the string lines up with the line on it, and is not kinked by, or hanging away from, the set square), slide the roller into the paint along the hypotenuse, and continue as before.

There, I think that covers every possible nitpick.

The kittens are saved! Good job to both Macbi (for seeing how to construct a sine wave) and jestingrabbit (for acing the implementation). Now you know how to make a sine wave with minimal effort.

Although,

Spoiler:

Since this particular world is obviously infinitely large and flat, and thus has perfectly uniform gravity, the surface of the paint in the bucket will indeed be planar. However, for the case of, let’s say, a nearly-spherical world, the surface of the paint would be ever-so-slightly curved (and all the more so by surface tension, Coriolis effects, etc.)

To get by those quibbles, and hence be able to make a truly perfect sine wave if you’re provided perfect tools to work with, one option is to slide the tubular “brush” part off the metal paint roller frame, use the 45º angle as a guide for the circular saw, and cut the tube in two at that angle. Then put one piece of the tube back on the paint roller, dip it in paint, and roll.

But yeah, dipping the paint roller at an angle is indeed the “pretty” solution.

[skeptical scientist]

I'm not sure why we care about the angle. My usage (and Wikipedia agrees) has always been that any equation of the form y = a sin(bx + c) graphs a sine wave.

[RFLS]

So, no one liked my answer at all? I believe it met the requirements.

[skeptical scientist]

No. A polar graph of r=sin(t) is not a sine wave.

[RFLS]

Can you explain that to me? I'm not asking in a sarcastic/belligerent tone, I would truly like to know. I've taken through college calculus 2 (derivatives and integration) and a physics engineering course on mechanics, in case that imposes limits on the extent of the answer you can give.

As far as I know, a sine wave is just a smooth, repetitive oscillation described by a mathematical function, with the following properties-

A, the amplitude, is the peak deviation of the function from its center position.
ω, the angular frequency, specifies how many oscillations occur in a unit time interval, in radians per second
φ, the phase, specifies where in its cycle the oscillation begins at t = 0.
When the phase is non-zero, the entire waveform appears to be shifted in time by the amount φ/ω seconds. A negative value represents a delay, and a positive value represents an advance.

The above is pretty much directly from Wikipedia.

I believe drawing it in a polar coordinate system still meets all of those requirements.

In fair warning, I don't take "because" as an answer.

[phlip]

I'd prefer a bizarre clockwork contraption that did the job of the gif I posted, but its a nice construction I guess.

Spoiler:

Use the pair of compasses to draw a large circle on the plywood slab - use the saw to cut that out to form a wheel, along with three long straight (as determined by the straightedge) square poles, and also one shorter one. Now, cut a small circular hole near the edge of the wheel, using the jacknife. Thread the shorter pole you made into the hole - it will form an axle that should freely spin within the hole. Attach one of the longer poles, at one end, to the axle with the string (carve a groove in each pole with the knife, put them together and tie it off, it should hold steady).

Next, hang the sheet of posterboard from one of the sawhorses, along with the second long pole (slightly to one side). Attach the bucket of water to the bottom of the pole to make sure it hangs vertically. If the sawhorse isn't high enough, stack it on top of the other sawhorse.

Now, cut out another piece of the plywood slab (it is, of course, a large enough slab for all of these pieces... if it's not, just scale everything down to fit). On one side, carve a horizontal groove, and on the other a vertical one. Use the compass and straightedge to make them perpendicular. Put the vertical pole you just attached to the sawhorse in one groove, and your third long pole in the other, and loosely tie them on with the string - tight enough that they can't fall out of the grooves, but loose enough that they can slide freely. Use the milk as a lubricant if necessary. This third pole should now be held horizontal and be able to move up and down.

Now, attach (with yet more string), the can of paint to the bottom of the first pole that's attached to the wheel we made earlier, so it will hang vertically too. Now, place this wheel against the vertical pole attached to the sawhorses, and roll it vertically along that pole, while moving the horizontal mobile pole to be touching the wheel from underneath at all times. Hold the pencil in the corner between the pole hanging from the wheel and the horizontal pole, and trace its movement on the posterboard.

Now, can you build all of that in 10 minutes? Doubtful. But still.

RFLS: that's the definition of a "sine wave" in the context of things actually oscillating - the movement of something whose position as a function of time is sinusoidal. In the context of a drawing, or geometry, "sine wave" refers specifically to the shape of a Cartesian graph of a sinusoidal function - that is, it's referring to the shape, not the function. And thus a plot of the same function in a different coordinate system is a different shape.

[RFLS]

Okay. Makes sense. I guess my answer was technically correct, but something no self-respecting math professor or Evil Mathematician would accept.

[mike-l]

I'm not sure why we care about the angle. My usage (and Wikipedia agrees) has always been that any equation of the form y = a sin(bx + c) graphs a sine wave.

And you could always just draw the axes afterwards...

[jestingrabbit]

cut the tube in two at that angle.

That will introduce far more error than the curavture of the surface of the paint: there'll be some roller fuzz that is unsupported by the roller, making the peaks of the sine wave wonky.

@phlip: yeah, I was thinking of a construction like that, but I just can't really see it working. No doubt one can make something similar that does work well, but it would need some gears, and probably a cunning spring contraction to keep the stylus firmly gliding along the board. Ideally it would trundle along under its own power, leaving a sine wave on the ground in its wake.

[WarDaft]

That will introduce far more error than the curavture of the surface of the paint: there'll be some roller fuzz that is unsupported by the roller, making the peaks of the sine wave wonky.

But isn't that a limit of the tools, rather than the method?

[jestingrabbit]

That will introduce far more error than the curavture of the surface of the paint: there'll be some roller fuzz that is unsupported by the roller, making the peaks of the sine wave wonky.

But isn't that a limit of the tools, rather than the method?

So, because of how the question is framed, you want me to consider that a less accurate method is better? Frankly, I could define the flatness of the paint surface as a tool aspect of the paint, in much the way that the straightness of a straight edge is an aspect of that tool.

Regardless, a question that wants a less accurate answer than can be provided, for whatever reason, isn't a good question.

[WarDaft]

Wait, never mind, I misread... you were talking about the actual paint holding fuzz on the roller, not plastic shavings making a rough edge, which is what I had thought.