Shadow Answer: Minotaur is being absurd.

[extropalopakettle]

First off, I knew this was coming. It was pretty obvious
that the person who devised this "puzzle" thought the right
answer was the naive answer, which is plainly wrong.
Let's go over this:

"As it turns out, at any altitude the Falcon's shadow will
be 195 square feet."

Just plain wrong. As the Minotaurs solution later states,
and as many puzzled puzzlers know, the umbra gets smaller,
and the penumbra gets larger. And by several square feet,
at 30 feet of altitude. By no way of measuring the area
(umbra, penumbra, whatever) does it remain 195 square feet,
and certainly not at "any altitude".

"Because of the extreme distance of the sun, the photons'
paths are effectively parallel when they meet the glider."

Just plain false. As much as 1/2 degree difference
between rays.

"The nit-pickers out there will recognize that this solution
is theoretically not completely correct."

While 1 plus 1 is 3, nit-pickers will recognize that,
theoretically, 1 plus 1 is 2.

It is not "theoretically not completely correct".
It is really, actually and truly totally incorrect. Is
that nit-picking? I don't think so.

How, please tell us, is what is theoretically true, in this
case, different from what is realistically true? It isn't.

Theoretically, lines that meet at at angle of 1/2 degree
are not parallel. Realistically, lines that meet at at angle
of 1/2 degree are also not parallel.

"Because the sun is actually a finite (though very large)
distance away, the shadow should be slightly larger than
the glider."

My opinion: This would be nitpicking. If the sun were a
point light source, then the rays would be so close to
parallel that the increase in size of the shadow would
be essentially negligible. I don't think anyone was
nit-picking about this point though. It has nothing to
do with the argument about why the umbra gets smaller, and
the penumbra gets larger.

"But because the sun isn't a point, but a large sphere,
light emits from many sources."

Yes, and those sources are seperated by as much as 1/2
degree from the vantage point of the earth.

"And since these sources cover an area larger than the
glider, the shadow should be smaller than the glider."

NO. It is because of this that the umbra WILL be
substantially smaller, and the penumbra WILL be
substantially larger.

Note Minotaurs use of "should be", as if to mean
"should be, but isn't". If it isn't smaller, why should
it be smaller? It should be, AND IT IS.

"Which is correct? Both are:"

Confusion. First he states a reason why the shadow
should be larger, then a different reason why it should be
smaller, and now says both are correct, and, in the same
sentence, as if to elaborate, says:

"Which is correct? Both are: the glider's umbra, the part
of the shadow where none of the sun's rays hit, is
smaller. The glider's penumbra, the part of the shadow
where some, but not all of the sun's rays are blocked by
the glider, is larger."

But both of these are because of the second reason he gives,
and neither has to do with the first reason. He talks about
both being correct, and then talks about phenomena which
relate to the second reason only.

"And both of these answers ignore the fact that photons
don't necessarily travel in straight lines. But that's
for another puzzle."

That's for another reason to call people nit-pickers when
they're reasoning is more on the mark than your own.

It's one thing to make a mistake, as Minotaur did in posting
this puzzle with the naive answer in mind. It's another to
stick to it in the face or reason, calling reason
"nit-picking".

[agraco]

For all the pleasure I have had out of this site, I really could not care less if the Minotaur made a mistake and had a lousy way of going about his "naive" answer.

His answer was already critized and you could have added your two cents to that thread. I don't see the point in berating the guy with a new thread "Minotaur is being absurd" in addition to the aggresive nature of your post.

regards,
-Alexandre.

[extropalopakettle.]

Sorry, but I find it quite annoying.

He posts a "puzzle" with a very simple answer in mind, and let's it sit there for two weeks. In that time, people realize and clearly demonstrate through their arguments that the simple answer is incorrect, not just in theory, but in fact. The correct answer differs significantly from his, but he decides that they're "nit-pickers". He get's to say what the right answer is, because he's "the Minotaur". Again, I'm sorry, but in fact, that's absurd.

[Ghost Post]

regardless of whether you agree with the minotaurs reasoning or his presentation, this puzzle and most of his others generated stimulating thought provoking discussions. I think that is what most of the visitors to the G.L. are looking/hoping for.

[AcidFast]

Extro, you definitely have a point, but I see no reason to go using words like, "absurd" and "naive." You seem really upset about this, and I am forced to wonder why. Its just a puzzle, man. On a free puzzle site, at that. Chill out.

AcidFast

[Ghost Post]

Extro, theoretically you are correct. And realistically, you are also correct. However, I think people would be a lot more impressed with your "correctness" if you were a little more diplomatic about it. Give the guy a break here. It's obvious most of the visitors to this site really appreciate him, imperfections and all.

[Buzzsaw]

I am lousy at math and am therefore a poor overall contributer but I recognize extropalopakettle's competitive nature and his frustration ventilated here. I think this site is the greatest and I am going out on a limb by assuming extropalopakettle does too. Otherwise why would he have such passion over this inadequate puzzle explanation. Read between the lines and you will see a guy frustrated by this particular puzzle answer who ultimatly appreciates this site as much as the rest of us. I don't hear him call the minotaur absurd rather being absurd by riding the fence in this isolated instance. He is a competitive guy who overall much appreciates this site, is a significant contributer and I say we let him blow off some steam. As many of you did he spent a lot of time on the puzzle to then get a blow off answer.

Minotaur, the last answer you gave was inadequate but you have done, and do a great job with this site and we regulars appreciate you and this site very, very much. Thanks for it's existence. Take care. Brillo

[This message has been edited by Buzzsaw (edited 11-03-1999).]

[Ghost Post]

At the risk of sending off a horde of replies....


absolutely.

[Andy]

I'm also disappointed by the Minotaur's answer - but not as much as extropalopakettle (BTW - is there an intropalomakettle?) seems to be. Firstly, I have read the postings in the discussion forum (I didn't work thru all the math, but it looks reasonable), so I have seen enough answers to be satisfied. I have enough confidence in myself and other forum members to accept the demonstrated answers even though the Minotaur gave a different one. Also, we're not being graded on how well our answers match the Minotaur's - in fact, in at least this case, we're grading his answer. I wonder if the Minotaur posted a previously-prepared answer without considering what we said about it, or if his answer was influenced by space considerations. Well, enough rambling for now.

[Alias_Sean]

Define "shadow" as the mean distance between the umbra and penumbra and then the Minotaur's answer is correct.

The average shadow remains the same.

[Ghost Post]

Well, no, because at some point the umbra disappears altogether, while the penumbra continually increases in area.
Of course, if you set out to come up with a definition of shadow from which you can show that the area doesn't change, you can, but it would be extremely contrived - so much so that it would be absurd (sorry if anyone is offended by that word) to use the word "shadow" in the statement of the puzzle and expect that that definition would be assumed by the readers.

Regarding my use of the word "naive" to describe the "area stays the same" solution:
It is not intended to be offensive. The word simply means overly simplistic.

There is a connection here with computer graphics and ray-tracing. People have been doing computer graphics using ray-tracing for many years now. Early on, outdoor images would be produced with a lighting model which assumed parallel light rays. It was a very simple, "naive" model. Naive is not necessarily bad. It's a good place to start. The results of this model, though, were highly unrealistic looking, sharp-edged shadows. Which is exactly what you would expect if the suns rays were parallel. It wasn't that they were theoretically wrong. Ultimately, nobody would care about that. But in computer graphics, you try to make it look real. And that's not how real shadows look.

It takes a bit more work (for both programmer and machine) to do ray-tracing with a more accurate model, such as one where the light source has some area, and produces non-parallel rays. But it pays off in producing shadows that look right - with an umbra and penumbra, resulting in a fuzzy, soft edge, and more so when the object is further from the object the shadow is cast on.

When I first read the puzzle, I instantly thought "obviously, the area of the shadow stays the same, because the suns rays are parallel". Then I tought "but that wouldn't make for much of puzzle, would it?" Than later I remembered those diagrams of lunar and solar eclipses with the umbra and penumbra, and realized it wasn't as simple as that.

[araya]

Good point about the computer modelling - I've done a fair bit of 3D modelling, and you're right, the parallel-ray theory is reasonable but it just doesn't look right, even in a computer-generated scene.

My reaction to the puzzle was the same as yours - I thought 'it stays about the same', then I thought, 'but the sun is very large, so it must get smaller', then I thought 'but as things near the sun, the shadow would get larger', then I realized, what does 'shadow' apply to - it's such a vague term. It certainly is contrived to take the average area of the umbra and penumbra and call that the shadow (as long as you're below the critical altitude).

[chief ten beers]

I think I'll stop posting to this thread now unless someone posts something interesting.
I hate arguing for argument's sake and that is clearly what this is turning into.

[CrystyB]

1. link

2. Extro, in fewer words, what is/was your problem with having the solution being at first overly simplistic, and then introducing two seemingly conflicting complex answers that are explained later?