# The Limits of Elven Vision

Kate’s Tolkien re-read has reached Rohan, and her latest re-read post includes a reference to a comment I made about Legolas’s improbable visual acuity:

Re: Legolas seeing the Riders: I have since been advised by the resident physicist that the size of a pupil is a limiting factor on the resolution possible–if I understood properly, basically it boils down to how much light can come in–and under the laws of physics as we know them, it is not actually physically possible for Legolas to have resolved that level of detail at 5 leagues, regardless of how good his brain is at decoding images or whatnot.

The passage in question is this, from The Two Towers:

“Riders!” cried Aragorn, springing to his feet. “Many riders on swift steeds are coming towards us!”

“Yes,” said Legolas,”there are one hundred and five. Yellow is their hair, and bright are their spears. Their leader is very tall.”

Aragorn smiled. “Keen are the eyes of the Elves,” he said.

“Nay! The riders are little more than five leagues distant,” said Legolas.”

So, how reasonable is this, in physics terms?

The key idea here is the notion of resolution, that is, how well can you distinguish between two objects separated by some distance. The limiting factor here is the wave nature of light– light passing through any aperture will interfere with itself, and produce a pattern of bright and dark spots. This means that even an infinitesimally small point source of light will appear slightly spread out, and two closely spaced point sources will begin to run into one another.

The usual standard for determining whether two nearby sources can be distinguished from one another is the Rayleigh criterion, which takes the mathematical form:

That is, the sine of the angular separation between two objects is equal to 1.22 multiplied by the ratio of the wavelength of the light being considered to the diameter of the (circular) aperture through which the light passes. To get better resolution, you need either a smaller wavelength or a larger aperture.

We can make a ballpark estimate of what Legolas’s eyes would need to look like, given the rest of the data from the passage. He says that the riders are “little more than five leagues distant.” A league is something like three miles, which would be around 5000 meters, so let’s call it 25,000 meters from Legolas to the Riders (that’s a bit low, but we’ll assume he’s bragging). Visible light has an average wavelength of around 500nm, which is a little more green than the blond hair of the Riders, but close enough for our purposes.

The sine of a small angle can be approximated by the angle itself, which in turn is given, for this case, by the size of the separation between objects divided by the distance from the objects to the viewer. Putting it all together, these figures suggest that, in order to distinguish between two point sources separated by one meter, Legolas’s pupils would need to be 0.015m in diameter. That’s a centimeter and a half, which is reasonable, provided he’s an anime character. I don’t think Tolkien’s Elves are described as having eyes the size of teacups, though.

We made some simplifying assumptions to get that answer, but relaxing them only makes things worse. Putting the Riders farther away, and using yellower light would require Legolas’s eyes to be even bigger. And the details he claims to see are almost certainly on scales smaller than one meter, which would bump things up even more.

So, we can say with some confidence that Tolkien was neither a physicist nor an astronomer.

(This is actually used as the basis for a problem in one of the Six Ideas That Shaped Physics textbooks– the Quantum volume, I think, but I don’t have the books here– which is where I first encountered it.)

1. #1 Kate Nepveu
June 23, 2009

Excellent, I have linked over there as well.

2. #2 Daniel Baker
June 23, 2009

I’ve also wondered about this sort of thing before. At one point (also in the Two Towers), Gandalf mentions that elves “can tell a sparrow from a finch a league off”. A league is 5.56km, sparrows and finches are roughly 0.1m in height. This means that they subtend 0.001deg visual angle, which is well over an order of magnitude smaller than anything humans can detect, and probably also implausible from a physics perspective.

Later in the book, Tolkien suggests that orcs have very good night vision (“There’s only one thing those maggots [orcs] can do: they can see like gimlets in the dark.”), but this is perhaps more plausible than the extreme acuity of Elven vision.

3. #3 Moopheus
June 23, 2009

I believe this is part of why we call it “fantasy.” Maybe elves do look like anime characters; I don’t recall if there’s a precise description. Maybe elves have magical invisible light-gathering projections from their eyes. Who knows? It’s hardly the least-likely thing in the book.

4. #4 john S. Wilkins
June 23, 2009

Unless Elves have telescopic lenses…

5. #5 cisko
June 23, 2009

“Yes,” said Legolas,”there are one hundred and five.

I can see figuring this based on some standard marching formation — say four squads of 25, plus squad leaders and a captain. Then Legolas could figure the number based on the number of squads, without specifically counting each person.

Yellow is their hair, and bright are their spears.

I can imagine specular reflections twinkling off brightly polished spearheads. For the hair, perhaps Legolas is seeing several heads close together, rather than each one individually.

This is the tough one, particularly because Eomer is mounted. Maybe he has a taller horse? Either way, Legolas has to be able to see his head (or spear) as distinct from the others. Hard to do unless he’s at least a meter taller than the others. I can buy the other two, but not the height thing. (Or the sparrow vs. finch; that would require resolving details around 1cm in size or less.)

Of course, we could just argue that a Middle-Earth league is a lot less than 3 miles. But if we want a pupil around 5mm, we get a league very close to 1 mile, which seems a real stretch.

Or, magic. *poof*

June 23, 2009

Maybe elves have phase-sensitive retinas and can do interferometry between their two eyes.

June 23, 2009

Maybe elves have phase-sensitive retinas and can do interferometry between their two eyes.

Somebody at in the comments to the original post suggested that as well. Using interferometry would give you an effective aperture size of the spacing between the eyes, maybe 10cm at the outside. This would give you resolution on the 10-20cm level, which is closer to working.

8. #8 mollishka
June 23, 2009

Clearly elves can see in the UV. Using a wavelength of 100nm gives him very reasonably sized pupils… and the atmosphere of Middle Earth is not opaque at such wavelengths.

9. #9 Daniel Baker
June 23, 2009

Of course, the implausibility of elven vision is nothing compared with the visual abilities of Sauron. Not only does the Eye of Sauron ‘see all’, suggesting infinite visual acuity, his gaze can “pierce all shadows of cloud, and earth, and flesh”. This implies that he can utilise the very short wavelength portions of the electromagnetic spectrum – Sauron has X-ray vision!

10. #10 John Novak
June 23, 2009

I am disappointed that there has been no attempt to calculate the F# of Legolas’ eyes.

(The interferometry idea is cute, but I gaurantee you that, if made out of anything resembling conventional nerve tissue, the act of shifting his eyes or moving his head will result in a complete decalibration of the system.)

11. #11 MikeMa
June 23, 2009

I agree with the shorter wavelengths to accomplish the trick. Remember elves live in ‘both worlds’ and can see the nine riders even when they are but shadows to the hobbits.

Anyway, seems more likely that the elven cones and rods are more plentiful, smaller or more varied allowing the discrimination of higher frequency, shorter wavelengths. Translation for color is done in the brain.

12. #12 Sylvain
June 23, 2009

Note that your calculation proves only that on Legolas’ retina the spreading of two points coming from two riders would be much larger than the distance between the two spots. Although, thank to your nice explanation, everyone will agree that Legolas measure is beyond diffraction limit, you could not conclude that Tolkien “describes” here an unrealistic situation.

There is several means to do better that diffraction limit and scientist have found several way to do so.
I’ll give two examples, one from microscopy another from astronomy.
In microscopy to reach the nanometer spatial resolution in real time a team at urbana-Champaign university has developed the FIONA (Fluorescent imagine with one-nanometer accuracy) technique. They have been able to monitor nanometer step of a myosin breaking with two orders of magnitude the diffraction limit.
Although, the size of a planet in a remote place of our galaxy is far beyond the diffraction limit, astronomers keep finding new exoplanets. Again, they have to come with some tricks in analyzing there data, but still there are breaking the diffraction limit.
The diffraction limit is not a limitation, for inspired mind!

As Legolas do not says how he came up with this figure and that it is well establish that you can go beyond diffraction limit, why would it be physically wrong to affirm that he is using a trick to come up with his 105 riders?
Sylvain

June 23, 2009

(The interferometry idea is cute, but I guarantee you that, if made out of anything resembling conventional nerve tissue, the act of shifting his eyes or moving his head will result in a complete decalibration of the system.)

Clearly Legolas’ optical system would have to consist not only of phase-sensitive retinas, but also fiber-optic delay lines and an optical correlator in his brain somewhere :)

If this were the case, turning his head or moving his eyes would actually help. With one baseline (2 elements), he only gets one fourier mode of the image and is thus only sensitive to one particular angular size. By turning his head away and looking at the army from the side, he can get more a more densely-sampled fourier plane. Of course, for completeness he’d also want to do the same thing with his head rotated 90 degrees so his eyes are oriented vertically and he can reconstruct a 2-D image.

Given, though, that Tolkien never mentions this curious ritual whenever Legolas looks at something, it’s pretty unlikely that Legolas was an interferometer.

14. #14 Anthony
June 23, 2009

The other problem with seeing riders at 15 miles is that they’re probably over the horizon at that point. If we assume that both of them are on a plain, Legolas is about 6′ tall, and the riders are about 8′ tall, maximum spotting distance is around 7 miles. To hit 15 miles, if the riders are on a flat surface, Legolas needs to be some 80′ above the ground. Possibly more, if there’s high grass.

One amusing solution I’ve seen to this is that to the elves, the earth is flat. This also accounts for how they can sail west and leave, whereas humans just go around the world.

15. #15 Lobster
June 23, 2009

It’s the quick calculation that still gets me. Maybe he did see four squads of 25 plus a squad leader plus a commander or whatever.

Why would he say 105 people? Why not just “four squads?” And that kind of precision sets him up for being incorrect if Kilroy the Horrible happened to sleep in and was late for the raid.

But really it’s that passage that illustrates what irritates me about Tolkien. The guy’s not a good writer. Good storyteller, but not a good writer.

Get that damn smile off your face, Aragorn, and quit it with the heavy handed exposition. 105 riders is a problem.

16. #16 Kate Nepveu
June 23, 2009

Weirdly, various people have addressed the horizon issue over at the original post and have told me that despite my American conception of “downs” as fairly flat (which is where the text says they are), there are specific examples that would fit the bill just fine.

(Also, I see my calculations about the horizon neglected to take into account the height of Legolas and the riders. Oops.)

17. #17 dave
June 23, 2009

The other option is that Legolas has been through (or near) Rohan before in his long lifespan and, when he sees a bunch of Riders in the distance knows that they’re most likely blonde, tall, spear-carrying and ladies with glued on beards. (Last point is just the movie, of course.)

18. #18 Anthony
June 23, 2009

In any slightly bowl-shaped area the horizon problem goes away. However, we then run into the opposite problem: while human vision won’t be able to resolve a hundred guys at fifteen miles, it won’t be that hard to see that there’s something there (a moving patch of different color, and most likely a cloud of dust).

19. #19 Anthony
June 23, 2009

Oops, never mind, Aragorn saw it too, though the dust is evidence enough by itself.

20. #20 tonylurker
June 23, 2009

Sylvan:In microscopy to reach the nanometer spatial resolution in real time a team at urbana-Champaign university has developed the FIONA (Fluorescent imagine with one-nanometer accuracy) technique. They have been able to monitor nanometer step of a myosin breaking with two orders of magnitude the diffraction limit.

But that’s just a near-field microscopy technique. The only reason it works is because the imaging device (a small scattering tip) is located within the region where the exponentially decaying evenescent component of the electro-magnetic field dominates over the radiating component (it’s held really, really, really close to the object being imaged); therefore it can sidestep the diffraction limit because it is operating in a regeme where the calculations used to determine that limit do not apply.
The riders are welll outside the near-field.

While there are tricks that allow you to beat the diffraction limit, they would require more than just legolas’s eye.

Heck, why not claim he has a magical connection to nearby animals/people and can construct a magical psyco-interferometric link to all the eyes around him that are pointed towards the riders. That could expand his effective aperature dramatically.

Or you could posit that he’s only putting on an act by pretending to look at the riders, but is instead getting his information from some other magical elven means. This is fantasy after all.

June 23, 2009

I’ll give two examples, one from microscopy another from astronomy.
In microscopy to reach the nanometer spatial resolution in real time a team at urbana-Champaign university has developed the FIONA (Fluorescent imagine with one-nanometer accuracy) technique. They have been able to monitor nanometer step of a myosin breaking with two orders of magnitude the diffraction limit.
Although, the size of a planet in a remote place of our galaxy is far beyond the diffraction limit, astronomers keep finding new exoplanets. Again, they have to come with some tricks in analyzing there data, but still there are breaking the diffraction limit.

tonylurker (#20) has already dealt with the microscopy objection, so let me just note that the detection of extrasolar planets does not have anything to do with “breaking the diffraction limit.” Most of the extra-solar planets that have been detected have been detected by measuring shifts in spectral lines due to the Doppler effect as the orbiting planet “tugs” on the star. Others have been found by observing the small decrease in the brightness of the star when a planet passes in front of it.

Neither of these techniques is affected by the diffraction limit in any meaningful way. The diffraction limit on resolution only comes in when trying to distinguish two sources that are close together. To date, there has been only one direct observation of a planet orbiting a distant star, which was done using a coronagraph to block the light from the star, allowing the astronomers to pick up the fainter light reflected by the planet. This is not a sub-diffraction-limit technique, either, at least as I understand it. The planet they observed was far enough away from the star to be resolvable with the optics they were using.

22. #22 Eofhan
June 23, 2009

Maybe Elves literally have “eyes like an eagle”? They could have color vision extending into the near ultraviolet, with oil drops on the ends of their cones, filtering incoming light & shifting it to longer wavelengths? (OK — I suspect the calculation would apply to the wavelength at the retina, not at the lens, but still, it’s an idea . . . )

Or maybe Elves have something like the nictating membrane of a cat? Only, instead of being a simple protective mechanism, it’s an oil-filled, variable-geometry lens giving them exceptional distance-sight “in the blink of an eye”. (sorry, bad pun)

23. #23 ObsessiveMathsFreak
June 23, 2009

While I like the presentation of this argument, I just don’t accept the final conclusion. Ultimately, my problem is with the assumptions.

Firstly and foremostly, the argument uses the Rayleigh Criterion, which has never made any sense to me. The distinction between “resolved” and “unresolved” seems like less of a mathematic terms and more a subjective one. Which mathematically two objects may fall inside the Rayleigh limit, our brains may still be quite able to determine things from context.

Secondly, as someone who wears glasses, I have difficulty believing that the eye behaves like a circular lens aperture. According to the above argument, I should be able to distinguish objects a meter apart from 10km away! But my bad eye can’t even do that at 100m. So the eye is a more complicated object that a simple lens.

Most importantly, the Rayleigh limit results from the diffraction of light at the edges of the circular aperture. But the edges of the lens in the eye are not so sharp. Moreover the entire structure of the eye is more complicated. This leads to the following consideration. Suppose there was a particular eye structure that allowed one to bypass or at least mitigate the Rayleigh criterion’s severity. Then, would evolution have happened upon it?

Finally, the argument leaves out a fairly important point. Legolas, along with many non-fictional creatures, has two eyes! The synthetic aperture of vision is therefore somewhat higher than that of one eye alone. I don’t know the details of that in this case, but I imageine it can’t be totally discounted.

24. #24 ildi
June 23, 2009

Too funny! I understand the dissection of sci-fi, where the laws of physics are supposed to apply

… in space no one can hear you scream…

but fantasy? Isn’t the answer always, “it’s magic!”

25. #25 csrster
June 23, 2009

I wrote a paper on the Elvish-eyesight Aperture Synthesis Hypothesis which was published in Anor in about 1984/5 (iirc it was later reprinted in Amon Hen). So far as I know this is the first published exposition of the idea. I also speculated that elves had fibre-optic visual systems which they used to form an interference pattern on an extra retina within their skull.

I also discussed the requirements this would place on elves to move their heads continually around three separated axes (nodding, shaking and cocking).

All in all, I had far too much time on my hands for a final-year science student at a prestigious university.

26. #26 Doug l
June 23, 2009

I think the elves had adaptable optics in their eye’s lenses capable changing contour to bring up pinpoint clarity, a finer resolution retina, and could image fainter images on it with a longer persistence of memory. T’was usefull in their knowledge and application of nuclear energy for good.

27. #27 Doug l
June 23, 2009

I think the elves had adaptable optics in their eye’s lenses capable changing contour to bring up pinpoint clarity, a finer resolution retina sensitive to UV, and could image fainter images on it with a longer persistence of memory. T’was usefull in their knowledge and application of nuclear energy for good.

28. #28 Jim
June 23, 2009

I assume that the original calculation is based on one exposure only?

For moving details I had the impression that several sequenced images/exposures (a film) could be extrapolated to higher resolution when using computers with cameras, and that the brain has a similar way to treat the information flow over time. Birds like owls and doves for instance use this technique to improve their sight, by moving their heads while observing something. I still havent sen Legolas turn and twist his head too much whil squinting towards the horizon though.

But still, and add cognitive associations (which has been mentioned in a few comments before mine) to this, aided by experience and knowledge of which elves have an abundance.

These are two more factors that ought to increase the chance of a correct assumption on Legolas part.

29. #29 jim
June 23, 2009

I don’t think we can postulate major anatomical differences between elves and humans. While elves aren’t human (they seem to have some sort of anti-aging gene, for example), there have been several attested successful elf-human crosses (we don’t know how many attempted crosses there have been; this is not the sort of thing that Tolkien talks about). So the two species can’t be genetically that far apart. Any organ that would permit interferometry has to be ruled out, and, I suspect, even sensitivity to frequencies much beyond human responses wouldn’t be possible.

Personally, I believe that the unit of distance in the Red Book which Tolkien translates as “league” is much shorter than the three miles that we usually think of as a league.

30. #30 MRW
June 23, 2009

Obsessivemathgeek –

On your second point… the aperture sets the *maximum* resolution, so your vision being poorer than it really doesn’t man anything for the calculation.

On the first point, I agree. I’ve been able to see details in a microscope that are smaller than the Rayleigh criterion. It’s good for a rough estimate, but you can distinguish things whose diffraction patterns overlap a bit more than is used for the Rayleigh criterion.

31. #31 Dan F
June 23, 2009

Actually, some Tolkeinists (read: nerds) have argued that Tolkein meant for the distances in Middle Earth to be shorter than ours. The unit “league” may have been significantly shorter than the league you were referring to. Perhaps you’ve just uncovered another bit of textual evidence for that case.

32. #32 psweet
June 23, 2009

Although several comments have made this point indirectly, it seems worth mentioning that the physics of the lens is not the only relevant factor. The resolution of the retina is equally important — think of the difference between an old 1 MP digital camera, and a new 8 MP one. Of course, while we’re at it, how about explaining Ents, Beor, and the physiology of elvish immortality?

33. #33 The Ridger
June 23, 2009

C’mon. He could walk on snow! He was thousands and thousands of years old. You’re quibbling about his eyesight?

34. #34 BlackGriffen
June 23, 2009

What if parts of his eyes have a negative index of refraction (across the whole visible spectrum)? If that’s not palatable in a world of magic (and elves are supposed to be semi-supernatural), then how about “invisible magical spectacles,” or “astral optics” or basically some mechanism that can extend beyond the visible boundaries of his eyes to bend light in a conventional fashion?

I know, it’s really just an excuse for giving a physics lecture. :)

@psweet: Increasing the pixel count only helps if the image isn’t blurry, and this post is basically discussing the minimum blurriness achievable under ideal circumstances.

35. #35 Anton Mates
June 23, 2009

Dermo-optical perception. The skin over the entirety of an elf’s face is photosensitive. That’s why they tend to have no facial hair; those few with beards are half-blind.

36. #36 photon
June 23, 2009

Of course, elves originally saw by starlight, and then by the light of bioluminescent plants and glowing jewels, so it would not be unreasonable to assume that their visual systems may have evolved/been created (and there’s a topic for another pointless thread) to respond to different wavelengths to humans’.

37. #37 Sean Barrett
June 23, 2009

The sine of a small angle can be approximated by the angle itself, which in turn is given, for this case, by the size of the separation between objects divided by the distance from the objects to the viewer.

Technically this is not quite right; the separation divided by the distance is not the angle itself, but rather the tangent of the angle. (SOHCAHTOA: sin = opposite/hypoteneuse; tangent = opposite/adjacent.)

Because the cos of a very small angle is very nearly 1, however, the tangent is just as nearly equal the angle as is the sine. But that doesn’t mean you can just skip that step. And there’s really not much need to even mention the angle; with such a very tiny angle, clearly the hypoteneuse and the adjacent side have essentially the same length (the distance to either separated point is also essentially 25K meters), and so you can correctly say that the sine itself is in this case approximated by the separation divided by the distance, and never mention the angle at all.

(You could break out a calculator to be on the safe side, but if you’re going to do that you need to know the actual formulation to compute the angle, not compute it as opposite/adjacent! But, yes, both angle (in radians) and the sine are also 1/25000 to about 10 sigfigs.)

38. #38 Bill Tozier
June 23, 2009

Hmm. Plains. It’s daylight, so there is some solar heating of the ground. Has anybody ever watched a telephoto image (or looked through a telescope) at a person or vehicle approaching over a heated flat ground in extreme distance?

A vacuum it ain’t. So one more word to add to the mix: mirage.

Now what? Are things better, or worse?

39. #39 ScentOfViolets
June 23, 2009

The other problem with seeing riders at 15 miles is that they’re probably over the horizon at that point.

One amusing solution I’ve seen to this is that to the elves, the earth is flat. This also accounts for how they can sail west and leave, whereas humans just go around the world.

Well, yes, at one point Middle Earth was flat. I forget exactly when it was curved ’round(this is starting to sound like Richard McKenna’s “Fiddler’s Green”.) And as you point out, the Elves can sail West, apparently without magical hyperdrives. So we already know from the start that the vision thing isn’t exactly pure physics.

40. #40 MattXIV
June 23, 2009

Doesn’t the Raleigh Criterion deal with the case where the light sources to be distinguished are at the same wavelength? The hair color of the rider’s could be deduced based on there being a yellow light source above what could be deduced to be the colors of their horses and torsos.

41. #41 cisko
June 23, 2009

The earth was curved when Ilúvatar bent the world, after Ar-Pharazôn sailed his fleet from Númenór and landed on the shores of Valinor. BIG no-no.

But the real important question is this: can Legolas see farther than Perrin Aybara?

“Put up your bows,”he said. “That’s Alliandre’s gelding. It must be our people. Can’t you see the Aiel are all Maidens?” Not a one was tall enough to be an Aielman.

“I can barely make out they’re Aiel,” Dannil muttered, giving him a sidelong look. They all took it for granted that his eyes were good, even took pride in it — or used to — but he tried to keep them from knowing how good. Right then, he did not care, though.

42. #42 Jonathan Vos Post
June 24, 2009

It is reductionist to attempt cramming Lord of the Rings Magic into modern Physics.

What makes you think that the Middle Earth of LOTR is the same as our Earth?

How can you not admit the possibility that this is an alternate universe, not merely an alternate history, but with different physical laws?

The creation myth in Silmarillion makes it clear that this universe is different, foundationally.

What is the atomic number of mithril? Wrong question.

Does the inverse square law apply to the Rings and the Palantir? Not a chance.

Better to try to reverse engineer the axioms of the Physics of LOTR. The mere fact that Tolkien didn’t know Physics is not the issue. He was relentlessly logical in areas where he had expertise.

43. #43 MarkusR
June 24, 2009

And why does Gandalf need a staff for walking if he is able to fight a Balrog?

44. #44 Rhett
June 24, 2009

@MarkusR,

I don’t need a belt to hold up my pants, but I wear one anyway. Gandalf either thinks the staff is cool or he is afraid to go against the fashion norms for wizards.

@Chad – this is a great post.

45. #45 Rhett
June 24, 2009

@MarkusR,

I don’t need a belt to hold up my pants, but I wear one anyway. Gandalf either thinks the staff is cool or he is afraid to go against the fashion norms for wizards.

@Chad – this is a great post.

46. #46 ScentOfViolets
June 24, 2009

And why does Gandalf need a staff for walking if he is able to fight a Balrog?

Posted by: MarkusR

Because he was an early adopter of the Roosevelt Doctrine: Speak softly and carry a big stick.

47. #47 Matt Springer
June 24, 2009

Why does Yoda need a staff if he can fight a Sith lord?

Actually Gandalf has an excuse: the staff itself is magic, as is made very clear when he’s hassling the Rohan Secret Service before going in to see Theoden.

48. #48 rob
June 24, 2009

many people think that the denizens of middle earth lived in a middle ages type technology. that was not so. middle earth actually was quite far along the tech tree. legolas simply tapped into sauron’s spy satellite using his iphone with a neat app he wrote a thousand years earlier.

also, last week when i had lunch with Tolkien and Presley, Tolkien mentioned that it was a typo in the book. “leagues” was supposed to be “kilometers” since middle earth had gone over to the metric system 31.4 years earlier. a type setter messed up and “leagues” has made it into every printed copy of the book instead of “kilometers.”

June 24, 2009

Because the cos of a very small angle is very nearly 1, however, the tangent is just as nearly equal the angle as is the sine. But that doesn’t mean you can just skip that step.

Yes, it does — we’re physicists, after all.

50. #50 Lab Lemming
June 25, 2009

I don’t see how else you’re gonna put 3 meter long horses within 1 meter of each other. Spread them 10 meters apart (e.g. 1 second following time for a 10m/s gallop), and suddenly normal eyes work just fine.

51. #51 Eofhan
June 25, 2009

And why does Gandalf need a staff for walking if he is able to fight a Balrog?

In Middle Earth, a Wizard’s staff is necessary to use magic. That’s why Gandalf broke Saruman’s staff when Saruman refused parole.

Gandalf refused to surrender his staff at Theoden’s door, and offered to release Saruman from ruined Isengard in exchange for Saruman’s staff (and the key of Orthanc). Those incidents tend to support the idea that the staff must also be in direct possession of the Wizard.

I don’t know if a staff has inherent magical power (like a D&D Artifact), or acts to focus the Wizard’s power (like Harry Dresden’s staff). If the latter, then I suppose it’s sort of a lens and we’re back to Rayleigh’s . . . ;)

I wonder if magic is quantized?

52. #52 skimitar
June 26, 2009

A mathematical analysis of Tolkien?

Remember Gandalf’s words to Saruman!

“He that breaks a thing to find out what it is has left the path of wisdom.”

53. #53 jason
June 26, 2009

Elves can simulate a larger eye by running back and forth… just like the telescopes that can be ‘as big as the earth’.

54. #54 jason
June 26, 2009

Elves can simulate a larger eye by running back and forth… just like the telescopes that can be ‘as big as the earth’.

55. #55 Jon H
June 29, 2009

Eagles.

I’m not sure how, but it involves giant eagles.