Mirror - mirror - mirror
 

This post is inspired by Ralph's previous one - I haven't found my double, but there will be mirrors.
Here is a math challenge for Sphereans which I sometimes use with little kids as well as with college math majors (all fail at their first try):

Walking with an iPhone in my hand through a modern art museum, I come to an installation at the corner of the hall: the floor and both walls are made of mirror. How many iPhones will I see?

A hint for the Sphereans is in the following poem, by Russian poet and translator Dmitry Usov (1896-1943), and in its English translation (by Elysée Wilson-Egolf and myself):

Translator

A peaceful evening with a book in hand.
The clock's tick tocks do not resemble racing.
Before me, figments from a poet's head
Lie represented in their author's phrasing:

“At dusk, the silence is more vibrant yet,
And streets before the night are growing quiet,
The moon puts on its windowpane lorgnettes,
But greets me through the looking-glass in private.”

From these four lines, I pull a guiding thread;
They're given – neither narrower, nor wider,
I cannot say directly what they said,
And nonetheless all four will be recited:

“In evening hours, utterings reverb,
And city noises, fading, disappear.
I'm facing – not the shining lunar orb,
But its representation in the mirror.”


Переводчик

Недвижный вечер с книгою в руках,
И ход часов так непохож на бегство.
Передо мною в четырех строках
Расположенье подлинного текста:

«В час сумерек звучнее тишина,
И город перед ночью затихает.
Глядится в окна полная луна,
Но мне она из зеркала сияет».

От этих строк протягиваю нить;
Они даны — не уже и не шире:
Я не могу их прямо повторить,
Но все-таки их будет лишь четыре:

«В вечерний час яснее каждый звук,
И затихает в городе движенье.
Передо мной — не лунный полный круг,
А в зеркале его отображенье».

15 февраля 1928

One, only one, forever just one iPhone, because ....

Eggsactly! It's one of the few numbers I can count on. But there are innumerable reflections of the phone?

The first attempted answers by Allen and Ralph are wrong (expectedly, as I said). But the correct one is indeed in the poems (and your answers are not even related to them).

Alexander, your solution may be convincing to the exclusion of all else, but right now it seems to me that Allen and Ralph have a valid alternative. And it is in the poem: the translator sees a representation of the moon, not the moon itself. “Four” is another answer suggested by the poem, but that’s too easy, so as usual I’m at a loss.

Zero. You have a very large hand, so the iPhone is entirely blocked from sight. Besides, your eyes are closed.

With all those mirrors, would not even one iPhone be travelling at the speed of light into infinity?

No, Alexander, you are wrong. Representations of the phone will be limited to one in each mirror because you have given only one mirror in each of the x,y, and z planes. There are no multiple reflections unless superior peripheral vision is included. Those are representations only anyhow. There is, definitely, only One phone. You did not include reflections or anything beyond in the problem statement.

If you have almost parallel mirrors and stand between there will be infinite regression. You have no parallel mirrors. Your problem as stated allows only one physical phone. Reflections do not count. I am not fooled.

But you are assuming that you're alone in the room. I live in NY, where the modern art museum is invariably crowded, and several people in the room are bound to be carrying iPhones.

And what does "both walls" mean? Most rooms have more than two walls.

A careful reading of the problem statement shows that this a standard corner reflector. We see them every time we shine our car headlights onto a modern STOP sign that is painted with paint containing crystals that are tiny 90 degree corner reflectors. Standing on the floor mirror facing the corner, there will be a mirror on each side that intersect at presumably 90 degrees, and of course the floor at 90 degrees. Even in a modern art museum that humans might visit, there won't be significantly curved or involuted space. Poetry does not supersede physical reality. I regret that I am not amused at all.

One . All others are reflections.

OK, in engineering, problems are often solved by rethinking the initial set up (the solution is then called "invention"). In math, problems are to be considered within the given, and somewhat idealized setting. In particular - no cheap cheating, such as "ah! reflections are not real iPhones", or "there could be other people with iPhones in the room", or "the hand blocks the view", or "in modern art the angles between the walls don't have to be 90 degrees", or "if I close my eyes ..." - no, the situation is meant to be the most natural (and I did see such an installation in a museum), and the question, of course, is about the number of my iPhones I see, including the reflections and the actual one.

Roger, by "both walls" I mean that at the corner of the room, (of course, parts of) the walls and the floor are mirrors - in other parts of the room, some other modern art pieces are positioned.

Allen, my answer cannot be wrong, since I haven't given any one yet. You and Carl gave the same answer, 4, but for different reasons: you - that there are 3 reflections, one in each of x-,y-,z- planes plus the actual iPhone, and Carl - based on the poems.
Carl, could you please clarify how the poem gives 4?

Two. The "iPhone" in the first sentence and the "iPhone" in the second.

The actual answer though is undefined as is the question.

Oh, I LOVE these kinds of puzzles.

I'd say 'none'. You are holding the iphone and looking in the mirror and the installation. You will see a representation of the iphone, not the phone itself.

(I also usually get them wrong. I still love them though)

Sarah-Jane

(I've just read the thread more carefully - I was in a rush before - and now realise that you don't want the representation answer. I guess then it depends what you are focussing on. Are you holding up the iphone in front of you and deliberately looking for reflections?)

Is there an absolute "right" answer?
Or is there a "consensus" answer?
Or is the best answer one that takes into account things like the carrier of the iPhone is at the Museum of Modern Art?
Was the mirrored wall and floor an art exhibit? Or part of one?
Was the iPhone held in a certain way as to prevent its reflection in one of the mirrors?

I still stick to my first answer: One (assuming the iPhone is visible in the hand). All the others are reflections.
.

I'll say four. If just two 90 degree walls had mirrors, there would be three. One on each wall, plus a half on each where they join. I'm not sure how the floor would figure in, though. Beyond my puny brain.

That was my line of reasoning too, along with the repeated “four” in the poem, but I dismissed it as too easy and too superficial a use of the poem. And if you’ve got your camera screen turned on … but that’s “cheap cheating” again. How different math is from philosophy! To say that it’s “cheating” to distinguish image from reality!

Orwn, you are talking about the word "iPhone" in the formulation of my question, but that's easy - and it wasn't the question.

The actual answer is not less undefined than that to 2+2=? (Most people would say 2+2=4, though there are some unusual situations where that's not true.)

The cheating is in the betting on the ambiguity of the words.
No, the iPhone has nothing to do with the question: when I ask 3rd graders, I draw a girl with a flower and ask how many flowers she sees.

"Four" indeed occurs in the poem, but this is not specific to the poem - any quatrain has 4 lines. To give you a further "poetic" hint:

There is an article by Gasparov about this poem of Usov (mostly he compares the lexicon of the "source" and "translation" to conclude that the mock translation is indeed a translation of the source, and not the other way around) but at the end he praises the poem for the excellent match between form and content.

You are right in trying first a simpler problem - with two mirrors instead of three. But isn't a translation akin to a mirror image? In my post, you can easily see at least two such "mirrors" (especially if you imagine the original and the English translation typeset side-by-side as it is done here:
https://math.berkeley.edu/~giventh/v...translator.pdf
But how many "mirror images" (including the source) are there?

There is a right answer in the same sense in which 2+2=4 is right.
But last time I tried it with college math majors, we had all kinds of hypothetical answers (all wrong: infinity, 3, 4, 11(?), ...), and then we took a vote (for, "that's how all scientific truth are decided," said I, hinting specifically at the global warming problem).

You are right, other iPhones are reflections, but how many of them are there? (OK, when you give me your answer, I'll take the burden of adding the iPhone itself on myself.)

Sarah-Jane,

Of course, I didn't mean any tricky interpretation of the question. If you wish, I can be more accurate and ask: what's the maximal number of images of an object (including the object itself) which one can see in a room whose 2 adjacent perpendicular (and straight) walls and the floor are made of mirrors?
And I understand that you like such puzzles for their own sake, but I can assure you that I wouldn't post this into a poetry forum if the situation was not perfectly illustrated by these poems.

Why then you don't object the phrase written on the right mirror of every car: "Objects in mirror are closer than they appear"?

So, if the images were parallel then there would be infinite images, but they aren't, so there isn't.

Without getting out the aluminium foil (I'm tempted), I'm guessing there might be two clear reflections in each 'side' and a ghost image in the middle where the mirrors meet - this is perhaps the 'guiding thread' pulling the four reflections together in the poem.

But then that doesn't account for the floor. That won't reflect the person
maybe but will account for another two reflections from each side panel taking it to eight.

But will it be nine with a further 'ghost' image bringing it together, or even ten if there are two ghost images where the floor panels meet the side panels.

The poem seems to imply just one 'thread', so I'm going to go for two images in each side panel, four at the bottom and one 'ghost image' where they all meet, making it an unlikey-to-be-correct nine!

Sarah-Jane
(thank you - I am enjoying this very much - also, I love 'window-pane lorgnettes' in the poem)

If you are going to tell us a poem with the line "represented in their author's phrasing" is a hint, then you can't dismiss my answer, which follows the train of thought you laid down the tracks for. The iPhone is represented twice in your word-problem. If you don't want it turned into a word problem, don't tell us a poem about linguistic reference is a clue–or better yet, acknowledge that the poem opens up the possibility for other answers. (I suspect you do not also give the poem to your math students).

Orwn, in another reply I mentioned an article of Gasparov about this poem, where he praises it for the coherence between form and content. So, the hint is not in some subtle connotations hidden in the poem but in the very essence of it: the quatrain being mock-translated in it is about a mirror reflection of the Moon, which serves at the same time as a metaphor for the very act of translating.

Sarah-Jane,
Congratulations (almost) -- you've figured out the correct answer, but then you confused yourself. To "unconfuse" you:

What was the reason you said first that if the mirrors were parallel, the number of images would be infinite? Because, I guess, you realize that images reflected in one mirror can, together with that mirror, be reflected in the other mirror, and so on. And this happens even though the mirrors themselves don't meet at all - they are parallel. If so, what makes you think that there is some kind of "ghost image" where two perpendicular mirrors meet?

Back to the poem, I guess you are treating two translations as two perpendicular mirrors (as they are!) and find there 4 versions of the same
"original" stanza: the original, its mock-translation from Russian into Russian, then its translation in English, and then the mock-translation of that English translation into English (or, maybe, it is the English translation of the Russian mock-translation). But why do you treat the last version as merely a "ghost image"? (Is our translation that bad :-?) I think there are 4 more-or-less equivalent versions of the stanza there, and the last one is not hiding in any line between mirrors, but a legitimate reflection of a reflection.

Thus, you are right that by bringing in the 3rd mirror, you'll find one more reflection of each of the four images, so totally eight (imagine a translation of the whole Russian-English thing to Esperanto). But they have nothing to do with the lines where the mirrors meet.

So, here is the next challenge for you: find the third mirror in the poem and figure out what should you count there to get the answer 8.

P.S. The "lorgnettes": in the Russian the image is different - there the full Moon is looking at herself in all windows (but shines at the author from a mirror).

This. Look at the first picture.

It is an unfortunate viewing angle: move the camera, and you'll see that the middle image is not at the intersection line at all.

I feel like that third-grader. You ask her how many flowers the girl sees. The third-grader answers, “One” (as some of us did), and you say, “No, eight!”

[Hadn't read the whole thread.]

Bevelled edges. I have a material mind. I was looking for the 'guiding thread' (S3) and 'what could not say directly' and some kind of metaphor for reflection and translation (refraction and prisms).


Sarah-Jane

That's not how it works. Third-graders give different answers: 1,4, infinity. Then we consider the case of 2 mirrors and draw pictures. Kids argue and eventually convince each other that it is 4: by someone explicitly drawing the 4 trajectories of the light ray from the flower to the eye (and finding a universal way of drawing them all). Then it becomes clear to everyone in class that in the 3-space the answer is 8 (though it is hard to draw pictures), and in the 4-space 16, and so on.

If you meant that bevels themselves are mirror-like surfaces, then they would make many more reflective surfaces. As the photo https://www.physicsclassroom.com/cla...-Angle-Mirrors
linked to by Roger clearly shows, the 4 images (of the candle, but also of the hand holding it) are not in bevels, but in the mirrors themselves. Imagine now that the table there is also a mirror surface. Then under the candle and each of its 3 reflections there will be one more, up-side-down copy of it.

P.S. The impossibility of saying "directly" what the source quatrain says is just to say that it needs to be a "translation" as if to a different language (because technically speaking, since the "translation" is in the same language, it is perfectly possible to just copy the source verbatim and say "this is my translation").

[misunderstanding]

I could say the same thing about how the line I quoted before--"represented in their author's phrasing"--also embodies the essence of your word-problem in that the words serve as a reflection of their author's meaning, which are in turn reflected in the mind of the reader, who translates the word into their personal idiom (Benjamin's whole theory of translation). Frankly this: "the original, its mock-translation from Russian into Russian, then its translation in English, and then the mock-translation of that English translation into English (or, maybe, it is the English translation of the Russian mock-translation)" is far more hidden in the poem than the line I quoted. (Also, this four-fold translation assumes a lot about psycholinguistics that not every linguist would agree with).

Which brings me to my first answer: that the problem and solution are undefined. You expect us to follow your train of thought precisely even though the words you have used to articulate the problem expand beyond your control. So Allen's answer is also correct; when you say "iPhone" there is no reason for anyone to consider that the phone and the reflection of the phone are the same thing, but your word-problem depends on our doing so. You even have to come back and further explain what you mean to set us within certain parameters of thought. Allen has translated your word "iPhone" in his head to mean the phone itself, unreflected, which he uses to find the answer. This is perfectly acceptable--you can't point us to some Platonic dictionary with all the "correct" meanings of the words!

Roger was on the right track. I see now that two mirrors at right angles produce three reflections. An explanation I found says a third perpendicular mirror reflects those three reflections and the actual iPhone (3 + 3 + 1 = 7), though my puny brain can’t visualize it either. I guess if we add in the original iPhone, we get 8. Is that the solution? Ok, but I don’t see what that has to do with the Russian, English and Esperanto translations (2 + 2 + 4 = 8) or with the girl holding a flower (1 = 1). And, like Walter, I think Allen’s philosophical answer distinguishing image from reality also makes perfect sense.

Carl,

Here's my thinking: Two perpendicular mirrors produce three images, that's one image per mirror and one "composite" image across the join.

If you have three mirrors, then you have, in effect 3 sets of paired mirrors. So presumably, each of those pairs will produce a composite image across the join, since we know that's what a pair of mirrors do. So that's 3 composite images.

Plus each of the 3 mirrors will produce a whole reflection. So, that would make 6 reflections in total. So, at least 6 is what I thought.

I have since looked on the internet, and as you say the answer given is 7 reflections, though I'm not clear how that works (no one gives a clear diagram). I guess the 7th reflection must be bounced off all three mirrors, and possibly is seen in the corner (where the three mirrors join).

As to whether we should also count the real iPhone or not, well, the question doesn't specify whether or not it's visible to me when I look in the mirror, only that it's in my hand. So, I guess it's up to us.

Matt

Carl: (1) the girl with a flower was standing in my mirror corner, so she saw 8 flowers.
(2) to see all 8 images, it suffices in the photo sent by Roger imagine that the table surface on which the candle is standing is reflective. Then you'll see the up-side-down image of the candle below the actual one, and the whole picture of this pair will be reflected 3 times in the mirrors.

Perhaps it is the time to replace hints with direct answers to my puzzle.

First, those who misunderstood the question, or misinterpreted the implicit assumptions, or proposed some interpretations which I dismissed as invalid, did nothing wrong: it is quite usual that when a subject is discussed within a group of people unfamiliar with the implicit assumptions of the subject, all kinds of variations in the interpretation of the situation can occur, and require clarification.

Nevertheless, as the picture https://www.physicsclassroom.com/cla...-Angle-Mirrors sent to us by Roger shows, my question makes quite straightforward practical sense and has a quite definite answer: 2 mirrors produce 3 mirror images, and not 2 how one might think.

A right way to think of mirror reflections in geometrical optics should be familiar from the story about Alice behind the looking glass. Instead of thinking that a light ray from the actual object reflects in a mirror before reaching our eye, one may equivalently think that it is the object that gets reflected in the mirror to occupy a new position behind the glass, and the light ray from that new position travels to our eye straight through the mirror surface. So, the question about the number of images becomes the question about the number of such "new positions".

In fact, if two mirrors in Roger's link were not perpendicular to each other, the number of reflections could be larger than 3 or even (theoretically speaking) infinite - depending on the angle between the mirrors. This is because one will see not only reflections of the object in each of the mirrors, but also reflections in the reflected mirrors, doubly-reflected mirrors and so on. But with the 90-degree angle there will be only 4 images (the object and 3 reflections) and with 3 pairwise perpendicular mirrors 8 (1+7).

To see why, imagine a rectangular box (to be closer to home, think of a cardboard box filled with yet unsold copies of your book). It is a very symmetric shape with 8 corners. It has 3 symmetry planes: one horizontal and two vertical (one parallel to the longer side of the box, the other parallel to the shorter side). Consider one of the 8 corners to be "the object" and start reflecting it in these 3 pairwise perpendicular symmetry planes. No matter how many times you reflect, you'll get just one of the 8 corners of the box. Looking at them from wherever you are you'll see the 8 identical images (the "actual" corner plus its 7 "reflections").

Now, let's return to the poem. In it, Dmitry Usov parallels an act of translating to reflecting in a mirror. So, as one expects, when an object (the original quatrain) is "reflected" (translated), we get two incarnations of it: the original and the translation.

But my post contains also an English translation of the poem (i.e. one more mirror), and - contrary to what one instinctively expects with two mirrors - my post contains not 3 (the source and two reflections) but 4
incarnations of the same quatrain. That was supposed to work as a hint for you (and it did work for some) that two mirrors produce not 1+2 images, but 1+3.

However, in addition to the two metaphorical mirrors (mock-translation from a language to itself, and translation between Russian and English), the poem also contains a third mirror, the looking glass through which the Moon is seen. So, if you count not how many copies of the same quatrains are there, but how many images of the Moon are mentioned there, you literally get the right answer for three mirrors: 8.

The number itself comes simply as the number of regions (they are suggestively called octants) into which three (pairwise perpendicular) planes divide space: each region will contain one incarnation of Alice who travelled across these three mirror planes.