[Matt Q]

[Hadn't read the whole thread.]

[Sarah-Jane Crowson]

If so, what makes you think that there is some kind of "ghost image" where two perpendicular mirrors meet?

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

[Alexander Givental]

Originally Posted by Carl Copeland

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!”

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.

[Alexander Givental]

Originally Posted by Sarah-Jane Crowson

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

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").

[Matt Q]

[misunderstanding]

[Orwn Acra]

Originally Posted by Alexander Givental

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.

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!

[Carl Copeland]

Originally Posted by Roger Slater

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.

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.

[Matt Q]

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

[Alexander Givental]

Originally Posted by Carl Copeland

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: (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.

[Alexander Givental]

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.