[Alexander Givental]

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

[Allen Tice]

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

[RCL]

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

[Alexander Givental]

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).

[Carl Copeland]

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.

[Roger Slater]

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

[Ann Drysdale]

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

[Allen Tice]

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.

[Roger Slater]

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.

[Allen Tice]

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.