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I'd like to hear insights and theory of the mirror numbers and their possible significance in mathematics and geometry. With mirror numbers I mean these four examples:

432 -> 234 123 -> 321 153 -> 351 987 -> 789 

Sum of 432 & 234 is 666 and sum of 153 & 351 is 504, which are famous numbers from historical perspective, namely from Pythagoras, Plato, Archimedes and Revelation of John.

Supplementing questions arose on a chat with Dan:

1) How to determine if a number x can be represented as n + rev(n)?

2) How to determine possible n & rev(n) for number x?

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    Um, a $4-3-2$ triangle is congruent to a $2-3-4$ triangle.2012-11-07
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    But of course. I was thinking triangles with 90 degree corner first, but it seems triangles are not very good geometrical presentation of these numbers as they look same, until orientation is regarded.2012-11-07
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    A $4-3-2$ triangle has no 90 degree angle.2012-11-07
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    Any scalene triangle could be presented with two right triangles. Would the properties of these two triangles in 4-3-2 be congruent with two right triangles of 2-3-4? I guess they would...2012-11-07
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    See [OEIS sequence A067030](https://oeis.org/A067030).2017-04-27
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    Seems to be an exact match, @Robert Thanks for reference and heads up for 5 years old issue.2017-04-28

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This is not terrifically profound, but if $y$ is the mirror of $x$ in base $b$, then $x \equiv y \mod b-1$, while $x \equiv \sigma y \mod b+1$ where $\sigma = 1$ if $x$ and $y$ have an odd number of base-$b$ digits and $-1$ if they have an even number of base-$b$ digits.

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    Looks interesting, but statement should be deciphered a little bit to be understandable for me :) What is sign ≡ in this case?2012-11-07
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    Look up modular arithmetic: http://en.wikipedia.org/wiki/Modular_arithmetic2012-11-07
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    That's a really nice hand. Mentioned numbers are supposedly precessional numbers, thus part of the cycle of the precessional year. So at least modulo 12, 72, 360 could be used here, if I understood correctly. 153 was the fractional part of the square root 3 used by Archimedes and I'm pondering if it really is just a random consequence, that it adds up with mirror number to the number Plato used for ideal citizenship units. Or if there is some other mathematical correlation between these. Geometrical topic I brought here for possible visual demonstration purposes.2012-11-07
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    PHPGAE, I think you are grasping at straws.2012-11-07
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    "grasping at straws. Fig. to depend on something that is useless; to make a futile attempt at something." Is that what you mean with the statement Dan?2012-11-07
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    Yes, you are referring to numerology which won't lead to any knowledge.2012-11-07
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    Well then you need to speak of usefulness to yourself only. I'm studying and applying numbers to my own interests which are useful to me, so please don't mind that.2012-11-07
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    @PHPGAE that is understandable. I meant mathematical knowledge as I see it.2012-11-07
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    let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/6362/discussion-between-dan-brumleve-and-phpgae)2012-11-07
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Such pairs of numbers have no intrinsic mathematical significance (although related numbers are studied; see palindromic numbers) due to the arbitrary choice of base $10$. I don't know of any geometric connection but I would expect it to be very subtle, not as direct as what you are suggesting. The fact that there is both a 2-3-4 triangle and a 6-6-6 triangle is a coincidence emboldened by the choice of a relatively large base.

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    No, palindromes are numbers that are their own mirrors.2012-11-07
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    Misread the question and jumped the gun. Fixed, I think.2012-11-07
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    Base 10 is actually a good point. How does situation change if we treat given numbers base 60? It's likely given numbers and results should be actually treated base 60, because ancient cultures used Sexagesimal system: http://en.wikipedia.org/wiki/Sexagesimal Of course for the question I'm looking for math connection on mirror numbers in general, but as talking number base here, I think it's good background information. I hope that people who don't like these historical connections, will just pass them and contribute from plain math view of point, if they have something to say.2012-11-07
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    Base 2 would be a better place to start with more fundamental meaning. The smaller the base the better the theorem. The historical connections are interesting but consider the context: 60 and 360 are used as bases because of their smoothness, not because of any digital consummations.2012-11-07
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Subtract the small mirror from the large mirror & divide by 9. It's fun and very symbolic. Mirrors without and within.

For example:

2559591 - 1959552 = 600039

600039 / 9 = 66671

8883 - 3888 = 4995

4995 / 9 = 555

5832 - 2385 = 3447

3447 / 9 = 383

25367473251-15237476352 = 10129996899

10129996899/9 = 1125555211

Try things like year of birth & another year:

5891 - 1985 = 3906

3906 / 9 = 434

7102 - 2017 = 5085

5085 / 9 = 565

565 + 434 = 999

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    There is a rule of nine. Transpose digits of the number in any base. Substract and result should be divisible by base - 1. Above mirror feature is a special case of such much more general feature.2017-08-07