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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. –  Robert Israel Nov 7 '12 at 7:50
    
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. –  PHPGAE Nov 7 '12 at 7:55
    
A $4-3-2$ triangle has no 90 degree angle. –  Robert Israel Nov 7 '12 at 8:04
    
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... –  PHPGAE Nov 7 '12 at 8:11

2 Answers 2

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? –  PHPGAE Nov 7 '12 at 8:18
    
Look up modular arithmetic: en.wikipedia.org/wiki/Modular_arithmetic –  Robert Israel Nov 7 '12 at 8:30
    
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. –  PHPGAE Nov 7 '12 at 8:44
    
PHPGAE, I think you are grasping at straws. –  Dan Brumleve Nov 7 '12 at 8:48
    
"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? –  PHPGAE Nov 7 '12 at 8:55

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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1  
No, palindromes are numbers that are their own mirrors. –  Robert Israel Nov 7 '12 at 7:44
    
Misread the question and jumped the gun. Fixed, I think. –  Dan Brumleve Nov 7 '12 at 7:45
    
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: 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. –  PHPGAE Nov 7 '12 at 9:18
    
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. –  Dan Brumleve Nov 7 '12 at 10:09

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