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 -> 789Sum 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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$\begingroup$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.
$\endgroup$ 9 $\begingroup$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.
$\endgroup$ 4 $\begingroup$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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