The digits of \(\pi\) and \(e\) were laid alongside each other. Of course in certain places the digits between the two match. These points were fixed. Then the minimum systematic list of transformations to make \(e\) into \(\pi\) was found up to \(~30\) terms. The method was such, starting with \(e^*=e\)
-Read along \(e^*\) until we find a digit that is different to that digit of \(\pi\). -Find the smallest digit that when swapped with this digit will also be the same as it’s \(\pi\) equivalent. -Swap them. Repeat.
The list so far is \[\begin{array}{|c|c|c|} \hline Move & Digits\;Swapped & Transform \\ \hline 1. & e029 <-> e001 & 2 \;became\; 3 \\ 2. & e121 <-> e002 & 7 \;became\; 1 \\ 3. & e104 <-> e003 & 1 \;became\; 4 \\ 4. & e085 <-> e004 & 8 \;became\; 1 \\ 5. & e077 <-> e005 & 2 \;became\; 5 \\ 6. & e036 <-> e006 & 8 \;became\; 9 \\ 7. & e041 <-> e007 & 1 \;became\; 2 \\ 8. & e106 <-> e008 & 8 \;became\; 6 \\ 9. & e094 <-> e009 & 2 \;became\; 5 \\ 10.& e162 <-> e010 & 8 \;became\; 3 \\ 11.& e152 <-> e011 & 4 \;became\; 5 \\ 12.& e091 <-> e012 & 5 \;became\; 8 \\ - & e013---e013 & 9 \;remain\; 9 \\ 13. & e265 <-> e014 & 0 \;became\; 7 \\ 14. & e184 <-> e015 & 4 \;became\; 9 \\ 15. & e144 <-> e016 & 5 \;became\; 3 \\ - & e017---e017 & 2 \;remain\; 2 \\ - & e018---e018 & 3 \;remain\; 3 \\ 16. & e257 <-> e019 & 5 \;became\; 8 \\ 17. & e026 <-> e020 & 3 \;became\; 4 \\ - & e021---e021 & 6 \;remain\; 6 \\ 18. & e066 <-> e022 & 0 \;became\; 2 \\ 19. & e150 <-> e023 & 2 \;became\; 6 \\ 20. & e035 <-> e024 & 8 \;became\; 4 \\ 21. & e <-> e025 & 7 \;became\; 3 \\ \hline \end{array}\]
Giving \(_{10}S^{e}_{\pi}=\)29,121,104,85,77,36,41,106,94,162,152,
91,13,265,184,144,17,18,257,26,21,66,150,35,... Where the bottom left index is the base, and the right two indices are the numbers that transform between each other. This exact transform can then be applied to other irrational numbers to map them to other constants. We have picked the transformation such that when applied to \(e\) we get \(\pi\) and when applied to \(\pi\) we get \(e\). When applied to \(\sqrt{2}\) for example we get the number \[\sqrt{1}\to 0.0000000000000000000000000001 \\\sqrt{2}\to 2.400176372943... \\
\sqrt{3}\to 5.961868679368... \\\sqrt{4}\to 0.0000000000000000000000000002 \\
\sqrt{5}\to 7.454836887489... \\\sqrt{6}\to 7.459399441233... \\\sqrt{7}\to 6.500222445164... \\\sqrt{8}\to 4.900253754886... \\
\sqrt{9}\to 0.0000000000000000000000000003 \\\sqrt{10}\to4.834235792188... \\
\sqrt{11}\to6.208080287795...\\\sqrt{12}\to0.922636248737...\\
\sqrt{13}\to4.654042489723...\\
\sqrt{14}\to3.59 \\ \sqrt{15}\to7.33
\sqrt{16}\to0.0000...04 \\ \sqrt{17}\to9.5576214029 \\
\sqrt{18}\to6.4013301 \\ \sqrt{19}\to8.96615081 \\
\sqrt{20}\to4.8086726
\sqrt{21}\to7.3647840
%...........s.dsstddd
\\
a=0.000000000001\cdot n \to a \cdot n , \; n \in [0,9]\]
An interesting test would be the base \(10\) Champernowne constant, however it is not clear which digit is \(1\) the \(0\) or the \(1\), so two possibilities are \[C_{10} = 0.1234567891011213... \\ C_{10}\to 9.557325558801 \\ C_{10}\to 1.664422715051\]
Try \[C=0.\overbar{123456789} \\ C \to 1.343484638793... \\ C \to 2.454595749814... \\ C=0.\overbar{01} \\ C \to\]
Noting that sqrt(5)/sqrt(3) is quite close to 7.454.../5.961... with a difference of \(0.040574942971...\). Noting that the transformations of sqrt(5) and sqrt(6) are close.
There will be infinite numbers that are invarient to the transform, but less that all numbers clearly. The transform when applied twice undoes itself.