May 25th, 2018, 6:59 am

[$]\sqrt{7}=2+\overline{1,1,1,4}[$] where the number under the bar is the periodic integer sequence in the continued fraction. ppauper's calculator confirms this. ppauper missed the 4. So if the continued fraction is indeed for [$]\sqrt{7}[$], there should have been a [$]4[$] below and to the right of the rightmost 1. There is no 4 to be found.

The continued fraction can only be represented by the whole not part of its periodic part. Otherwise it would be absurd. That would have meant [$](1,1,1)[$] represents [$]\overline{1,1,1,2}[$] and [$]\overline{1,1,1,3}[$] and [$]\overline{1,1,1,15,3,7}[$] which are all different numbers.

Therefore the continued fraction in the painting can only represent [$]\overline{1}=\frac{\sqrt5-1}{2}[$] the golden ratio as I have said. This is easily checked by [$]\frac{\sqrt5-1}{2}=\frac{1}{1+\frac{\sqrt5-1}{2}}[$] with period [$]1[$]. And [$]\sqrt7\ne 2+\frac{\sqrt5-1}{2}[$]. The expression in the painting is thus wrong.

As for catastrofa's derivation, the obvious mistake is that just as in the painting, no 4 is to be found there. Also the derivation has no distinct natural numbers [$]i, j[$] shown such that [$]P_i=P_j[$]. In other words, there is no periodicity shown. So the derivation is incomplete and wrong.