Period of Reciprocal of 729 is 81

Theorem
The decimal expansion of the reciprocal of $729$ has $\dfrac 1 9$ the maximum period, that is, $81$:


 * $\dfrac 1 {729} = 0 \cdotp \dot 00137 \, 17421 \, 12482 \, 85322 \, 35939 \, 64334 \, 70507 \, 54458 \, 16186 \, 55692 \, 72976 \, 68038 \, 40877 \, 91495 \, 19890 \, 26063 \, \dot 1$

The recurring part can be arranged in groups of $9$ digits each, revealing an interesting pattern:

... that is, each row (apart from the last) can be obtained from the previous one by adding $111 \, 111 \, 111$ to it.

Proof
Performing the calculation using long division:

0.000137174211248285322359396433470507544581618655692729766803840877914951989026063100  --- 729)1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000      729      ---      2710      ....