The square root of 2, and multiples of 7

The square root of 2 has a mathematical curiosity. It starts with a long string of multiples of 7.

For reference
\[ \sqrt{2} = 1.4142135623730950\dots \]
\[ \frac{\sqrt{2}}{7} = 0.2020305089104421\dots \]

I've known about this for years. Chalked it up to a mathematical coincidence, and not really thought about it. What connects two, seven and ten (where ten comes from the numeric base used)? The answer is so simple

\[ \sqrt{\frac{1}{2}} = 0.7071067811865475 \dots \approx 0.7 \]