  In this paper, I present a way to compile the surface code into two-body
parity measurements ("pair measurements"), where the pair measurements run
along the edges of a Cairo pentagonal tiling. The resulting circuit improves on
prior work by Chao et al. by using fewer pair measurements per four-body
stabilizer measurement (5 instead of 6) and fewer time steps per round of
stabilizer measurement (6 instead of 10). Using Monte Carlo sampling, I show
that these improvements increase the threshold of the surface code when
compiling into pair measurements from $\approx 0.2\%$ to $\approx 0.4\%$, and
also that they improve the teraquop footprint at a $0.1\%$ physical gate error
rate from $\approx6000$ qubits to $\approx3000$ qubits. However, I also show
that the teraquop footprint of Chao et al's construction improves more quickly
than mine as physical error rate decreases, and is likely better below a
physical gate error rate of $\approx 0.03\%$ (due to bidirectional hook errors
in my construction). I also compare to the planar honeycomb code, showing that
although this work does noticeably reduce the gap between the surface code and
the honeycomb code (when compiling into pair measurements), the honeycomb code
is still more efficient (threshold $\approx 0.8\%$, teraquop footprint at
$0.1\%$ of $\approx 1000$).
