  Temporally periodic solutions are extracted numerically from forced box
turbulence with high symmetry. Since they are unstable to small perturbations,
they are not found by forward integration but can be captured by Newton-Raphson
iterations. Several periodic flows of various periods are identified for the
micro-scale Reynolds number $R_{\lambda}$ between $50$ and $67$. The
statistical properties of these periodic flows are compared with those of
turbulent flow. It is found that the one with the longest period, which is two
to three times the large-eddy-turnover time of turbulence, exhibits the same
behaviour quantitatively as turbulent flow. In particular, we compare the
energy spectrum, the Reynolds number dependence of the energy-dissipation rate,
the pattern of the energy-cascade process, and the magnitude of the largest
Lyapunov exponent. This periodic motion consists of high-activity and
low-activity periods, which turbulence approaches, more often around its
low-activity part, at the rate of once over a few eddy-turnover times. With
reference to this periodic motion the Kaplan-York dimension and the
Kolmogorov-Sinai entropy of the turbulence with high symmetry are estimated at
$R_{\lambda}=67$ to be $19.7$ and $0.992$ respectively. The significance of
such periodic solutions, embedded in turbulence, for turbulence analysis is
discussed.
