Analysis of the recent long exposure Chandra X-ray observation of the early-type O star zeta Pup shows clear variability with a period previously reported in optical photometric studies. These 813 ks of HETG observations taken over a roughly one year time span have two signals of periodic variability: a high significance period of 1.7820 +/- 0.0008 day, and a marginal detection of periodic behavior close to either 5 day or 6 day period. A BRITE-Constellation nanosatellite optical photometric monitoring, using near-contemporaneous observations to the Chandra data, confirms a 1.78060 +/- 0.00088 day period for this star. The optical period coincides with the new Chandra period within their error ranges, demonstrating a link between these two wavebands and providing a powerful lever for probing the photosphere/wind connection in this star. The phase lag of the X-ray maximum relative to the optical maximum is approximately phi=0.45, but consideration of secondary maxima in both datasets indicates possibly two hot spots on the star with an X-ray phase lag of phi=0.1 each. The details of this periodic variation of the X-rays are probed by displaying a phased and trailed X-ray spectrum and by constructing phased light curves for wavelength bands within the HETG spectral coverage, ranging down to bands encompassing groups of emission lines. We propose that the 1.78 day period is the stellar rotation period and explore how stellar bright spots and associated co-rotating interacting regions or CIRs could explain the modulation of the optical and X-ray output for this star and their phase difference.