Where the numbers come from
The orbits are the stable, coplanar solutions in Table 4 of Jason Wang and colleagues' 2018 paper on HR 8799, fitted to Gemini Planet Imager astrometry: semi-major axes of 70.8, 43.1, 26.2 and 16.2 au, and a star of 1.47 solar masses. I computed the periods from those numbers with Kepler's third law; the page recomputes them as you load it.
The paper's own period ratios are 2.11 for b to c, 2.10 for c to d and 2.06 for d to e, each a little above 2. Only orbits that were both coplanar and near 1:2:4:8 stayed stable in large numbers when they ran the system forward for its 40 million years.
The twelve years
A Hacker News post tonight about a 12-year image sequence of a star with four planets sent me here. Jason Wang's 2023 time-lapse of HR 8799 shows 12 years of Keck images in 4.5 seconds. The shaded arcs on the plate are those 12 years: a fifth of the way round for the inner planet, a fortieth for the outer one.
The planets start where they are tonight. For 3 October 2026, whereistheplanet.com (Jason Wang, Matas Kulikauskas, Sarah Blunt) predicts b at 1,718.7 milliarcseconds from the star and position angle 74.2°, c at 959.6 mas and 347.9°, d at 701.7 mas and 248.3°, e at 408.8 mas and 343.1°. In the Earth view those are the hollow rings. On the sky the planets move counter-clockwise: the same site puts e 27° further round by 2031. From there the page moves them on circular orbits, so it drifts from the real predictions as the years run.
Northwestern's write-up says the innermost planet takes about 45 years; the Table 4 orbit gives about 54. Different fits give different numbers, and I am using the paper's table rather than the press release. The face-on view draws the orbits as circles. The Earth view tilts them by the paper's 26.8° with the line of nodes at 67.9°, north up and east to the left as in a telescope image. The real orbits are nearly circular, with eccentricities up to 0.13.