  The geometric arrangement of planet and moon orbits into a regularly spaced
pattern of distances is the result of a self-organizing system. The positive
feedback mechanism that operates a self-organizing system is accomplished by
harmonic orbit resonances, leading to long-term stable planet and moon orbits
in solar or stellar systems. The distance pattern of planets was originally
described by the empirical Titius-Bode law, and by a generalized version with a
constant geometric progression factor (corresponding to logarithmic spacing).
We find that the orbital periods $T_i$ and planet distances $R_i$ from the Sun
are not consistent with logarithmic spacing, but rather follow the quantized
scaling $(R_{i+1}/R_i) = (T_{i+1}/T_i)^{2/3} = (H_{i+1}/H_i)^{2/3}$, where the
harmonic ratios are given by five dominant resonances, namely $(H_{i+1} :
H_{i}) = (3:2), (5:3), (2:1), (5:2), (3:1)$. We find that the orbital period
ratios tend to follow the quantized harmonic ratios in increasing order. We
apply this harmonic orbit resonance model to the planets and moons in our solar
system, and to the exo-planets of 55 Cnc and HD 10180 planetary systems. The
model allows us a prediction of missing planets in each planetary system, based
on the quasi-regular self-organizing pattern of harmonic orbit resonance zones.
We predict 7 (and 4) missing exo-planets around the star 55 Cnc (and HD 10180).
The accuracy of the predicted planet and moon distances amounts to a few
percents. All analyzed systems are found to have $\approx 10$ resonant zones
that can be occupied with planets (or moons) in long-term stable orbits.
