Early-type galaxies -- slow and fast rotating ellipticals (E-SRs and E-FRs) and S0s/lenticulars -- define a Fundamental Plane (FP) in the space of half-light radius $R_e$, enclosed surface brightness $I_e$ and velocity dispersion $sigma_e$. Since $I_e$ and $sigma_e$ are distance-independent measurements, the thickness of the FP is often expressed in terms of the accuracy with which $I_e$ and $sigma_e$ can be used to estimate sizes $R_e$. We show that: 1) The thickness of the FP depends strongly on morphology. If the sample only includes E-SRs, then the observed scatter in $R_e$ is $sim 16%$, of which only $sim 9%$ is intrinsic. Removing galaxies with $M_*<10^{11}M_odot$ further reduces the observed scatter to $sim 13%$ ($sim 4%$ intrinsic). The observed scatter increases to the $sim 25%$ usually quoted in the literature if E-FRs and S0s are added. If the FP is defined using the eigenvectors of the covariance matrix of the observables, then the E-SRs again define an exceptionally thin FP, with intrinsic scatter of only $5%$ orthogonal to the plane. 2) The structure within the FP is most easily understood as arising from the fact that $I_e$ and $sigma_e$ are nearly independent, whereas the $R_e-I_e$ and $R_e-sigma_e$ correlations are nearly equal and opposite. 3) If the coefficients of the FP differ from those associated with the virial theorem the plane is said to be `tilted. If we multiply $I_e$ by the global stellar mass-to-light ratio $M_*/L$ and we account for non-homology across the population by using Sersic photometry, then the resulting stellar mass FP is less tilted. Accounting self-consistently for $M_*/L$ gradients will change the tilt. The tilt we currently see suggests that the efficiency of turning baryons into stars increases and/or the dark matter fraction decreases as stellar surface brightness increases.
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