How well do planetary orbits fit with Johannes Kepler's in- & circumscribed Platonic solids?
It is easy enough to do the calculations, formulae for the in and circum radii of the Platonic solids can be found here which give ratios of circ to in radii of (note formulea for the radii have dropped common factor of the side length, which we don't need as we are interested in the ratios):

&gt;ri4=sqrt(6)/12,rc4=sqrt(6)/4
     0.204124 
     0.612372 
&gt;
&gt;ri6=1/2,rc6=sqrt(3)/2
          0.5 
     0.866025 
&gt;
&gt;ri8=sqrt(6)/6,rc8=sqrt(2)/2
     0.408248 
     0.707107 
&gt;
&gt;ri12=sqrt(250+110*sqrt(5))/20,rc12=(sqrt(15)+sqrt(3))/4
      1.11352 
      1.40126 
&gt;
&gt;ri20=(3*sqrt(3)+sqrt(15))/12,rc20=sqrt(10+2*sqrt(5))/4
     0.755761 
     0.951057 
&gt;
&gt;rho4=rc4/ri4
            3 
&gt;rho6=rc6/ri6
      1.73205 
&gt;rho8=rc8/ri8
      1.73205 
&gt;rho12=rc12/ri12
      1.25841 
&gt;rho20=rc20/ri20
      1.25841 


Which may be compared with the orbital radius ratios from here (radii in km)

&gt;RMecury=57.9e6;
&gt;RVenus=108.2e6;
&gt;REarth=149.6e6;
&gt;RMars=227.9e6;
&gt;RJupiter=778.3e6;
&gt;RSaturn=1426.7e6;


Now we can compare the corresponding radii ratios:

&gt;[RVenus/RMecury,rho8]
      1.86874       1.73205 
&gt;[REarth/RVenus,rho20]
      1.38262       1.25841 
&gt;[RMars/REarth,rho12]
       1.5234       1.25841 
&gt;[RJupiter/RMars,rho4]
      3.41509             3 
&gt;[RSaturn/RJupiter,rho6]
       1.8331       1.73205 


Which, as these things go, is not bad.