How well do planetary orbits fit with Johannes Kepler's in- & circumscribed Platonic solids? In Mysterium Cosmographicum (1596) Johannes Kepler proposed that the relative distances between the orbits of the six ancient planets (six because heliocentrism had recently added Earth as one of the planet) correspond to the geometry of the five Platonic solids. Each planetary orbit was assumed to be the grand circle of a sphere. Each Platonic solid would circumscribe an inner planet's orbitally defined sphere, and in same position inscribe the next outer planet's "orbital sphere".


How well does it fit?
Is it somehow mathematically related to the equally spurious Bode's law?
I would appreciate some hints or links to the geometric calculation itself, for me to compare with my own exercise trying to solve part of it.


Johannes Kepler proposed this order of matching planets with Platonics (I suppose because this order gives the best fit):

Mercury &lt;-- planet

octahedron &lt;-- Platonic solid

Venus

icosahedron

Earth

dodecahedron

Mars

tetrahedron

Jupiter

cube

Saturn
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.