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Was Kepler aware that Venus and Mars have a golden section relationship other than that he attributed to them in his 5 perfect solids model of the Solar System? The dodecahedron and the icosahedron, which Kepler placed between the Earth and Mars and Venus, both have the golden section in their structure.

The value for Venus's semi-major axis (in AU) to the power of 21 multiplied by that of Mars to the power of 55 equals $\varphi$ (the golden section) to the power of 34! These are Fibonacci numbers and any such sequence means that using powers above or below in the Fibonacci sequence (or the Lucas sequence) will give an approximate same answer.

From NASA: Venus's semi-major axis (0.72332102 AU) to the power of 21 times Mars's semi-major axis (1.52371243 AU) to the power of 55 equals $\varphi$ to the power of 34. The discrepency is 1.000088909, but it would be the 76th root of this, 1.00000117, to correct the equation. Venus's semi-major axis to the power of 13 times Mars's semi-major axis to the power of 34 equals $\varphi$ to the power of 21, which is is 1.002919389. The 47th root is 1.000062026. Venus's semi-major axis to the power of 18 times Mars's semi-major axis to the power of 47 is $\varphi$ to the power of 29, which is 1.008694078. The 65th root is 1.000133186.

What understanding of these relationships and calculations did Kepler demonstrate in his published works?

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    $\begingroup$ There are currently 2 close votes as "off-topic" on this question. I don't know if this question is off-topic, but that's primarily because I don't understand what is being asked. This question would be much easier to understand if you use MathJax to format your post (see e.g. math.stackexchange.com/help/notation) and explain what the numbers are that you're using. $\endgroup$
    – Logan M
    Jan 28 '15 at 19:32
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    $\begingroup$ Along with what Logan said: The fact that this is just one big block of text doesn't help. Links would also be nice, as would complete sentences and perhaps a better list for the numbers at the end. $\endgroup$
    – HDE 226868
    Jan 28 '15 at 21:41
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    $\begingroup$ Seems ok now. Kepler couldn't use NASA data of course, but he must have had some estimates for semi-axes of his ellipses. One problem seems to be that he'd have had to relate them to the Earth-Sun distance to see numerical coincidences in AU. $\endgroup$
    – Conifold
    Jan 29 '15 at 0:18
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    $\begingroup$ Why in the world would raising such numbers to those powers have any real meaning? You consider these numbers up to 8 digits after the decimal and then declare high powers of them are "equal" when they are off by a factor that is not really 1 to such accuracy. $\endgroup$
    – KCd
    Jan 29 '15 at 8:06
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    $\begingroup$ In his book Harmonies of the World the only A.U. values I could find were .721 for venus and 1.523 for mars venus power 2 times mars power 5 =Phi power 3 1.0055 What I am asking is has anyone come across this idea before me? $\endgroup$ Jan 30 '15 at 3:23

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