Extending Kepler’s MysteriumCosmographicum

Matthew Dockrey

Agora Research Institute

May 25, 2017

Abstract

An attempt is made to extend Kepler’s classic model of the solar system to account for Uranus and Neptune.Possible additions to the five Platonic solids are examined, with the choice finally being made to use a d10(pentagonal trapezohedron) and a Utah Teapot. The relative orbit sizes thus produced are compared to modernfigures and Kepler’s original results.

I. Introduction

Mysterium Cosmographicum was published byJohannes Kepler in 1596. In it, the relative spac-ing of the orbits of the six planets then known(Mercury, Venus, Earth, Mars, Jupiter, Saturn)were explained in terms of a nested series ofPlatonic solids (octohedron, icosahedron, do-decahedron, tetrahedron, cube).[Kepler, 1596]While the fit was far from perfect, Kepler neverentirely dropped the concept.[Kepler, 1618]

Not actually being nessecary to a scien-tific understanding of the solar system, Ke-pler’s Mysterium Cosmographicum model didnot prove nearly as influential as his laws ofplanetary motion. Outside of the scientificworld, however, his imagery and mysticismdid have a lasting impact.

In order to explore this strange, beautiful,flawed model, the author set about extendingit to account for Uranus and Neptune. It washoped that this effort would help modern read-ers appreciate some of the significant differ-ences in the scientific mindset of Kepler’s age.

II. Methods

There are two components to the Keplerianmodel: the spherical shells representing theorbits of the planets, and the Platonic solids

between them which provide the spacing. Thespherical shells have a thickness based on theeccentricity of that planet’s orbit. Thus, Mer-cury and Mars each have a relatively thick shell,while those for Venus and Earth are much thin-ner.

i. Selection of solids

One major reason Kepler was a proponentof the Mysterium Cosmographicum model wasthe apparent numerological significance of sixplanets with five Platonic solids to providespacing between them. This poses a particularchallenge when trying to extend the model toanother two planets, as a total of seven Platonicsolids would be needed.

One option would be to explore higherdimensional spaces. In four dimensions,for instance, there are six regular poly-topes. However, in all dimensions higherthan four there are only three regularpolytopes.[Coxeter, 1973] Thus, even if onewere to find the use of higher dimensionalobjects in this context acceptable, it isn’t clearwhich two should be chosen. This option wasrejected as being contrary to the numerologicalbiases of the original model.

Without real Platonic solids available, the au-thor was forced to approach the problem more

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Figure 1: The Six Platonic Solids, James Arvo and DavidKirk, 1987 [Avro, 1987]

abstractly. Luckily, two options immediatelypresented themselves.

Utah teapot

The Utah teapot is a classic computer graphicsobject dating back to 1975. It has been includedas a pre-defined primitive in basically everygraphics package ever since. Having a simpleobject with relatively complicated geometryhas proved to be very valuable, and the Utahteapot has become the default for that role.Because of its ubiquitous inclusion amongstother basic solids, it has often been referred toas the “sixth Platonic solid”.1 The author feltthis amply justified its use here.

d10

To the best of the author’s knowledge, theonly context in real life where all the Platonicsolids are commonly seen is in a set of gamingdice. These normally include a tetrahedron(d4), cube (d6), octahedron (d8), dodecahedron(d12), and icosahedron (d20). They also includea pentagonal trapezohedron (d10).2 While ad10 is not technically a Platonic solid, beingpart of a group otherwise made up of Platonicsolids makes it good candidate for the seventh.

Figure 2: A set of gaming dice

ii. Modeling

The 3D modelling was all performed inBlender, a GPLv2 licensed free software pack-age. Sizing of the elements was performedmanually, using the renderer itself to check forcollisions between the objects.

The model was built starting with the shellof Jupiter being set to an arbitrary diameter of10 units. Its thickness was set based on modernnumber for Jupiter’s perihelion (4.95029 AU)and aphelion (5.45492 AU)[Simon, 1994], for apercentage of 9.241%. From this base, the nextsolid outward (cube) and inward (tetrahedron)were placed, and scaled to just touch the shellof Jupiter. The shells for Saturn and Mars wereadded and their thickness set, again using themodern perihelial and aphelial measurements.This was repeated for the entire system. Forthe Utah teapot, the shell was fitted only to therounded belly of the teapot. The handle andspout were ignored.3

III. Results

After building up the model, the relative sizesof the orbits could be compared. It is importantto remember that relative sizes is all that Keplerwas interested in, as absolute measurementsweren’t possible yet. (This is still reflected in

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Figure 3: The extended Keplerian model

the use of the AU, or astronomical unit. Defin-ing that unit as the size of the Earth’s orbitwasn’t an act of supreme cosmological arro-gance, it was simply using the only yardstickavailable.) For each pair of neighboring plan-ets, the percentage size of the outer orbit interms of the inner orbit is given, for both themean and aphelial diameters.1

While the fit isn’t very good by modern stan-dards, the addition of Uranus and Neptunedoesn’t stand out as particularly terrible com-pared to Kepler’s original results. The fit forUranus-Neptune, in fact, using the Utah teapotsolid, provides one of the most accurate fits ofall.2

It is clear that the d10 is a particularly poorfit for the Saturn-Uranus pair. It can at leastbe said that the result here when looking atthe aphelial ratios is still better than Kepler’soctohedron between Mercury and Venus. Thiscould be seen a cherry-picking, but the au-thor feels it is well within the spirit of Kepler’soriginal work, which freely switched betweenperihelial, mean and aphelial numbers as con-venient. That said, further research is neededto find a more appropriate Platonic solid alter-native.

IV. Discussion

An extended Mysterium Cosmographicum modelof the solar system was never going to providenew scientific insights, but it does have valuein exploring the mindset of the 17th century.

It can be challenging for the modern reader

to think in their terms, particularly the biastowards using ratios and proportions insteadof absolute measurements. This was the stan-dard until well after Kepler, largely becausethe metrology simply didn’t exist yet to sup-port absolute measurements of any quality. Inaddition, geometry was still the predominantmathematics of the era. The nature of geo-metric proofs naturally leads one to think pri-marily in terms of relative proportions. Thisnow-foreign assumption, that the answer to aproblem isn’t a number but a ratio, is one ofthe more subtle impediments to understand-ing historical scientific publications. A modernreader can follow the proofs without too muchtrouble, even though a deep familiarity withEuclid and Appolonius is now rare. But with-out understanding the why of the proofs, onecan quickly get lost. Building and exploringmodels based on this worldview is essential totruly understanding the foundations of mod-ern science.

Kepler also represents a critical turning-point in the history of science, as we movedtowards a purely mathematical understandingof the universe and away from a syllogisticone. Part of this change was a radical shiftin how the beauty of scientific propositionswere understood. We still rely on beauty asan important heuristic for evaluating hypothe-ses, of course. The preference for certain formsof equations or for theories with few specialcases is no less of an aesthetic judgment thantrying to find the Platonic solids embodied inthe structure of the solar system. It wasn’t re-lying on beauty that led Kepler wrong, it wasrelying on the wrong kind of beauty. That wasa lesson which took at least another centuryto learn, but Kepler was a critical part of thechange. While he never did fully give up onthe Mysterium Cosmographicum model, he stillspent decades processing Tycho Brahe’s obser-vations, slowly disproving his earlier theory.The result, Kepler’s laws of planetary motion,was the first model of orbital motion based pri-marily on data, not a philosophical ideal. Thiswas a crucial step towards Newton’s hypothesesnon fingo.

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Planet pair Real (mean) Real (aphelion) Model (mean) Model (aphelion)Mercury-Venus 1.846 2.339 2.124 1.776Venus-Earth 1.389 1.369 1.294 1.306Earth-Mars 1.520 1.405 1.414 1.522Mars-Jupiter 3.421 3.583 3.466 3.322Jupiter-Saturn 1.835 1.823 1.922 1.935Saturn-Uranus 2.011 2.0312 1.664 1.650Uranus-Neptune 1.567 1.626 1.638 1.579

Table 1: Ratios of orbital sizes using real world and model data

Planet pair Difference (mean) Difference (aphelion)Mercury-Venus 0.278 -0.563Venus-Earth -0.095 -0.063Earth-Mars -0.106 0.116Mars-Jupiter 0.045 -0.261Jupiter-Saturn 0.088 0.112Saturn-Uranus -0.346 -0.381Uranus-Neptune 0.070 -0.04

Table 2: Differences of real world and model ratios

References

[Kepler, 1596] J. Kepler. Mysterium Cosmograph-icum. Tübingen, 1596.

[Kepler, 1618] J. Kepler. Epitome AstronomiaeCopernicanae. Linz, 1618.

[Coxeter, 1973] H. S. M. Coxeter. Regular Poly-topes (3rd ed.). New York: Dover Publica-tions, 1973.

[Avro, 1987] James Arvo and David Kirk. "Fastray tracing by ray classification." ACMSiggraph Computer Graphics. Vol. 21. No.4. ACM, 1987.

[Simon, 1994] Simon, J.L.; Bretagnon, P.;Chapront, J.; Chapront-Touzé, M.; Fran-cou, G.; Laskar, J. (February 1994). "Nu-merical expressions for precession formu-lae and mean elements for the Moon andplanets". Astronomy and Astrophysics.282 (2): 663-683

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