Kepler's Triangles

From the article by M. Beech in the Journal of recreational Mathematics 29 (2), 114 - 120.


Introduction: Johannes Kepler's (1571 - 1630) lasting and indelible contribution to humanity is encapsulated within his three fundamental laws of planetary motion. The story surrounding Kepler's derivation of his laws and his diligent study of Tycho Brahe's observations are well known and repeated in all introductory astronomy texts [1, 2].

As with many new discoveries in science, Kepler did not simply discover his three laws of planetary motion by following a linear path of reasoning. Indeed, his long journey to enlightenment was punctuated by many time-consuming sojourns down scientifically barren cul-de-sacs. However, just as a car ride is often the more memorable because of the sights seen on the way, so to do a few of Kepler's 'wrong-turns' offer some interesting intellectual food for thought.

Kepler revealed the discovery of his first two laws of planetary motion in his Astronomia Nova, published in 1609. The announcement of his third law was not made until 1618. Prior to the release of Astronomia Nova, however, Kepler had published a text in 1597 called Mysterium Cosmographicum (literally, the Cosmic Mystery). In his Mysterium, Kepler attempted to show that planetary orbits were arranged in accordance with the regular, or Platonic polyhedra [3]. In this way, for example, Kepler argued that the ratio of orbital radii for Venus and Mercury was the same, or nearly so, as the ratio of the radii for the circumscribed and inscribed spheres of an octahedron.

Kepler's initial inspiration to study planetary geometry was conceived while teaching an astronomy class at the University of Turbingen, in Germany, on July 9th, 1595 [2]. During the class Kepler had drawn for his students the inscribed and circumscribed circles to an equilateral triangle. In an instant of appreciation and with a feeling of deep significance, he realized that the ratio of the radii for the inscribed and circumscribed circles was the same as the ratio of orbital radii of Saturn and Jupiter.

Figure 1. The inscribed and circumscribed triangles to the orbits of Jupiter and Saturn

Kepler could not believe that the coincidence of ratios was purely fortuitous, there had, he reasoned, to be meaning behind such a result. Kepler argued that given that Saturn and Jupiter were the first two planets of the Solar System and that the triangle was the simplest polygon, then the ratio of the orbital radii of Jupiter and Mars should be in the same ratio as the radii of the circumscribed and inscribed circles to a square, the second simplest polygon. The orbital radii of Mars and Earth, by the same measure, should be related to the circumscribed and inscribed circles to a pentagram. Kepler's reasoning was logical enough but false in reality. The planets simply do not arrange themselves in accordance with Kepler's polygonal scheme. Having found that the planets could not be organized according to a progression of polygons, Kepler turned to three dimensional objects and developed the nested polyhedral model which he described in the Mysterium Cosmographicum.

It seems that Kepler never completely abandoned the ideas presented in his Mysterium, although he did admit that his nested polyhedron theory had its weaknesses [2]. Below, we return to Kepler's original idea of describing planetary orbits in terms of circles inscribed and circumscribed to various regular polygons - we shall refer to these polygons as basis polygons. In particular we shall investigate just how well planetary orbits can be described as inscribed and circumscribed circles to triangles, squares and pentagrams.

Inscribed and Circumscribed circles: The formula for the radii of inscribed and circumscribed circles to an n-sided regular polygon are straightforward to derive and we simply state the result required. If the circumscribed circle has a radius a, and the inscribed circle has radius b, then the ratio b/a is

b / a = Cos [p / N]

where N is the number of equal length polygonal segments. For a triangle (N = 3) we obtain the relation b/a = 1/2, and for a square (N = 4) we obtain b/a = 1/Sqrt[2].

Our Solar System: In the analysis that follows we assume, just as Kepler did, that the planets move along circular, coplanar orbits. Column 2 of table 1 below shows the average distance of the planets from the Sun in astronomical units (AU). The average Sun-Earth distance is defined as 1 AU. Column 3 gives the ratio of successive orbital radii. With the exception of a break between Mars and Jupiter, it can be seen that each planet is approximately 64% further from the Sun than the planet immediately interior to it. Columns 4 and 5 of table 1 show the conditions required of the circumscribed orbital radius if each planet is assumed to be inscribed on a triangles and a square. Several nested pairs of planets are apparent from table 1, and these are listed in table 2 (see figure). We see, for example, that the ratio of the orbital radii of Venus and mars is nearly the same as the ratio for the circumscribed and inscribed circles to an equilateral triangle. Likewise, the ratio of the orbital radii of Earth and Mars is nearly in the same ratio of radii for the circles inscribed and circumscribed to a square. Neptune and Pluto are the only planets for which a pentagram forms a basis polygon.


Planet b = r (AU) a / b 2 b b Sqrt[2]
Mercury 0.39 ---- 0.78 0.552
Venus 0.72 1.846 1.44 1.018
Earth 1.00 1.389 2.00 1.414
Mars 1.50 1.50 3.00 2.121
Jupiter 5.20 3.467 10.40 7.354
Saturn 9.50 1.827 19.00 13.435
Uranus 19.2 2.021 38.40 27.153
Neptune 30.1 1.568 60.20 42.568
Pluto 39.4 1.309 ---- ----

Table 1. Solar System data table.

We define a goodness of fit parameter in terms of the polygon about which the orbits are apparently inscribed and circumscribed. If the perimeter of the inscribed polygon is Pi and the perimeter of the circumscribed polygon is Pc, the goodness of fit is taken as

GOF ( %) = (1 - |Pi / Pc|) * 100

A GOF of 0% corresponds to a perfect fit. In table 2 we see that the GOF for Kepler's original pair is 9.4%. From table 2 we find that the best-fit example of an inscribed and circumscribed pair of planetary orbits is that between Saturn and Uranus (GOF = 1.04%), with the second best-fitting pair being Venus and Earth (GOF = 1.82%).


Planetary pair Basis polygon GOF (%)
Mercury -Venus Triangle 8.33
Venus - Mars Triangle 4.00
Venus - Earth Square 1.82
Earth - Mars Square 6.06
Jupiter - Saturn Triangle 9.47
Saturn - Uranus Triangle 1.04
Uranus - Neptune Square 10.85
Uranus - Pluto Triangle 2.54
Neptune - Pluto Pentagram 5.57

Table 2. Planetary pairings and their basis polygons.

Figure 2: The inscribed and circumscribed triangles and square for the orbits Mercury to Mars

Figure 3. The inscribed and circumscribed triangles, square and pentagram to the orbits of Jupiter to Pluto.

Discussion: The astute reader will have noticed that we have not included the asteroids in the analysis presented above. The main reason for this is that no truly representative orbital radius can be associated with the main-belt asteroids - their orbital radii vary between 2 and 5 AU.

It seems interesting to me that, given the apparent ease with which the orbital radii of the planets can be arranged as the inscribed and circumscribed circles to triangles, squares and pentagrams (albeit to a GOF < 11%), it is slightly odd that Kepler did not make more of the issue. Certainly, Kepler had the same information on average orbital radii as we used in table 1, out to the planet Saturn that is, and it is well known that Kepler was a diligent and thorough mathematician. It does not seem unreasonable to me, therefore, to assume that Kepler had some inkling of the relations described above for the planets interior to Saturn. This apparently missed chance by Kepler suggests the not quite trivial comment that Kepler wanted to forge, for presumably esthetic reasons, a clear link between planetary radii and a progression of basis polygons. In other words, he was looking for evidence of a 'grand' and systematic design, and was not remotely interested in simply arranging planetary orbits in terms of similar polygons - one pair of planets corresponding to the inscribed and circumscribed circles about a triangle offered a hint of design, two or three such correspondences suggested mere serendipity.

References: [1]. Zeilig, M. Astronomy: The Evolving Universe, 7th ed., John Wiley and Sons, Inc., New York, 1994. [2]. Koestler, A. The Sleepwalkers, Penguin Books, Middlesex, UK., 1986. [3] Hollingdale, S. H. The Bulletin of the Institute of Mathematics and its Applications, 22, (3/4), 1986, 34 - 37.

Figure 4. All inscribed and circumscrined polygons to the orbits of the planets. A triangle has also been inscribed and circumscribed between the orbits of Uranus and Pluto in this diagram.

This page last modified July 1st, 2004

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