THE SOUTHWORTH PLANETARIUM
207-780-4249 www.usm.maine.edu/planet
70 Falmouth Street Portland, Maine 04103
43.6667° N 70.2667° W
Founded January 1970
Julian date: 2457905.16
"Think of intelligence as being similar to temperature. Every mind is actuated by intelligence just as molecules are energized by heat induced kinetic energy. Just as molecules impart energy to one another through interaction, a single mind can be all the more empowered through interaction with all the other minds, through conversation, books, lectures, and ideas. If you open yourself up to every mind, without regard to perceived IQ, then your mind becomes all the more energized. Draw from the intellectual heat energy of everyone around you...even if they don't know you're doing it."
THE DAILY ASTRONOMER
Thursday, June 1, 2017
Earth's Dodecahedron
It was perhaps the most elegantly beautiful idea that ever turned out to be wholly misguided. A notion that the planetary motions and relative distances were determined by a contrivance of perfect geometries. It is appropriate that late 16th-early 17th century mathematician Johannes Kepler (1571-1630) would have crafted such a intricate model. He possessed a mystic's sense that sublime harmonies under gird the Universe with a devout theologian's confidence that nothing is purposeless.
Tetrahedron
Kepler, then a mathematics teacher in what is now Graz, Austria, happened upon this epiphany one afternoon while describing a series of rotating triangles formed by successive conjunctions* of Jupiter and Saturn. Each such conjunction is displaced slightly along the ecliptic relative to the one preceding it. Were one to plot a series of conjunctions, one would draw a pinwheel array of turning triangles. Yielding such a construct was evidence to Kepler's mind, that Euclidean geometry, a fundamental branch of mathematics whose name literally means "Earth-measurement," governs celestial motions.
Octahedron
The coupling of Earth-based mathematics and cosmic dynamism was not an unexpected marriage. The heavens were largely perceived as perfect and immutable: a precisely ordered arrangement of pristine spheres: ephemeral orbs traversing circular paths around either a static Earth or angelically fired Sun. (In Kepler's time, the proper solar system arrangement, either geocentric or Copernican, had not yet been settled.) When Kepler, then a mathematics instructor rather bored with his vocation, scrutinized the conjunction wheel, he fashioned a solar system model that he instinctively knew had to be correct. In fact, his theory was so compelling, it induced him to abandon his ecclesiastical ambitions in favor of astronomy, an enterprise in which he then had only a cursory knowledge.
Cube
In this theory Kepler asserted that the six known planets (Mercury, Venus, Earth, Mars, Jupiter and Saturn) occupied spheres separated by the five Platonic solids, The Five Platonic solids were the tetrahedron, cube, octahedron, dodecahedron, and the icosahedron. The tetrahedron consists of four equilateral triangles formed a pyramid. Six perfect squares comprise the square. The octahedron is essentially two tetrahedrons that shared the same base so that their respective apexes were 180 degrees apart. The dodecahedron is formed from twelve regular hexagons; the icosahedron by twenty equilateral triangles. These are the only** geometric solids that, if embedded in a sphere, would have each vertex touch the sphere. And, if a sphere were inscribed within these solids, the sphere would touch the midpoint of each sides.
Dodecahedron
Kepler reasoned that the solar system was restricted to having six planets because only five platonic solids existed to separate them. (Uranus was discovered more than a century after Kepler's death.) The solids served as a celestial lattice work that lent structure and stability to the planetary orbits. A cube, if situated between Saturn and Jupiter's sphere, determined their separation distance.
Or, as Kepler, himself, wrote
"The Earth's orbit, being the measure of all things, should have circumscribed around it a dodecahedron. The circle containing this shape will hold the planet Mars. Around Mars circumscribe a tetrahedron; the circle containing it will be Jupiter. Beyond Jupiter circumscribe a cube; the circle containing it will be Saturn. Within Eath's sphere is a icosahedron; the circle contained within it will be Venus. Inscribe within Venus the octahedron and the circle containing it will be Mercury."
Icosahedron
Therefore, the number of planets was readily explicable. It was ordained that the solar system was limited to its attendant planets, for the geometrical constructs within the cosmos would have been unable to accommodate any more.
Unfortunately, Kepler's construct was doomed. Kepler derived a distance-period formula based on these solids. He calculated that the increase in period between planets was equal
to twice the differences of their distances from the Sun. Eventually, as Kepler would learn much later as he meticulously analyzed Tycho Brahe's Mars observations, this relation was incorrect. So, too, was the belief that the planets were enclosed in a rigid system of the five only perfect solids. The final relation he did discover, that the square of a planet's period was related to the cube of its mean distance from the Sun, was correct. He dubbed it the "harmonic law," perhaps in tribute to the beautiful geometries that drove him to become one of the greatest astronomers of all time.
*A conjunction occurs when two planets are "aligned," along the same line of right ascension (celestial equivalent to longitude). Despite notions to the contrary, conjunctions are not precise alignments of two planets, with one blocking another. These events, called planet-planet occultations, are exceedingly rare.
**Euclid of Alexandria, the architect of the branch of geometry now known as "Euclidean," proved that the tetrahedron, cube, octahedron, dodecahedron, and icosahedron are the only solids that exhibit the properties described above.