THE SOUTHWORTH PLANETARIUM
207-780-4249 <%28207%29%20780-4249> www.usm.maine.edu/planet
70 Falmouth Street Portland, Maine 04103
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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.
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