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	SparkNotes: Johannes Kepler: General Summary
		Johannes Kepler was born in Germany in 1571, in the middle of the Scientific Revolution.
		Kepler's theory was incorrect, but the book was the first major work in support of the Copernican system since Copernicus's death fifty years before.
		Kepler's work on Mars led him to discover his first two planetary laws: that the planets travel in elliptical orbits and that they sweep out equal areas of their orbits in equal times.
	www.sparknotes.com /biography/kepler/summary.html (1092 words)

	Skyscript: JOHANNES KEPLER And the Music of the Spheres by David Plant
		Kepler's, First Law states that the planets move in ellipses and that the Sun is not at the exact centre of their orbits.
		Kepler said his MC was in the 22nd degree of Aquarius, and his AS was in the 25th degree of Gemini, so his given time of 2:30 pm checks out.
		Kepler' s belief in the ancient doctrine that the Earth as a whole may be regarded as a living entity is echoed in the 'Gaia principle' popularised by James Lovelock during the 1980s.
	www.skyscript.co.uk /kepler.html (4380 words)

	Harmonies of the World Index
		By means of your concords of various voices, and through your ears, she has whispered to the human mind, the favorite daughter of God the Creator, how she exists in the innermost bosom.
		Johannes Kepler, who originally studied theology, was introduced to the Copernican world-view while studying for his Master's degree in Philosophy at the University of Tübingen.
		He became convinced that there was a relationship between the five regular solids and the structure of the known solar system.
	www.sacred-texts.com /astro/how/index.htm (560 words)

	Reference.com/Encyclopedia/Regular polyhedron
		All the solid angles of the polyhedron are congruent.
		In the 17th century, Johannes Kepler studied data on planetary motion compiled by Tycho Brahe and for a decade tried to establish the Pythagorean ideal by finding a match between the sizes of the polyhedra and the sizes of the planets' orbits.
		His search failed in its original objective, but out of this research came Kepler's discoveries of the Kepler solids as regular polytopes, the realisation that the orbits of planets are not circles, and the laws of planetary motion for which he is now famous.
	www.reference.com /browse/wiki/Regular_polyhedron (1291 words)

	Schiller Institute Kepler Translations
		Many know of Harmonice Mundi as the work in which Kepler announced the third of his laws of planetary motion: the ratio of the cube of the (average) radius of the planet's orbit to the square of its periodic time is equal to a constant for all planets.
		Kepler considers that this process is absolutely key to the understanding of the regular polygons, which in turn he considers as the basis for consonant (sweet-sounding) intervals in music as well as for the five Platonic Solids which he used to explicate the number of the planets and the relative sizes of their orbits.
		Thus, from the investigation of constructable numbers, Kepler moves to the construction of polygons and their star figures, and contrasts those which can be constructed with others which cannot, whether or not they can be approximated.
	www.schillerinstitute.org /transl/trans_kepler.html (2659 words)

	Kepler
		Kepler sent copies of his first major work to a number of scientists, including Tycho Brahe, who was soon to become the imperial mathematician of the Holy Roman Empire.
		Kepler now had access to Brahe's incomparable collection of astronomical observations, the result of decades of unremitting and painstaking toil by the greatest naked-eye observer of the heavens and the leader of a highly qualified team of astronomers.
		Kepler extended Copernicus' reasoning to the other planets and was the first to declare that the other planets resemble the Earth in being material bodies.
	abyss.uoregon.edu /~js/glossary/kepler.html (2144 words)

	The Galileo Project \| Science \| Johannes Kepler
		Johannes Kepler was born in Weil der Stadt in Swabia, in southwest Germany.
		Kepler remained in Graz until 1600, when all Protestants were forced to convert to Catholicism or leave the province, as part of Counter Reformation measures.
		Kepler served as Tycho Brahe's assistant until the latter's death in 1601 and was then appointed Tycho's successor as Imperial Mathematician, the most prestigious appointment in mathematics in Europe.
	galileo.rice.edu /sci/kepler.html (1275 words)

	Kepler solid
		A Kepler solid is a regular nonconvex polyhedron, all the faces of which are regular polygons and which has the same number of faces meeting at all its vertices.
		The Kepler solids were defined by Johannes Kepler in 1619, when he noticed that the stellated dodecahedrons (there are two, a greater and a lesser) were composed of "hidden" dodecadrons (with pentagonal faces) that have faces composed of triangles, and thus look like stylized stars.
		Kepler's contribution was in recognizing that they fit the definition of regular solids, even though they were concave rather than convex, as the traditional Platonic solids were.
	www.ebroadcast.com.au /lookup/encyclopedia/ke/Kepler_solid.html (259 words)

	Johannes Kepler - RecipeFacts (Site not responding. Last check: )
		Kepler disdained astrologers who pandered to the tastes of the common man without knowledge of the abstract and general rules, but he saw compiling prognostications as a justified means of supplementing his meager income.
		Kepler is known to have compiled prognostications for 1595 to 1606, and from 1617 to 1624.
		As court mathematician, Kepler explained to Rudolf II the horoscopes of the Emperor Augustus and Muhammad, and Kepler gave astrological prognosis for the outcome of a war between the Republic of Venice and Paul V.
	www.recipeland.com /facts/Johannes_Kepler (3404 words)

	Malaspina Great Books - Johannes Kepler (1571)
		Kepler was born on the 27th of December 1571, at Veil, in the duchy of Wurttemberg, of which town his grandfather was burgomaster.
		Kepler thus found that the first duties required of him were of an astrological nature, and set himself with characteristic alacrity to master the rules of the art as laid down by Ptolemy and Cardan.
		Kepler immediately hastened to Wurttemberg, and owing to his indefatigable exertions she was acquitted after having suffered thirteen month's imprisonment, and endured with undaunted courage the formidable ordeal of territion, or examination under the imminent threat of torture.
	www.malaspina.org /home.asp?topic=./search/details&lastpage=./search/results&ID=125 (4108 words)

	Kepler solid
		It is clear from the general arrangement of the book that he regards only the five Platonic solids as regular, and does not understand the regular nature of his great dodecahedron.
		The Kepler solids were discovered by Johannes Kepler in 1619.
		Kepler's final step was to recognize that these polyhedra fit the definition of regular solids, even though they were not convex, as the traditional Platonic solids were.
	www.algebra.com /algebra/about/history/Kepler-solid.wikipedia (1053 words)

	Kepler, Johannes (1571-1630)
		Born in Weil der Stadt, southwest Germany, Kepler studied at the university of Tübingen and, as a graduate, was tutored by Michael Maestlin who introduced him to the heliocentric concepts of Copernicus.
		Because of the mathematical skills Kepler showed in his Mysterium cosmographicum, he was invited by Tycho Brahe to Prague to become his assistant and to calculate new orbits for the planets from Tycho’s observations.
		Kepler exemplifies the resistance shown by some of the leading Renaissance scientists (including also Copernicus and Galileo) to the idea that there might be innumerable inhabited worlds.
	www.daviddarling.info /encyclopedia/K/KeplerJ.html (882 words)

	ipedia.com: Johannes Kepler Article (Site not responding. Last check: )
		Kepler was a professor of mathematics at the University of Graz, court mathematician to Emperor Rudolf II, and court astrologer to General Wallenstein.
		Johannes Kepler was born on December 27, 1571 at the Imperial Free City of Weil der Stadt (now part of the Stuttgart Region in the German state of Baden-Württemberg, 30 km west of Stuttgart's city center).
		The smallest orbit, that of Mercury, was the innermost sphere.
	www.ipedia.com /johannes_kepler.html (1708 words)

	The Start of Scientific Cosmology (Cosmology: Ideas)
		Johannes Kepler as imperial mathematician to Rudolph II, the Emperor of the Holy Roman Empire.
		Kepler was committed to Copernicus's heliocentric system both for its technical advantages, which made it possible to dispense with some of the complications of Ptolemy's system, and on philosophical grounds, including a symbolic identification of the Sun with God at the center of all things.
		Kepler's colorful life was marred in the end by, among other things, his mother's trial for witchcraft, and a periodically nomadic and economically uncertain existence under the stress of wars and political upheavals.
	www.aip.org /history/in-progress/cosmology/ideas/start-of-scientific-cosmology.htm (2044 words)

	PowerPedia:Johannes Kepler - PESWiki
		Through his career Kepler was a mathematics teacher at a Graz seminary school (later the University of Graz, Austria), an assistant to Tycho Brahe, court mathematician to Emperor Rudolf II, mathematics teacher in Linz, Austria, and court astrologer to General Wallenstein.
		Kepler did not understand why his laws were correct; it was Isaac Newton who discovered the answer to this more than fifty years later.
		As court mathematician, Kepler explained to Rudolf II the horoscopes of the Emperor Augustus and the Prophet Muhammad, and Kepler gave astrological prognosis for the outcome of a war between the Republic of Venice and Paul V.
	peswiki.com /index.php/PowerPedia:Johannes_Kepler (4784 words)

	Biography of Kepler
		Kepler was born on December 27, 1571 in Weil der Statt, near Germany.
		Kepler next studied Mars for a number of years and hoped to calculate its orbit, but this was a big task with immense calculations that take time without the help of mechanical aids.
		Kepler's third law of planetary motion: The cubes of the mean distances of the planets of the Sun are proportional to the squares of their revolution periods.
	www.andrews.edu /~calkins/math/biograph/199899/biokeplr.htm (1310 words)

	Johannes Kepler - Kepler's 5 Regular Solids - Dr Robert A. Hatch (Site not responding. Last check: )
		Among his ideas not found in modern textbooks is the belief that the Five Regular Solids (three-dimensional geometrical objects with identical sides, for example a cube) account for the five intervals between the six known planets.
		Surprisingly (or not) Kepler was able to make the ratios work with fair accuracy, though the failure with one of the planets seems to have been a motive for his accepting a position with Tycho Brahe, the Prince of Astronomers.
		Evidence suggests Kepler believed in the efficacy of this theory all of his life, that is, it held equal weight with his so-called three laws of planetary motion.
	www.clas.ufl.edu /users/rhatch/pages/03-Sci-Rev/SCI-REV-Home/resource-ref-read/chief-systems/kepler-prob/08-0KEPL-SOLIDS.html (292 words)

	News \| Gainesville.com \| The Gainesville Sun \| Gainesville, Fla. (Site not responding. Last check: )
		The Kepler solids were discovered by Johannes Kepler in 1619.
		Kepler's final step was to recognize that these polyhedra fit the definition of regular solids, even though they were not convex, as the traditional Platonic solids were.
		A Kepler-Poinsot solid covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the solids with pentagrammic faces and the vertices in the others.
	www.gainesville.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Kepler-Poinsot_solid (1057 words)

	What are Kepler's laws of motion and what exactly do they mean?
		In the second law, Kepler used the idea of radius vector, which is the line drawn from the sun to any point in a planet's orbit, and stated that the radius vector from the sun to a planet sweeps out equal areas in equal intervals of time.
		Kepler labored intensely to find a way to use the five solids to describe planetary motion, for he believed that some nested arrangement of the solids would show how the Creator meant Nature to be simple and elegant and geometrically symmetric.
		Kepler's Third Law states the relationship between the period of motion for a planet and the semi-major axis of the planet's elliptical orbit.
	www.physlink.com /Education/AskExperts/ae613.cfm (1029 words)

	Kepler_solids - SiteTracer.com
		It has been proven that these four solids, together with the 5 platonic solids, are the only regular solids possible.
		Another wonderful truth is that the Kepler solids are the duals of the Poinsot...
		Kepler saw them from a deeper perspective, and in recognition, we now refer to these as Kepler solids.
	www.sitetracer.com /search/Kepler_solids (230 words)

	Johannes Kepler's Polyhedra
		Kepler's logical approach to polyhedra does not mean that he was free of the mysticism of the day.
		Kepler's important contribution was to define this class of polyhedra and systematically explore it, to find all its members and prove his set was complete.
		Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids.
	www.georgehart.com /virtual-polyhedra/kepler.html (612 words)

	The Kepler Solids
		This solid makes a lovely Christmas decoration, and is often seen as such in store windows and commercial displays during the holiday season.
		In their colored versions, each solid consists of ten golden triangles of each of six different colors - sixty triangles in all.
		In the second of the solids, the great stellated dodecahedron, the twenty triangular pyramids are given the color scheme below.
	britton.disted.camosun.bc.ca /jbkeplersolids.htm (565 words)

	Glossary
		The pentagram is a non-convex polygon; the Kepler-Poinsot solids are non-convex polyhedra.
		Standardly, there are nine regular polyhedra: the five Platonic solids and the four Kepler-Poinsot solids, but others might be allowed, depending on the definition of polyhedron.
		This includes the Platonic solids, the Archimedean solids, the prisms and antiprisms, and the nonconvex uniform solids.
	www.georgehart.com /virtual-polyhedra/glossary.html (724 words)

	The Renaissance Man's Polyhedra Weeb Site: Polyhedron Models for the Classroom by Magnus J. Wenninger
		(Kepler discovered two about 1619, and Poinsot rediscovered these and discovered the two others in 1809.) These solids are all the more interesting because they were unknown to the ancient world.
		First it may be noted that the five-pointed star, or pentagram, arises first by producing the sides of a pentagon or by drawing all the diagonals (see Figure 18 and Figure19).
		A basic solid could of course be used—for example, beginning with a dodecahedron, you could cement the vertex parts, twelve pentagonal pyramids, one onto each face, and thus obtain a small stellated dodecahedron.
	www.theweebsite.com /polyhedra/pmftc/pmftc4.html (2303 words)

	Kepler-Poinsot Solids (Site not responding. Last check: )
		A natural extension is to extend the faces of a three-dimensional solid (polyhedron) until they meet.
		The Kepler-Poinsot solids are stellations of the dodecahedron and icosahedron.
		The solid has the edges and vertices of an icosahedron, but instead of triangular faces has triangular dimples.
	www.uwgb.edu /dutchs/symmetry/kpsolid.htm (522 words)

	American Mathematical Society :: Featured Column
		Kepler used some of the data to generate proposed orbits and then used other data values as "proof" that the formulas he was generating were correct.
		From a modern perspective the regular convex polyhedra (also known as the Platonic solids) can be thought of as solids where all the faces are congruent regular polygons with the same number of sides, and where the faces all have the same pattern at each vertex.
		Once Kepler completed the formulation of his three laws of motion for the planets, it became possible for Newton's new model (universal gravitation with an inverse square law) to become the new "standard model," replacing Ptolemy and confirming Copernicus' and Kepler's views.
	www.ams.org /featurecolumn/archive/cosmology.html (3871 words)

	Definition of Kepler-Poinsot solid
		A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids).
		The Kepler solids were defined by Johannes Kepler in 1619, when he noticed that the stellated dodecahedra (there are two, the great and the small) were composed of "hidden" dodecahedra (with pentagonal faces) that have faces composed of triangles, and thus look like stylized stars.
		Wentzel Jamnitzer actually found the great stellated dodecahedron and the great dodecahedron in the 1500s, and Paolo Uccello discovered and drew the small stellated dodecahedron in the 1400s.
	www.wordiq.com /definition/Kepler-Poinsot_solid (384 words)

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		Kepler sent copies of his first major work to a number of scientists, including Tycho Brahe, who was soon to become the imperial mathematician of the Holy Roman Empire.
		In his Astronomia Nova ("New Astronomy") of 1609, Kepler had demonstrated that the orbit of the planet Mars is an ellipse.
		Following the model established by Archimedes, the most talented mathematician of antiquity, Kepler, in his volumetric researches, investigated the properties of nearly 100 solids of revolution--made by rotating a two-dimensional surface on one of its axes--that had not been considered by Archimedes.
	www.phy.bg.ac.yu /web_projects/giants/kepler.html (2137 words)

	Geometric Solids, Montessori World Educational Institute
		Apex and apices or apexes (plural) - the vertex of an angle.
		solid is regular if the spices are the same.
		There are nine regular solids: the five Platonian, pictured above, and the four polyhedra described by Kepler-Poinsot.
	www.montessoriworld.org /sensfile/sgeosoli.html (398 words)

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