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Kepler Insert Englisch 4 investigations into flat tessellations with pentagons, pentagons, with tessellations flat into investigations Internet: www.zometool.com Internet: with regular pentagons (Zometool’s red plane). His His plane). red (Zometool’s pentagons regular with Tel.: +49 (0)8324-9336040 · Mail: [email protected] [email protected] Mail: · (0)8324-9336040 +49 Tel.: ) and of tessellations of and ) kit Ice Crystals and Stars and Crystals Ice Kepler’s Kosmos Kepler’s hexagonal structure of snowflakes (see Zometool (see snowflakes of structure hexagonal Zometool Europa UG · Rainbühlgasse 7 · D 87541 Bad Hindelang Bad 87541 D · 7 Rainbühlgasse · UG Europa Zometool b c a Kepler’s further works include the study of the the of study the include works further Kepler’s Zometool Universum. Zometool in the extensive and versatile and extensive the in composition, Harmonie der Welt. der Harmonie composition, “: just a small taste of the many possibilities many the of taste small a just “: „ and “ „ “, „ Creator 3 Creator 5 design Hyperdo a “Copernican revolution” in the material sciences. material the in revolution” “Copernican a created a musical monument to Kepler with his 1957 1957 his with Kepler to monument musical a created with Dan Schechtman’s discovery of quasicrystals, which prompted which quasicrystals, of discovery Schechtman’s Dan with mechanics and the “music” of the universe. Paul Hindemith Hindemith Paul universe. the of “music” the and mechanics blocks have experienced an exciting revival during the 20th century, 20th the during revival exciting an experienced have blocks in which Kepler explains not only the mathematics, but also the the also but mathematics, the only not explains Kepler which in Penrose tiles can cover a plane surface quasiperiodically. Kepler quasiperiodically. surface plane a cover can tiles Penrose is a brilliant and daring “Theory of Everything”, Everything”, of “Theory daring and brilliant a is Harmonicus Mundi Harmonicus Kepler blocks will fill space according to matching rules, just as just rules, matching to according space fill will blocks Kepler first systematic documentation of mathematical tessellations. mathematical of documentation systematic first constructed with two types of “squashed” cubes (parallelepipeds). cubes “squashed” of types two with constructed discovered two new non-convex polyhedrons and compiled the compiled and polyhedrons non-convex new two discovered “squashed” squares (parallelograms), the triacontahedron can be can triacontahedron the (parallelograms), squares “squashed” He proved that there are 13 half-regular Archimedean solids, Archimedean half-regular 13 are there that proved He variations of Richert tilings. While Richert’s tiles are two types of types two are tiles Richert’s While tilings. Richert of variations treasure trove of both 2-dimensional and 3-dimensional geometry. 3-dimensional and 2-dimensional both of trove treasure its sub-units (Kepler blocks), which are a 3-dimensional a are which blocks), (Kepler sub-units its 1619. It contains a more precise model of the cosmos and is a is and cosmos the of model precise more a contains It 1619. Kepler also discovered the rhombic triacontahedron and and triacontahedron rhombic the discovered also Kepler (The Harmony of the World), which he published in published he which World), the of Harmony (The Mundi Richert and Sir Roger Penrose in the 20th century. 20th the in Penrose Roger Sir and Richert Sun Johannes Kepler described his third theory in in theory third his described Kepler Johannes other Zometool kits. kits. Zometool other Harmonicus Harmonicus decagons and stars heralded the tilings of Clark Clark of tilings the heralded stars and decagons The components of Kepler’s Kosmos are naturally compatible with all with compatible naturally are Kosmos Kepler’s of components The Richart and Penrose. Penrose. and Richart b c , Kepler, with large and small Richart diamonds (yellow), (yellow), diamonds Richart small and large with Kepler, , below: below: Tiling a TOOL ZOME TOOL ZOME , + arranged to form a rhombic triacontahedron. rhombic a form to arranged + , above: 3 Kepler blocks Kepler 2 1 to the cube of the semi-major axis of its orbit. its of axis semi-major the of cube the to The square of the orbital period of a planet is directly proportional directly is planet a of period orbital the of square The . 3 ® ® equal intervals of time. of intervals equal art and science at play at science and art art and science at play at science and art A line joining a planet and the sun sweeps out equal areas during areas equal out sweeps sun the and planet a joining line A . 2 The orbit of every planet is an ellipse with the sun at one focus. one at sun the with ellipse an is planet every of orbit The . 1 3 2 1 : : Kepler’s Laws of Planetary Motion Planetary of Laws Kepler’s art and science at play The flash of light: Kepler believed that God had given ® him a sign and that he had discovered the “key to the universe”. Kepler summised that the numerical ratio between the orbits of neighbouring planets would be ZOMETOOL similar to the sphere radius of the regular multi-sided solids (polyhedrons) - ie: that of the inscribed sphere At the beginning of the 17th. century, the world was above: Kepler’s illustrations of the Platonic Solids from the Mysterium Cosmographicum, 1596: above: Zometool models of the five Platonic Solids, corresponding to Kepler’s vision of the universe: to the circumscribed sphere. in turmoil. The Thirty Years’ War was decimating Cube - earth, Tetrahedron - fire, Dodecahedron - universum, Icosahedron - water, Octahedron - air. Icosahedron, Octahedron, Tetrahedron, Cube und Dodecahedron. Germany and the Netherlands were embroiled in far left: quill illustration of the moon, from Sidereus Nuncius, Galileo Galilei, 1610. the struggle for independance from Spain; ancient Kepler was born in Weil der Stadt; a sickly child, whose sight was Kepler’s theory stated that if a particular regular polyhedron were Kepler in Prag practices such as herbology, witchcraft and astrology marred by smallpox at an early age. He worked at his mother’s to by placed within the orbit of a planet, the sphere which exactly The Danish astrologer, Tycho Brahe, invited Kepler to work with continued to flourish amidst breathtaking advances inn, drew maps and attended a school for latin and the grammar surrounded this solid would then, by definition, describe the orbit him in Prag. It is probable that Brahe was interested in Kepler’s in science and technology. Hugo Grotius’ book, Das school in Maulbronn. He was a gifted child and graduated from of the next planet. Kepler took the orbit of the earth as the deter- mathematical skills, hoping that these would complement his freie Meer (The Freedom of the Seas) - the basis of our his theology course at the University of Tübingen at the early mining factor for all subsequent orbits. It was an elegant solution: own precise observations. Following Brahe’s death a year later, current conception of civil-rights - was banned by the Pope age of 20. During this time, he studied Copernicus’ vision of at that time, there were only six known planets and the Greeks the Holy Roman Emporer Rudolf II, a Habsburger, appointed immediately after publication, whilst in England Shakespeare’s the world and, at 23, renounced his plan to become a protestant had discovered that there are only five polyhedrons. Kepler Kepler to be his court mathematician in Prag. Kepler therefore Sonnettes were being received with great acclaim. priest in order to take up a professorship for mathematics at tested this theory, using Copernicus’ calculations as a basis and “inherited” Brahe’s extensive works, with which he hoped to Graz. He became more and more interested in the work of the result showed a discrepancy of less than 10%... Even by improve his own earlier theories. The measurements did not In March 1609 in Prag, Johannes Kepler, mathematician to the Copernicus and realised that he had stumbled upon an today’s standards, a quite stunning success. comform and he was forced to abandon his attempt. He began Holy Roman Emporer Rudolf II, presented his master with a heavy untapped treasure. Kepler began searching for a system work on an astrological system which was not based on the book. His work, Astronomia Nova, subtitled Physica Coelestis, behind the relationships of the distances between the Kepler’s cosmological theory, later published under the planetary orbits. In 1609, Kepler published Astronomia Nova, (Physics of the Heavens), marked the beginning of a new era. planets and the sun, hoping to discover a “construction title Mysterium Cosmographicum (The Secret of the a book which contains the first and second of Kepler’s Laws. plan of the world”. He sought harmony and intrinsic Universe) was born. Despite a life filled with trouble and personal tragedy, Kepler’s coherence, believing that God’s works, like our architecture, Kepler’s Kosmos contribution in many very different fields, such as astronomy, must be subject to rules and regulations. He hoped to find the It is the beginning cosmology, optics, arithmetic and geometry, is significant. solution in geometry. of a new era..
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