Johannes Kepler - Wikipedia, the Free Encyclopedia
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Essays-Mechanics / Electrodynamics/Download/8831
WHAT IF THE GALILEO AFFAIR NEVER HAPPENED? ABSTRACT From the 5th Century A.D. to the advent of Scholastic Aristotelianism, the curriculum in Roman Catholic universities in Europe taught a Rotating Geocentric Earth. St. Thomas Aquinas (1225 A.D.-1274 A.D.) introduced the writings of Aristotle (384 B.C.-322 B.C.) and of Claudius Ptolemy (100 A.D.-170 A.D.) in the 13th Century A.D., and the curriculum changed to teach a Non-Rotating Geocentric Earth. In this GSJ presentation, an Alternate History is presented. The famous Trial of Galileo never happened and the book by Nicolaus Copernicus was never banned by the Vatican of the Roman Catholic Church. The Roman Catholic Church simply ignored Galileo & his insulting book of 1632 A.D. The Roman Catholic Church Astronomers produced the Gregorian Calendar in 1583 A.D. The Roman Catholic Astronomers had abandoned the Aristotle/Ptolemy Model of Astronomy and had replaced it with the Non-Rotating Tychonian Model. This is NOT Alternate History. Johann Kepler published. That is NOT Alternate History. Isaac Newton published Principia, too. Christian Huygens contributed to the technology of the Mechanical Clock. Mechanical Time and Sundial Time demonstrate two similar but not identical paths. This is NOT Alternate History, either. The writer that is known as Voltaire (1694-1778) campaigned for Newton’s Principia as did Willem Gravesande (1688-1742). Pierre-Simon LaPlace (1748-1827) rescued Newton’s Celestial Mechanics of Universal Gravitation and Kepler’s Elliptical Orbits from disequilibrium issues in 1804A.D. Friedrich Bessel (1784-1846) discovered the first star to exhibit Stellar Parallax in 1838. -
Kepler and Hales: Conjectures and Proofs Dreams and Their
KEPLER AND HALES: CONJECTURES AND PROOFS DREAMS AND THEIR REALIZATION Josef Urban Czech Technical University in Prague Pittsburgh, June 21, 2018 1 / 31 This talk is an experiment Normally I give “serious” talks – I am considered crazy enough But this is Tom’s Birthday Party, so we are allowed to have some fun: This could be in principle a talk about our work on automation and AI for reasoning and formalization, and Tom’s great role in these areas. But the motivations and allusions go back to alchemistic Prague of 1600s and the (un)scientific pursuits of then versions of "singularity", provoking comparisons with our today’s funny attempts at building AI and reasoning systems for solving the great questions of Math, the Universe and Everything. I wonder where this will all take us. Part 1 (Proofs?): Learning automated theorem proving on top of Flyspeck and other corpora Learning formalization (autoformalization) on top of them Part 2 (Conjectures?): How did we get here? What were Kepler & Co trying to do in 1600s? What are we trying to do today? 2 / 31 The Flyspeck project – A Large Proof Corpus Kepler conjecture (1611): The most compact way of stacking balls of the same size in space is a pyramid. V = p 74% 18 Proved by Hales & Ferguson in 1998, 300-page proof + computations Big: Annals of Mathematics gave up reviewing after 4 years Formal proof finished in 2014 20000 lemmas in geometry, analysis, graph theory All of it at https://code.google.com/p/flyspeck/ All of it computer-understandable and verified in HOL Light: -
The Astronomers Tycho Brahe and Johannes Kepler
Ice Core Records – From Volcanoes to Supernovas The Astronomers Tycho Brahe and Johannes Kepler Tycho Brahe (1546-1601, shown at left) was a nobleman from Denmark who made astronomy his life's work because he was so impressed when, as a boy, he saw an eclipse of the Sun take place at exactly the time it was predicted. Tycho's life's work in astronomy consisted of measuring the positions of the stars, planets, Moon, and Sun, every night and day possible, and carefully recording these measurements, year after year. Johannes Kepler (1571-1630, below right) came from a poor German family. He did not have it easy growing Tycho Brahe up. His father was a soldier, who was killed in a war, and his mother (who was once accused of witchcraft) did not treat him well. Kepler was taken out of school when he was a boy so that he could make money for the family by working as a waiter in an inn. As a young man Kepler studied theology and science, and discovered that he liked science better. He became an accomplished mathematician and a persistent and determined calculator. He was driven to find an explanation for order in the universe. He was convinced that the order of the planets and their movement through the sky could be explained through mathematical calculation and careful thinking. Johannes Kepler Tycho wanted to study science so that he could learn how to predict eclipses. He studied mathematics and astronomy in Germany. Then, in 1571, when he was 25, Tycho built his own observatory on an island (the King of Denmark gave him the island and some additional money just for that purpose). -
Rosseau, Brendan 2019 Astronomy Thesis Title
Rosseau, Brendan 2019 Astronomy Thesis Title: The Intellectual Marketplace: The Evolution of Space Exploration from Copernicus to von Braun & Beyond Advisor: Jay Pasachoff Advisor is Co-author: None of the above Second Advisor: Released: release now Authenticated User Access: Yes Contains Copyrighted Material: No The Intellectual Marketplace: The Evolution of Space Exploration from Copernicus to von Braun & Beyond by Brendan L. Rosseau Dr. Jay Pasachoff, Advisor A thesis submitted in partial fulfillment of the requirements for the Degree of Bachelor of Arts with Honors in Astronomy WILLIAMS COLLEGE Williamstown, Massachusetts May 8, 2019 TABLE OF CONTENTS Acknowledgments ……………………………………………………………………………………………….. 2 Abstract ………………………………………………………………………………………………………………. 3 Thesis Prologue ………………………………………………………………………………………... 4 Introduction …………………………………………………………………………………... 6 Part I: Early Astronomy ………………………………………………………………….. 11 Part II: Space Exploration in the New World ……………………………….….... 25 Part III: Spaceflight ………………………………………………………………………... 43 Looking Back & Looking Ahead ….………………………………………………….… 60 Appendix Endnotes ………………………………………………………………………………………. 64 About the Author …………………………………………………………………………… 68 Citations ……………………………………………………………………………………….. 69 1 ACKNOWLEDGMENTS I would like to express my great appreciation to Professor Jay Pasachoff, Field Memorial Professor of Astronomy at Williams College, for his invaluable contributions to this thesis and to my education in astronomy. I am particularly grateful for the assistance given by -
Hints Into Kepler's Method
Stefano Gattei [email protected] Hints into Kepler’s method ABSTRACT The Italian Academy, Columbia University February 4, 2009 Some of Johannes Kepler’s works seem very different in character. His youthful Mysterium cosmographicum (1596) argues for heliocentrism on the basis of metaphysical, astronomical, astrological, numerological, and architectonic principles. By contrast, Astronomia nova (1609) is far more tightly argued on the basis of only a few dynamical principles. In the eyes of many, such a contrast embodies a transition from Renaissance to early modern science. However, Kepler did not subsequently abandon the broader approach of his early works: similar metaphysical arguments reappeared in Harmonices mundi libri V (1619), and he reissued the Mysterium cosmographicum in a second edition in 1621, in which he qualified only some of his youthful arguments. I claim that the conceptual and stylistic features of the Astronomia nova – as well as of other “minor” works, such as Strena seu De nive sexangula (1611) or Nova stereometria doliorum vinariorum (1615) – are intimately related and were purposely chosen because of the response he knew to expect from the astronomical community to the revolutionary changes in astronomy he was proposing. Far from being a stream-of-consciousness or merely rhetorical kind of narrative, as many scholars have argued, Kepler’s expository method was carefully calculated both to convince his readers and to engage them in a critical discussion in the joint effort to know God’s design. By abandoning the perspective of the inductivist philosophy of science, which is forced by its own standards to portray Kepler as a “sleepwalker,” I argue that the key lies in the examination of Kepler’s method: whether considering the functioning and structure of the heavens or the tiny geometry of the little snowflakes, he never hesitated to discuss his own intellectual journey, offering a rational reconstruction of the series of false starts, blind alleys, and failures he encountered. -
Februar 2019
Februar 2019 Vor 551 Jahren geboren JOHANNES WERNER (14.02.1468 – Mai 1522) Im Jahre 1472 wurde in Ingolstadt die erste Universität Bayerns durch Herzog LUDWIG IX von Bayern-Landshut ge- gründet. Nach der Kirchenspaltung ab 1530 entwickelte sich die Hochschule unter dem Einfluss des Jesuitenordens zu einem der Zentren der Gegenreformation. Kurfürst MAXI - MILIAN , der spätere bayerische König MAXIMILIAN I, ver- legte 1800 die Universität zunächst nach Landshut, dann nach München – seit 1802 trägt sie den heutigen Namen: LUDWIG -MAXIMILIAN s-Universität (LMU). 1484 schreibt sich der in Nürnberg geborene 16-jährige JOHANNES WERNER an der theologischen Fakultät der Ingolstädter Hochschule ein. Auch wenn es von Kind an sein Wunsch gewesen ist, Mathematiker zu werden, verfolgt er nun doch konsequent den Weg zum Priesterberuf: 1490 wird er als Kaplan in Herzogenaurach tätig, verbringt einige Jahre in Rom, wechselt 1498 an eine Pfarrei in Wöhrd bei Nürnberg, bis er schließlich Pfarrer der Johanniskirche in Nürnberg wird. Die Pflichten dieses Amtes nimmt er gewissenhaft bis zu seinem Tod wahr. Seine Zeit in Rom hatte WERNER bereits zu intensiven Studien der Mathematik und der Astronomie genutzt. Nach Nürnberg zurückgekehrt, vertieft er sich in eigene Forschungen. So beo- bachtet er im Jahr 1500 die Bewegung eines Kometen, führt Messungen mit selbstgebauten Instrumenten aus und dokumen- tiert alle Daten mit Sorgfalt. Mit großem Geschick baut er Astrolabien (Sternhöhen- messer) und Sonnenuhren, konstruiert einen besonderen Jakobsstab mit Winkeleinteilung. 1514 erscheint WERNER s Übersetzung der Geographia des CLAUDIUS PTOLEMÄUS (In Hoc Opere Haec Continentur Nova Translatio Primi Libri Geographicae Cl. Ptolomaei ). Zusätzlich zu umfangreichen Kommentaren entwickelt er eigene Ideen, die in Astronomie und Geografie angewandt werden können. -
Space Travel to the Moon and Kepler's Dream
Proceedings of the Iowa Academy of Science Volume 79 Number Article 15 1972 Space Travel to the Moon and Kepler's Dream Paul B. Selz Parsons College Let us know how access to this document benefits ouy Copyright ©1972 Iowa Academy of Science, Inc. Follow this and additional works at: https://scholarworks.uni.edu/pias Recommended Citation Selz, Paul B. (1972) "Space Travel to the Moon and Kepler's Dream," Proceedings of the Iowa Academy of Science, 79(1), 47-48. Available at: https://scholarworks.uni.edu/pias/vol79/iss1/15 This General Interest Article is brought to you for free and open access by the Iowa Academy of Science at UNI ScholarWorks. It has been accepted for inclusion in Proceedings of the Iowa Academy of Science by an authorized editor of UNI ScholarWorks. For more information, please contact [email protected]. Selz: Space Travel to the Moon and Kepler's Dream SPACE TRAVEL & KEPLER'S DREAM 47 Space Travel to the Moon and Kepler's Dream PAUL B. SELZ1 PAUL B. SELZ. Space Travel to the Moon and Kepler's Dream. that the same conclusions would follow. This involved concepts Proc. Iowa Acad. Sci., 79(1):47-48, 1972. ~£ mass, inertia, gravity, acceleration, velocity, and the driving SYNOPSIS: Johann Kepler advocated Copernicus's heliocentric force in a trip to the moon. Twelve years before Newton's birth theory in his Dream and Notes. He imagined how a moon dweller in 1642, Kepler published in this little known dream ideas which would see the solar system and the conclusions he would draw. -
A Computer Verification of the Kepler Conjecture
ICM 2002 Vol. III 1–3 · · A Computer Verification of the Kepler Conjecture Thomas C. Hales∗ Abstract The Kepler conjecture asserts that the density of a packing of congruent balls in three dimensions is never greater than π/√18. A computer assisted verification confirmed this conjecture in 1998. This article gives a historical introduction to the problem. It describes the procedure that converts this problem into an optimization problem in a finite number of variables and the strategies used to solve this optimization problem. 2000 Mathematics Subject Classification: 52C17. Keywords and Phrases: Sphere packings, Kepler conjecture, Discrete ge- ometry. 1. Historical introduction The Kepler conjecture asserts that the density of a packing of congruent balls in three dimensions is never greater than π/√18 0.74048 .... This is the oldest problem in discrete geometry and is an important≈ part of Hilbert’s 18th problem. An example of a packing achieving this density is the face-centered cubic packing (Figure 1). A packing of balls is an arrangement of nonoverlapping balls of radius 1 in Euclidean space. Each ball is determined by its center, so equivalently it is a arXiv:math/0305012v1 [math.MG] 1 May 2003 collection of points in Euclidean space separated by distances of at least 2. The density of a packing is defined as the lim sup of the densities of the partial packings formed by the balls inside a ball with fixed center of radius R. (By taking the lim sup, rather than lim inf as the density, we prove the Kepler conjecture in the strongest possible sense.) Defined as a limit, the density is insensitive to changes in the packing in any bounded region. -
The Sphere Packing Problem in Dimension 8 Arxiv:1603.04246V2
The sphere packing problem in dimension 8 Maryna S. Viazovska April 5, 2017 8 In this paper we prove that no packing of unit balls in Euclidean space R has density greater than that of the E8-lattice packing. Keywords: Sphere packing, Modular forms, Fourier analysis AMS subject classification: 52C17, 11F03, 11F30 1 Introduction The sphere packing constant measures which portion of d-dimensional Euclidean space d can be covered by non-overlapping unit balls. More precisely, let R be the Euclidean d vector space equipped with distance k · k and Lebesgue measure Vol(·). For x 2 R and d r 2 R>0 we denote by Bd(x; r) the open ball in R with center x and radius r. Let d X ⊂ R be a discrete set of points such that kx − yk ≥ 2 for any distinct x; y 2 X. Then the union [ P = Bd(x; 1) x2X d is a sphere packing. If X is a lattice in R then we say that P is a lattice sphere packing. The finite density of a packing P is defined as Vol(P\ Bd(0; r)) ∆P (r) := ; r > 0: Vol(Bd(0; r)) We define the density of a packing P as the limit superior ∆P := lim sup ∆P (r): r!1 arXiv:1603.04246v2 [math.NT] 4 Apr 2017 The number be want to know is the supremum over all possible packing densities ∆d := sup ∆P ; d P⊂R sphere packing 1 called the sphere packing constant. For which dimensions do we know the exact value of ∆d? Trivially, in dimension 1 we have ∆1 = 1. -
A Triangle with Sides Lengths of a Rational Power of the Plastic Constant
A TRIANGLE WITH SIDES LENGTHS OF A RATIONAL POWER OF THE PLASTIC CONSTANT KOUICHI NAKAGAWA Abstract. Using the golden ratio φ, a triangle whose side length ratio can be expressed as p 2 1 : φ : φ represents a rightp triangle since the golden ratio has the property of φ = φ + 1 and therefore satisfies 12 + ( φ)2 = φ2. This triangle is called the Kepler triangle. As in the case of the Kepler triangle, in this study we determine triangles where the length of the three sides is expressed using only the constant obtained from the linear recurrence sequence (golden ratio, plastic constant, tribonacci constant and supergolden ratio). 1. Introduction The Fibonacci numbers are defined by the recurrence Fn+2 = Fn+1 + Fn with the initial values F0 = 0, F1 = 1, and the limit of the ratios of successive terms converges to p F 1 + 5 lim n = φ = ≈ 1:61803 ··· ; n!1 Fn−1 2 φ is called the golden ratio. One property of the golden ratio is φ2 = φ + 1; it is also an algebraic integer of degree 2. By the property of the golden ratio and the Pythagorean theorem, we have φ2 = φ + 1 () 12 + (pφ)2 = φ2: p Since the ratio of the lengths of the three sides is 1 : φ : φ, this triangle is a right triangle, called the Kepler triangle. In addition, the following are known as characteristic triangles. An isosceles triangle whose three side length ratios can be expressed as φ : φ : 1 is called a golden triangle. The three angles are 36° − 72° − 72°. -
An Overview of the Kepler Conjecture
AN OVERVIEW OF THE KEPLER CONJECTURE Thomas C. Hales 1. Introduction and Review The series of papers in this volume gives a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than π/√18 0.74048 . This is the oldest problem in discrete geometry and is an important≈ part of Hilbert’s 18th problem. An example of a packing achieving this density is the face-centered cubic packing. A packing of spheres is an arrangement of nonoverlapping spheres of radius 1 in Euclidean space. Each sphere is determined by its center, so equivalently it is a collection of points in Euclidean space separated by distances of at least 2. The density of a packing is defined as the lim sup of the densities of the partial packings formed by spheres inside a ball with fixed center of radius R. (By taking the lim sup, rather than lim inf as the density, we prove the Kepler conjecture in the strongest possible sense.) Defined as a limit, the density is insensitive to changes in the packing in any bounded region. For example, a finite number of spheres can be removed from the face-centered cubic packing without affecting its density. Consequently, it is not possible to hope for any strong uniqueness results for packings of optimal density. The uniqueness established by this work is as strong arXiv:math/9811071v2 [math.MG] 20 May 2002 as can be hoped for. It shows that certain local structures (decomposition stars) attached to the face-centered cubic (fcc) and hexagonal-close packings (hcp) are the only structures that maximize a local density function. -
Borromini and the Cultural Context of Kepler's Harmonices Mundi
Borromini and the Dr Valerie Shrimplin cultural context of [email protected] Kepler’sHarmonices om Mundi • • • • Francesco Borromini, S Carlo alle Quattro Fontane Rome (dome) Harmonices Mundi, Bk II, p. 64 Facsimile, Carnegie-Mellon University Francesco Borromini, S Ivo alla Sapienza Rome (dome) Harmonices Mundi, Bk IV, p. 137 • Vitruvius • Scriptures – cosmology and The Genesis, Isaiah, Psalms) cosmological • Early Christian - dome of heaven view of the • Byzantine - domed architecture universe and • Renaissance revival – religious art/architecture symbolism of centrally planned churches • Baroque (17th century) non-circular domes as related to Kepler’s views* *INSAP II, Malta 1999 Cosmas Indicopleustes, Universe 6th cent Last Judgment 6th century (VatGr699) Celestial domes Monastery at Daphne (Δάφνη) 11th century S Sophia, Constantinople (built 532-37) ‘hanging architecture’ Galla Placidia, 425 St Mark’s Venice, late 11th century Evidence of Michelangelo interests in Art and Cosmology (Last Judgment); Music/proportion and Mathematics Giacomo Vignola (1507-73) St Andrea in Via Flaminia 1550-1553 Church of San Giacomo in Augusta, in Rome, Italy, completed by Carlo Maderno 1600 [painting is 19th century] Sant'Anna dei Palafrenieri, 1620’s (Borromini with Maderno) Leonardo da Vinci, Notebooks (318r Codex Atlanticus c 1510) Amboise Bachot, 1598 Following p. 52 Astronomia Nova Link between architecture and cosmology (as above) Ovals used as standard ellipse approximation Significant change/increase Revival of neoplatonic terms, geometrical bases in early 17th (ellipse, oval, equilateral triangle) century Fundamental in Harmonices Mundi where orbit of every planet is ellipse with sun at one of foci Borromini combined practical skills with scientific learning and culture • Formative years in Milan (stonemason) • ‘Artistic anarchist’ – innovation and disorder.