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And His Mystic Hexagram B Blaise Pascal and His Mystic Hexagram B C A D F E John Martin Santa Rosa Junior College Great Mathematicians of the 17th Century Great Mathematicians of the 17th Century Paris Blaise Pascal Clermont 1623 – 1662 The Early Years Paris Blaise Pascal Clermont 1623 – 1662 The Early Years A CC B Blaise Pascal 1623 – 1662 The Early Years y x Étienne Pascal “Limaçon of Pascal” 1588 – 1651 The Early Years Pascal Mersenne Desargues Pascal’s Mystic Hexagram If six arbitrary points are chosen on a conic section and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon meet in three points that lie on a straight line. B C A D F E Pascal the Inventor The “Pascaline” Pascal the Inventor The “Pascaline” Pascal the Scientist Vacuum 36 in. 30 in. Torricelli’s Experiment Pascal the Scientist “Pascal has too much New Experiments vacuum in his head.” Concerning Vacuums R. Descartes October 1647 Pascal the Scientist Aristotle 384 BC – 322 BC “Nature abhors a vacuum.” Pascal the Scientist Puy de Dôme Change in elevation ~1000 m Change in level of Hg ~85 mm Pascal the Scientist “This knowledge can be very useful to farmers, travelers, etc., to learn the present state of the weather, and that which is to follow immediately but not to know that which is to come in three weeks.” Pascal’s Principle: A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid. Pascal the Theologian Tenets of Jansenism: Original sin Human depravity Necessity of divine grace Predestination Cornelius Jansen 1585 – 1638 Pascal the Theologian Pascal the Theologian Memorial The year of grace 1654, Monday, November 23. From about half past ten at night until about half past midnight, FIRE. GOD of Abraham, GOD of Isaac, GOD of Jacob not of the philosophers and of the learned. Certitude. Certitude. Feeling. Joy. Peace. GOD of Jesus Christ. My God and your God. Your GOD will be my God. Forgetfulness of the world and of everything, except GOD. He is only found by the ways taught in the Gospel. Grandeur of the human soul. Righteous Father, the world has not known you, but I have known you. Joy, joy, joy, tears of joy. I have departed from him: They have forsaken me, the fount of living water. My God, will you leave me? Let me not be separated from him forever. This is eternal life, that they know you, the one true God, and the one that you sent, Jesus Christ. Jesus Christ. Jesus Christ. I left him; I fled him, renounced, crucified. Let me never be separated from him. He is only kept securely by the ways taught in the Gospel: Renunciation, total and sweet. Complete submission to Jesus Christ and to my director. Eternally in joy for a day’s exercise on the earth. May I not forget your words. Amen. Pascal the Theologian Memorial The year of grace 1654, Monday, November 23. From about half past ten at night until about half past midnight, FIRE. GOD of Abraham, GOD of Isaac, GOD of Jacob not of the philosophers and of the learned. Certitude. Certitude. Feeling. Joy. Peace. GOD of Jesus Christ. My God and your God. Your GOD will be my God. Forgetfulness of the world and of everything, except GOD. He is only found by the ways taught in the Gospel. Grandeur of the human soul. Righteous Father, the world has not known you, but I have known you. Joy, joy, joy, tears of joy. I have departed from him: They have forsaken me, the fount of living water. My God, will you leave me? Let me not be separated from him forever. This is eternal life, that they know you, the one true God, and the one that you sent, Jesus Christ. Jesus Christ. Jesus Christ. I left him; I fled him, renounced, crucified. The Monastery at Let me never be separated from him. He is only kept securely by the ways taught in the Gospel: Port-Royal des Champs Renunciation, total and sweet. Complete submission to Jesus Christ and to my director. Eternally in joy for a day’s exercise on the earth. May I not forget your words. Amen. Pascal the Theologian Antoine Arnauld 1616 – 1698 Pascal the Theologian “The first work of genius in prose that we find is the collected edition of the Letters. All kinds of eloquence are displayed here, it is in this work that our language takes its final François-Marie Arouet form.” Voltaire 1694 – 1778 Pascal the Philosopher Pascal the Philosopher Pascal’s Wager: “Belief is a wise wager. Granted that faith cannot be proved, what harm will come to you if you gamble on its truth and it proves false? If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation, that He exists.” Pascal the Mathematician The Generation of Conic Sections Circle Ellipse Parabola Hyperbola Pascal the Mathematician Fair Division of the Stakes A game of chance between two players Each player equally like to win each round Each player contributes equal amounts First player to win certain number of rounds takes the prize. Pascal the Mathematician Fair Division of the Stakes In a game where one player needs m rounds to win and the other needs n rounds, divide the stakes in the ratio n−1 ⎛ m + n −1⎞ m+n−1⎛ m + n −1⎞ to ∑ ⎜ k ⎟ ∑ ⎜ k ⎟ k=0 ⎝ ⎠ k=n ⎝ ⎠ Pascal the Mathematician For Example: Suppose that two players agree that the first player to win seven rounds of a game will collect the prize. How should the stakes be divided among them if one has won five rounds and the other has won three? 3 ⎛ 5⎞ 5 ⎛ 5⎞ Divide the stakes in the ratio: to ∑ ⎜ k⎟ ∑ ⎜ k⎟ k=0 ⎝ ⎠ k=4 ⎝ ⎠ ⎛ 5⎞ ⎛ 5⎞ ⎛ 5⎞ ⎛ 5⎞ ⎛ 5⎞ ⎛ 5⎞ + + + to + ⎝⎜ 0⎠⎟ ⎝⎜1⎠⎟ ⎝⎜ 2⎠⎟ ⎝⎜ 3⎠⎟ ⎝⎜ 4⎠⎟ ⎝⎜ 5⎠⎟ Or 26 to 6 Pascal the Mathematician “The legacies the two men left are enormous. Disregarding everything else they did in their highly productive lives, the one mathematical breakthrough contained in their brief exchange of letters in 1654 started humanity on the path toward scientific risk management and thereby changed human life forever.” Pascal the Mathematician Treatise on the Arithmetic Triangle, Published in 1665 Pascal the Mathematician 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 The triangle in modern form. Pascal the Mathematician 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 The triangle in modern form. Pascal the Mathematician 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 The triangle in modern form. Yang Hui’s version appeared in 1261. Pascal the Mathematician The Cycloid The curve traced out by a point on the circumference of a rolling circle. The History of the Cycloid, Published in 1658 The Theorem of the Mystic Hexagram If six arbitrary points are chosen on a conic section and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon meet in three points that lie on a straight line. B C A D F E The Theorem of the Mystic Hexagram If six arbitrary points are chosen on a conic section and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon meet in three points that lie on a straight line. For any fixed configuration of six points, how many different hexagons are possible? How Many Hexagons? B The first point is arbitrary. C 5 choices for the second point. A D 4 choices for the third point. 3 choices for the fourth point. 2 choices for the fifth point. E F 1 choice for the sixth point. Hexagon ABCDEF is the same as Hexagon AFEDCB 5 × 4 × 3 × 2 × 1 Number of Hexagons = = 60 2 Sixty Hexagons Other Points of Interest Jakob Steiner Thomas Kirkman Arthur Cayley George Salmon Pascal’s Legacy “He lived most of his adult life in great pain, suffering even in his youth from migraines.” Gilberte Pascal Blaise Pascal 1623 – 1662 Pascal’s Legacy One pascal is the SI unit of pressure defined as one newton per square meter. Blaise Pascal 1623 – 1662 Pascal’s Legacy Niklaus Wirth invented the computer programming language Pascal in 1970. program HelloWorld (output); begin writeln (‘Pascal, I love you!’) end. Blaise Pascal 1623 – 1662.
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