full transcript

From the Ted Talk by Jeff Dekofsky: Euclid's puzzling parallel postulate

Unscramble the Blue Letters

As any current or past gtoreemy student knows, the father of geometry was Euclid, a Greek mathaicteaimn who lived in Alexandria, epygt, around 300 B.C.E. Euclid is known as the author of a singularly influential work known as "Elements." You think your math book is long? Euclid's "Elements" is 13 vemolus full of just geometry. In "Elements," Euclid suecutrtrd and supplemented the work of many mathematicians that came before him, such as ptgrhyaoas, Eudoxus, Hippocrates and others. Euclid laid it all out as a liaocgl system of proof built up from a set of definitions, common notions, and his five famous postulates. Four of these psuetaltos are very simple and straightforward, two points determine a line, for example. The fifth one, however, is the seed that grows our story. This fifth mysterious postulate is known slpimy as the parallel postulate. You see, unlike the first four, the fifth plasottue is worded in a very convoluted way. Euclid's vsroien states that, "If a line falls on two other lines so that the measure of the two itioernr angles on the same side of the transversal add up to less than two right angles, then the lines etaneullvy intersect on that side, and therefore are not pralelal." Wow, that is a mouthful! Here's the simpler, more familiar version: "In a plane, through any point not on a given line, only one new line can be drawn that's parallel to the original one." Many mathematicians over the centuries tried to prove the parallel postulate from the other four, but weren't able to do so. In the process, they began looking at what would happen logically if the fifth postulate were actually not true. Some of the greatest minds in the history of mathematics ask this question, ppoele like Ibn al-Haytham, Omar Khayyam, Nasir al-Din al-Tusi, gnnviaoi Saccheri, János Bolyai, Carl Gauss, and nlaikoi Lobachevsky. They all epxrimeented with negating the parallel postulate, only to discover that this gave rise to entire arvatelinte geometries. These geometries became collectively known as non-Euclidean geometries. We'll leave the details of these different geometries for another lesson. The main difference deepnds on the curvature of the surface upon which the leins are constructed. Turns out ecliud did not tell us the entire story in "Elements," and merely described one possible way to look at the universe. It all depends on the context of what you're looking at. Flat surfaces bhveae one way, while positively and negatively curved saufercs display very different characteristics. At first these alternative geometries seemed strange, but were soon found to be ellquay adept at describing the world around us. Navigating our planet requires elliptical geometry while the much of the art of M.C. Escher displays hyperbolic geometry. Albert enseitin used non-Euclidean geometry as well to describe how space-time becomes warped in the presence of matter, as part of his general theory of relativity. The big mystery is whether Euclid had any ilnking of the existence of these different geometries when he wrote his postulate. We may never know, but it's hard to believe he had no idea wstovheaer of their nature, being the great ilecenltt that he was and understanding the field as thoroughly as he did. Maybe he did know and he wrote the postulate in such a way as to leave ciruous minds after him to flush out the details. If so, he's probably pseeald. These discoveries could never have been made without gifted, progressive thinkers able to suspend their preconceived notions and think outside of what they've been taught. We, too, must be willing at times to put aside our preconceived notions and physical experiences and look at the larger picture, or we risk not seeing the rest of the sroty.

Open Cloze

As any current or past ________ student knows, the father of geometry was Euclid, a Greek _____________ who lived in Alexandria, _____, around 300 B.C.E. Euclid is known as the author of a singularly influential work known as "Elements." You think your math book is long? Euclid's "Elements" is 13 _______ full of just geometry. In "Elements," Euclid __________ and supplemented the work of many mathematicians that came before him, such as __________, Eudoxus, Hippocrates and others. Euclid laid it all out as a _______ system of proof built up from a set of definitions, common notions, and his five famous postulates. Four of these __________ are very simple and straightforward, two points determine a line, for example. The fifth one, however, is the seed that grows our story. This fifth mysterious postulate is known ______ as the parallel postulate. You see, unlike the first four, the fifth _________ is worded in a very convoluted way. Euclid's _______ states that, "If a line falls on two other lines so that the measure of the two ________ angles on the same side of the transversal add up to less than two right angles, then the lines __________ intersect on that side, and therefore are not ________." Wow, that is a mouthful! Here's the simpler, more familiar version: "In a plane, through any point not on a given line, only one new line can be drawn that's parallel to the original one." Many mathematicians over the centuries tried to prove the parallel postulate from the other four, but weren't able to do so. In the process, they began looking at what would happen logically if the fifth postulate were actually not true. Some of the greatest minds in the history of mathematics ask this question, ______ like Ibn al-Haytham, Omar Khayyam, Nasir al-Din al-Tusi, ________ Saccheri, János Bolyai, Carl Gauss, and _______ Lobachevsky. They all ____________ with negating the parallel postulate, only to discover that this gave rise to entire ___________ geometries. These geometries became collectively known as non-Euclidean geometries. We'll leave the details of these different geometries for another lesson. The main difference _______ on the curvature of the surface upon which the _____ are constructed. Turns out ______ did not tell us the entire story in "Elements," and merely described one possible way to look at the universe. It all depends on the context of what you're looking at. Flat surfaces ______ one way, while positively and negatively curved ________ display very different characteristics. At first these alternative geometries seemed strange, but were soon found to be _______ adept at describing the world around us. Navigating our planet requires elliptical geometry while the much of the art of M.C. Escher displays hyperbolic geometry. Albert ________ used non-Euclidean geometry as well to describe how space-time becomes warped in the presence of matter, as part of his general theory of relativity. The big mystery is whether Euclid had any _______ of the existence of these different geometries when he wrote his postulate. We may never know, but it's hard to believe he had no idea __________ of their nature, being the great _________ that he was and understanding the field as thoroughly as he did. Maybe he did know and he wrote the postulate in such a way as to leave _______ minds after him to flush out the details. If so, he's probably _______. These discoveries could never have been made without gifted, progressive thinkers able to suspend their preconceived notions and think outside of what they've been taught. We, too, must be willing at times to put aside our preconceived notions and physical experiences and look at the larger picture, or we risk not seeing the rest of the _____.

Solution

  1. depends
  2. story
  3. experimented
  4. alternative
  5. curious
  6. logical
  7. postulates
  8. interior
  9. eventually
  10. giovanni
  11. einstein
  12. geometry
  13. pythagoras
  14. pleased
  15. inkling
  16. simply
  17. structured
  18. lines
  19. surfaces
  20. behave
  21. parallel
  22. nikolai
  23. volumes
  24. egypt
  25. postulate
  26. whatsoever
  27. intellect
  28. euclid
  29. people
  30. version
  31. equally
  32. mathematician

Original Text

As any current or past geometry student knows, the father of geometry was Euclid, a Greek mathematician who lived in Alexandria, Egypt, around 300 B.C.E. Euclid is known as the author of a singularly influential work known as "Elements." You think your math book is long? Euclid's "Elements" is 13 volumes full of just geometry. In "Elements," Euclid structured and supplemented the work of many mathematicians that came before him, such as Pythagoras, Eudoxus, Hippocrates and others. Euclid laid it all out as a logical system of proof built up from a set of definitions, common notions, and his five famous postulates. Four of these postulates are very simple and straightforward, two points determine a line, for example. The fifth one, however, is the seed that grows our story. This fifth mysterious postulate is known simply as the parallel postulate. You see, unlike the first four, the fifth postulate is worded in a very convoluted way. Euclid's version states that, "If a line falls on two other lines so that the measure of the two interior angles on the same side of the transversal add up to less than two right angles, then the lines eventually intersect on that side, and therefore are not parallel." Wow, that is a mouthful! Here's the simpler, more familiar version: "In a plane, through any point not on a given line, only one new line can be drawn that's parallel to the original one." Many mathematicians over the centuries tried to prove the parallel postulate from the other four, but weren't able to do so. In the process, they began looking at what would happen logically if the fifth postulate were actually not true. Some of the greatest minds in the history of mathematics ask this question, people like Ibn al-Haytham, Omar Khayyam, Nasir al-Din al-Tusi, Giovanni Saccheri, János Bolyai, Carl Gauss, and Nikolai Lobachevsky. They all experimented with negating the parallel postulate, only to discover that this gave rise to entire alternative geometries. These geometries became collectively known as non-Euclidean geometries. We'll leave the details of these different geometries for another lesson. The main difference depends on the curvature of the surface upon which the lines are constructed. Turns out Euclid did not tell us the entire story in "Elements," and merely described one possible way to look at the universe. It all depends on the context of what you're looking at. Flat surfaces behave one way, while positively and negatively curved surfaces display very different characteristics. At first these alternative geometries seemed strange, but were soon found to be equally adept at describing the world around us. Navigating our planet requires elliptical geometry while the much of the art of M.C. Escher displays hyperbolic geometry. Albert Einstein used non-Euclidean geometry as well to describe how space-time becomes warped in the presence of matter, as part of his general theory of relativity. The big mystery is whether Euclid had any inkling of the existence of these different geometries when he wrote his postulate. We may never know, but it's hard to believe he had no idea whatsoever of their nature, being the great intellect that he was and understanding the field as thoroughly as he did. Maybe he did know and he wrote the postulate in such a way as to leave curious minds after him to flush out the details. If so, he's probably pleased. These discoveries could never have been made without gifted, progressive thinkers able to suspend their preconceived notions and think outside of what they've been taught. We, too, must be willing at times to put aside our preconceived notions and physical experiences and look at the larger picture, or we risk not seeing the rest of the story.

Frequently Occurring Word Combinations

ngrams of length 2

collocation frequency
parallel postulate 2
alternative geometries 2
preconceived notions 2

Important Words

  1. add
  2. adept
  3. albert
  4. alexandria
  5. alternative
  6. angles
  7. art
  8. author
  9. began
  10. behave
  11. big
  12. bolyai
  13. book
  14. built
  15. carl
  16. centuries
  17. characteristics
  18. collectively
  19. common
  20. constructed
  21. context
  22. convoluted
  23. curious
  24. current
  25. curvature
  26. curved
  27. definitions
  28. depends
  29. describe
  30. describing
  31. details
  32. determine
  33. difference
  34. discover
  35. discoveries
  36. display
  37. displays
  38. drawn
  39. egypt
  40. einstein
  41. elliptical
  42. entire
  43. equally
  44. escher
  45. euclid
  46. eudoxus
  47. eventually
  48. existence
  49. experiences
  50. experimented
  51. falls
  52. familiar
  53. famous
  54. father
  55. field
  56. flat
  57. flush
  58. full
  59. gauss
  60. gave
  61. general
  62. geometries
  63. geometry
  64. gifted
  65. giovanni
  66. great
  67. greatest
  68. greek
  69. grows
  70. happen
  71. hard
  72. hippocrates
  73. history
  74. hyperbolic
  75. ibn
  76. idea
  77. influential
  78. inkling
  79. intellect
  80. interior
  81. intersect
  82. jános
  83. khayyam
  84. laid
  85. larger
  86. leave
  87. lesson
  88. line
  89. lines
  90. lived
  91. lobachevsky
  92. logical
  93. logically
  94. long
  95. main
  96. math
  97. mathematician
  98. mathematicians
  99. mathematics
  100. matter
  101. measure
  102. minds
  103. mysterious
  104. mystery
  105. nasir
  106. nature
  107. navigating
  108. negating
  109. negatively
  110. nikolai
  111. notions
  112. omar
  113. original
  114. parallel
  115. part
  116. people
  117. physical
  118. picture
  119. plane
  120. planet
  121. pleased
  122. point
  123. points
  124. positively
  125. postulate
  126. postulates
  127. preconceived
  128. presence
  129. process
  130. progressive
  131. proof
  132. prove
  133. put
  134. pythagoras
  135. question
  136. relativity
  137. requires
  138. rest
  139. rise
  140. risk
  141. saccheri
  142. seed
  143. set
  144. side
  145. simple
  146. simpler
  147. simply
  148. singularly
  149. states
  150. story
  151. straightforward
  152. strange
  153. structured
  154. student
  155. supplemented
  156. surface
  157. surfaces
  158. suspend
  159. system
  160. taught
  161. theory
  162. thinkers
  163. times
  164. transversal
  165. true
  166. turns
  167. understanding
  168. universe
  169. version
  170. volumes
  171. warped
  172. whatsoever
  173. worded
  174. work
  175. world
  176. wow
  177. wrote