full transcript

## 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
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

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
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
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