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Seeing is believing Euclid’s wake In perspective
Renaissance imagination: geometry
Paul Hewitt
University of Toledo
Math 3510, 5 October 2010
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Seeing is believing Euclid’s wake In perspective
1. According to Stillwell, why is the geometry of vision “harderthan it looks”?
A. The eye sees points that aren’t there.B. The eye fails to see points that are there.C. BothD. Neither.
Bonus: When (approximately) were the Crusades?1095–1291
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Seeing is believing Euclid’s wake In perspective
1. According to Stillwell, why is the geometry of vision “harderthan it looks”?
A. The eye sees points that aren’t there.B. The eye fails to see points that are there.C. BothD. Neither.
Bonus: When (approximately) were the Crusades?1095–1291
![Page 4: Renaissance imagination: geometry - Live Toadlivetoad.org/Courses/Documents/9eb4/Notes/lecture_20101005.pdf · Seeing is believingEuclid’s wakeIn perspective Renaissance imagination:](https://reader031.vdocuments.mx/reader031/viewer/2022022517/5b05b4b07f8b9a58148bbd70/html5/thumbnails/4.jpg)
Seeing is believing Euclid’s wake In perspective
1. According to Stillwell, why is the geometry of vision “harderthan it looks”?
A. The eye sees points that aren’t there.B. The eye fails to see points that are there.C. BothD. Neither.
Bonus: When (approximately) were the Crusades?
1095–1291
![Page 5: Renaissance imagination: geometry - Live Toadlivetoad.org/Courses/Documents/9eb4/Notes/lecture_20101005.pdf · Seeing is believingEuclid’s wakeIn perspective Renaissance imagination:](https://reader031.vdocuments.mx/reader031/viewer/2022022517/5b05b4b07f8b9a58148bbd70/html5/thumbnails/5.jpg)
Seeing is believing Euclid’s wake In perspective
1. According to Stillwell, why is the geometry of vision “harderthan it looks”?
A. The eye sees points that aren’t there.B. The eye fails to see points that are there.C. BothD. Neither.
Bonus: When (approximately) were the Crusades?1095–1291
![Page 6: Renaissance imagination: geometry - Live Toadlivetoad.org/Courses/Documents/9eb4/Notes/lecture_20101005.pdf · Seeing is believingEuclid’s wakeIn perspective Renaissance imagination:](https://reader031.vdocuments.mx/reader031/viewer/2022022517/5b05b4b07f8b9a58148bbd70/html5/thumbnails/6.jpg)
Seeing is believing Euclid’s wake In perspective
2. What theorem of plane geometry requires Euclid’s fifthpostulate?
A. Rectangles exist.B. The pythagorean theorem.C. The angles in a triangle sum to π.D. All of the above.E . None of the above.
Bonus: When (approximately) did Genghis Khan rule?1206–1227
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Seeing is believing Euclid’s wake In perspective
2. What theorem of plane geometry requires Euclid’s fifthpostulate?
A. Rectangles exist.B. The pythagorean theorem.C. The angles in a triangle sum to π.D. All of the above.E . None of the above.
Bonus: When (approximately) did Genghis Khan rule?1206–1227
![Page 8: Renaissance imagination: geometry - Live Toadlivetoad.org/Courses/Documents/9eb4/Notes/lecture_20101005.pdf · Seeing is believingEuclid’s wakeIn perspective Renaissance imagination:](https://reader031.vdocuments.mx/reader031/viewer/2022022517/5b05b4b07f8b9a58148bbd70/html5/thumbnails/8.jpg)
Seeing is believing Euclid’s wake In perspective
2. What theorem of plane geometry requires Euclid’s fifthpostulate?
A. Rectangles exist.B. The pythagorean theorem.C. The angles in a triangle sum to π.D. All of the above.E . None of the above.
Bonus: When (approximately) did Genghis Khan rule?
1206–1227
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Seeing is believing Euclid’s wake In perspective
2. What theorem of plane geometry requires Euclid’s fifthpostulate?
A. Rectangles exist.B. The pythagorean theorem.C. The angles in a triangle sum to π.D. All of the above.E . None of the above.
Bonus: When (approximately) did Genghis Khan rule?1206–1227
![Page 10: Renaissance imagination: geometry - Live Toadlivetoad.org/Courses/Documents/9eb4/Notes/lecture_20101005.pdf · Seeing is believingEuclid’s wakeIn perspective Renaissance imagination:](https://reader031.vdocuments.mx/reader031/viewer/2022022517/5b05b4b07f8b9a58148bbd70/html5/thumbnails/10.jpg)
Seeing is believing Euclid’s wake In perspective
3. What was the first ever purely projective theorem?
A. Pythagoras’ Theorem.B. Pappus’ Theorem.C. Desargues’ Theorem.D. None of the above.
Bonus: When (approximately) did the Black Plague reachEurope?1340s
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Seeing is believing Euclid’s wake In perspective
3. What was the first ever purely projective theorem?
A. Pythagoras’ Theorem.B. Pappus’ Theorem.C. Desargues’ Theorem.D. None of the above.
Bonus: When (approximately) did the Black Plague reachEurope?1340s
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Seeing is believing Euclid’s wake In perspective
3. What was the first ever purely projective theorem?
A. Pythagoras’ Theorem.B. Pappus’ Theorem.C. Desargues’ Theorem.D. None of the above.
Bonus: When (approximately) did the Black Plague reachEurope?
1340s
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Seeing is believing Euclid’s wake In perspective
3. What was the first ever purely projective theorem?
A. Pythagoras’ Theorem.B. Pappus’ Theorem.C. Desargues’ Theorem.D. None of the above.
Bonus: When (approximately) did the Black Plague reachEurope?1340s
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Seeing is believing Euclid’s wake In perspective
4. What are the points of the projective plane?
A. Lines of sight.B. Homogeneous coordinates.C. Incidence axioms.D. All of the above.E . None of the above.
Bonus: When (approximately) did Tamerlane rule?1370–1405
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Seeing is believing Euclid’s wake In perspective
4. What are the points of the projective plane?
A. Lines of sight.B. Homogeneous coordinates.C. Incidence axioms.D. All of the above.E . None of the above.
Bonus: When (approximately) did Tamerlane rule?1370–1405
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Seeing is believing Euclid’s wake In perspective
4. What are the points of the projective plane?
A. Lines of sight.B. Homogeneous coordinates.C. Incidence axioms.D. All of the above.E . None of the above.
Bonus: When (approximately) did Tamerlane rule?
1370–1405
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Seeing is believing Euclid’s wake In perspective
4. What are the points of the projective plane?
A. Lines of sight.B. Homogeneous coordinates.C. Incidence axioms.D. All of the above.E . None of the above.
Bonus: When (approximately) did Tamerlane rule?1370–1405
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Seeing is believing Euclid’s wake In perspective
5. What is the point at infinity on the curve whose equation inhomogeneous coordinates is v2w = u3 − uw2?
A. (0,0,0)B. (0,1,1)C. (0,1,0)D. All of the above.E . None of the above.
Bonus: When (approximately) did the Ottomans conquerConstantinople?1453
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Seeing is believing Euclid’s wake In perspective
5. What is the point at infinity on the curve whose equation inhomogeneous coordinates is v2w = u3 − uw2?
A. (0,0,0)B. (0,1,1)C. (0,1,0)D. All of the above.E . None of the above.
Bonus: When (approximately) did the Ottomans conquerConstantinople?1453
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Seeing is believing Euclid’s wake In perspective
5. What is the point at infinity on the curve whose equation inhomogeneous coordinates is v2w = u3 − uw2?
A. (0,0,0)B. (0,1,1)C. (0,1,0)D. All of the above.E . None of the above.
Bonus: When (approximately) did the Ottomans conquerConstantinople?
1453
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Seeing is believing Euclid’s wake In perspective
5. What is the point at infinity on the curve whose equation inhomogeneous coordinates is v2w = u3 − uw2?
A. (0,0,0)B. (0,1,1)C. (0,1,0)D. All of the above.E . None of the above.
Bonus: When (approximately) did the Ottomans conquerConstantinople?1453
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Seeing is believing Euclid’s wake In perspective
An elliptic curve
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Seeing is believing Euclid’s wake In perspective
Asia in year 1200